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+{"text":"\\section{Introduction}\n\t\n\t\\IEEEPARstart{T}{he} cell voltages in a series-connected battery cell stack become unequal due to manufacturing tolerances, non-uniform aging, and unequal temperature distribution. The differences among the cell voltages increase with the number of charge-discharge cycles, leading to over-charge and over-discharge of some of the cells. A voltage equalizer is essential to improve the charge-capacity and the cycle life of the cell stack by avoiding such a situation.  \n\tThe passive voltage equalizers, which are the simplest and cheapest equalizers, dissipate a significant amount of stored energy\\cite{li_diss}.\n\tOn the other hand, the active equalizers transfer charge from the over-charged cells to under-charged cells to equalize all the cell voltages\\cite{14}. The operation of the active equalizers can be of two types: simultaneous equalization of all the cells and serial equalization of the selected cells, as shown in Fig.\\,\\ref{class}. \n\t\n\tThe first type of active equalizer uses multiple power converters or a multi-port power converter so that all the cells can take part in voltage equalization simultaneously. These equalizers can be classified into three categories: adjacent cell\\cite{lee_int,lee_quasi,park_des,cassani_top,ye_zero,hua_cap}, multi-cell to stack\\cite{einhorn,uno_double,chen,uno_single,hua_lifepo4,lim,hua_apwm,hua_rect,shang_mod,li}, and multi-cell to multi-cell\\cite{ling,evzelman,yelaverthi,wang,shang_auto,ye_model,shang_delta,zeltser,ye_star,TPEL} equalizers, among which the multi-cell to multi-cell equalizers offer fastest voltage equalization. Each of these equalizers dedicates one converter port to each cell even though many of the cells in the cell stack may not require voltage equalization at a given point of time. Such dedicated connections result in a higher component count, under-utilization of converter components, and the requirement of many high-frequency isolated gate drivers. \n\t\n\t\\begin{figure}[h!]\n\t\t\\centering\n\t\t\\includegraphics[width=7.0cm]{Drawings\/classification.pdf}\n\t\t\\caption{Classification of active voltage equalization methods.}\n\t\t\\label{class}\n\t\\end{figure}\t\n\t\n\tThe second type of active equalizer uses only one dual-port dc-dc converter shared by all the cells to reduce the component count and several selection switches for cell selection. \t\n\tWhen the number of cells increases, the dc-dc converter's component count remains the same, and only the number of selection switches increases. The charge transfer method in these equalizers is either cell-to-stack or cell-to-cell. The cell-to-stack equalizers \\cite{imtiaz,nazi,lee_double,raber,    park_mod,kim_auto,kim_mod2,hannan,zhang_interleaved,lin} selects only one cell is at a time, and charge transfer takes place between that cell and the entire cell stack. On the other hand, cell-to-cell equalizers \\cite{park_c2c,lee_cell_res,lee_cell_tr,yu  ,  xiong,pham,shang_cell} achieve a direct charge transfer from the most over-charged cell to the most under-charged cell. Thus, the cell-to-cell equalizers offer about twice the equalization speed, but they require twice as many selection switches. \n\t\n\tAs the number of selection switches is proportional to the number of cells, any simplification or cost-reduction of the selection switches and their drive circuits is valuable in case of a higher number of cells in the stack. As discussed in Section \\ref{sec_drive_ckt} of this paper, a MOSFET, switched at a high-frequency, requires a complex gate-driver, which is often costlier than the MOSFET itself in case of low-power applications. Hence, the cell-to-stack equalizers in \\cite{park_mod,kim_auto,kim_mod2,hannan,zhang_interleaved,lin} and cell-to-cell equalizers in\\cite{xiong,pham,shang_cell} have been proposed with low-frequency selection switches so that simpler drive circuits can be used. This work aims to propose a Low-Frequency Selection Switch based Cell-to-Cell (LFSSCC) voltage equalizer with high efficiency, simple implementation, and lower switch count. The advantages of the proposed equalizer compared to existing LFSSCC equalizers\\cite{xiong,pham,shang_cell} are discussed below.    \t\n\t\n\t\n\tThe dual-port dc-dc converters in \\cite{xiong,pham} use diodes with significant conduction loss and transformer with higher high-frequency losses and size, leading to lower conversion efficiency and increased equalizer size. The dc-dc converter in \\cite{shang_cell} achieves high-efficiency by achieving soft-switched operation with a higher component count. Thus, the dc-dc converters in the existing LFSSCC equalizers suffer from lower efficiency or higher circuit complexity. This work uses a capacitively level-shifted bidirectional Cuk converter, originally proposed for a multi-cell to multi-cell topology in \\cite{ling}. The equalizer in \\cite{ling} uses one closed-loop controlled Cuk converter for each cell, leading to a high component count and control requirements. On the other hand, this work uses only one Cuk converter for a large number of cells, reducing circuit complexity and control effort significantly. Thus, this converter is more suitable for a cell-to-cell equalizer and offers simpler implementation and high efficiency, as no transformer or diode is required in the conduction path. Section \\ref{topology} discusses the converter topology.\n\t\n\tThe LFSSCC equalizers\\cite{xiong,pham,shang_cell} can use either low frequency switched MOSFETs or relays as selection switches. The selection switch networks in these equalizers require $2n$ Double Pole Double Throw (DPDT) switches for $n$ series-connected cells. These switches can be implemented with either $8n$ low-frequency MOSFETs or $2n$ DPDT relays. This work proposes a low-frequency selection switch network with $(n+2)$ DPDT and $2$ SPST switches, leading to a significant reduction in the number of switches and drive circuits.\n\t\n\tThe voltage drop in cell impedance causes cell voltage recovery after equalization current is stopped. This voltage recovery leads to error in detecting end-of-equalization and a large number of switching of the selection switches. For a low-frequency selection switch based equalizer, a higher number of switching leads to longer equalization time and lower reliability. Different methods to reduce the number of selection switching are proposed based on voltage drop estimation\\cite{hannan,pham} and self-learning fuzzy logic based method\\cite{zhang_interleaved}. These methods require additional computation resources and cell charge-discharge characteristics. A simpler method with cell voltage recovery compensation, which does not require charge-discharge characteristics, is proposed to reduce the number of switchings of selection switches significantly.      \n\t\n\tThis work in the enhanced version of the work published in \\cite{ecce_volt_eq} with additional theoretical explanations and experimental results.\n\tSection \\ref{selection} explains the proposed low-frequency cell selection network, and Section \\ref{sec_volt_comp} discusses the proposed cell voltage recovery compensation. Sections \\ref{sec_comp} and \\ref{experiment} provides a comparison with existing equalizers and experimental validation of the proposed equalizer respectively.\n\t\n\t\n\n\t\n\t\\section{Drive Circuits for Selection Switches}\\label{sec_drive_ckt}\t\n\tIf a selection switch is operated at high frequency, it is implemented with MOSFETs. In contrast, a low-frequency selection switch is implemented with either MOSFETs or relay with significantly simpler and cheaper driver circuits.\t\n\t\n\t\\subsubsection{ High-Frequency MOSFET Drive Circuit} A high-frequency switched MOSFET for cell selection requires isolated gate driver IC, capable of providing a peak current of a few Ampere to achieve quick turn-on and turn-off, as shown in Fig.\\,\\ref{drivers}(a). Such a driver IC requires an isolated power supply and is often costlier than a low-power MOSFET. \n\t\n\t\\begin{figure}[h!]\n\t\t\\centering\n\t\t\\begin{subfigure}[]{\\includegraphics[width=2.1cm]{Drawings\/mosfet_driver_high_freq.pdf}}\n\t\t\\end{subfigure}\n\t\t\\hspace{0.4cm}\n\t\t\\begin{subfigure}[]{\\includegraphics[width=2.1cm]{Drawings\/mosfet_driver_low_freq.pdf}}\n\t\t\\end{subfigure}\n\t\t\\hspace{0.4cm}\n\t\t\\begin{subfigure}[]{\\includegraphics[width=2.3cm]{Drawings\/relay_driver.pdf}}\n\t\t\\end{subfigure}\t\t\n\t\t\\caption{(a) High-frequency gate driver for MOSFET, (b) low-frequency gate driver for MOSFET, (c) driver for relay.}\n\t\t\\label{drivers}\n\t\\end{figure}    \n\t\n\t\\subsubsection{  Low-Frequency MOSFET Drive Circuit} Longer turn-on and turn-off times are acceptable for a low-frequency switched MOSFET. The gate drive resistor $R_g$ is in the order of a few kilo-Ohms, and the driver needs to supply only a few milli-Ampere peak current. Thus, a low-cost digital isolator can replace the costly gate driver IC, as shown in Fig.\\,\\ref{drivers}(b).\n\t\n\t\\subsubsection{ Relay Drive Circuit} The driver circuit for a relay is simpler and cheaper than a MOSFET driver. Fig.\\,\\ref{drivers}(c) shows a relay driver, which requires a signal MOSFET $S$, a gate resistor $R_g$, and a free-whiling diode $D$. These components have to carry only a few tens of milli-Amperes of peak current, leading to a very low-cost implementation compared to high-frequency MOSFET drivers.\n\t\n\tThus, the use of low-frequency selection switches reduces the complexity and cost of the equalizer significantly.\n\t\n\t\n\t\\section{Dc-dc Converter Topology}\\label{topology}\n\tThe capacitively level-shifted Cuk converter allows variable voltage difference between the input and output ports' reference terminals without using any isolation transformer.  This converter is used here to transfer charge from the most over-charged cell to the most under-charged cell, as shown in Fig.\\,\\ref{dc_dc_sch}.\n\t\n\t\\begin{figure}[h!]\n\t\t\\centering\n\t\t\\includegraphics[width=5.5cm]{Drawings\/dc_dc_sch.pdf}\n\t\t\\caption{Schematic diagram of the capacitively level shifted Cuk converter connected to two cells of the stack.}\n\t\t\\label{dc_dc_sch}\n\t\\end{figure}\n\t\n\t\n\n\tThis converter has an additional capacitor $C_2$ compared to a conventional bidirectional Cuk converter to block the voltage difference between the grounds of the two ports. Using KVL in Fig.\\,\\ref{dc_dc_sch} and assuming small voltage ripple, the voltage of the capacitors $C_1$ and $C_2$ are given by,\n\t\\begin{eqnarray}\n\t\\label{vc1}\n\tV_{C1}=\\sum_{j=l}^{k}V_{bj} \\qquad \\text{and} \\qquad V_{C2}=\\sum_{j=l+1}^{k-1}V_{bj}\n\n\n\t\\end{eqnarray}    \n\tEach inductor connected to the input or output port is split into two coupled inductors to make the circuit symmetric for reducing common mode oscillation. A more detailed discussion on the converter topology is provided in \\cite{ecce_volt_eq}.\n\tA PI controller controls the cell current in one of the converter ports in closed-loop. \n\tThe converter is enabled when at least one cell voltage is out of the acceptable voltage range.\n\t\n\tFor a predetermined tolerance voltage $V_{tol}$ and the average cell voltage $V_{avg}$, the $k^{th}$ cell voltage $V_{bk}$ is in the acceptable voltage range if it satisfies the following condition, \n\t\\begin{eqnarray}\n\tV_{avg}-V_{tol}\\le V_{bk} \\le V_{avg}+V_{tol} \\quad \\text{for all } k\\in [1,n]\n\t\\end{eqnarray}\n\t\n\t\n\t\n\t\n\n\t\n\t\n\t\n\t\n\t\\section{Low-frequency Cell-to-cell Selection Network}\\label{selection}\n\tIn an LFSSCC equalizer topology, it is possible to connect the dc-dc converter between any two cells at a time to achieve power transfer between them.  The existing LFSSCC equalizers\\cite{shang_cell,xiong,pham} require $2n$ DPDT switches to implement such a cell selection network for $n$ series-connected cells, as shown in Fig.\\,\\ref{dpdt}(a), where the voltage rails \\textbf{\\textit{a}}, \\textbf{\\textit{b}}, \\textbf{\\textit{c}}, and \\textbf{\\textit{d}} are of fixed polarity. \t\n\tThis work proposes a new low-frequency cell-to-cell selection network with bipolar voltage rails to reduce the selection switch count. \n\t\n\t\\begin{figure}[h!]\n\t\t\\centering\n\t\t\\begin{subfigure}[]{\\includegraphics[width=3.8cm]{Drawings\/sel_net_dpdt_conventional.pdf}}\n\t\t\\end{subfigure}\\hspace{0.4cm}\n\t\t\\begin{subfigure}[]{\\includegraphics[width=3.3cm]{Drawings\/sel_net_dpdt.pdf}}\n\t\t\\end{subfigure}\n\t\n\t\t\\caption{Schematic diagram of the low-frequency cell selection network in (a) existing and (b) proposed cell-to-cell equalizers.}\n\t\t\\label{dpdt}\n\t\\end{figure}\n\t\n\t\n\t\n\t\n\t\\begin{figure}[h!]\n\t\t\\centering\n\t\t\\begin{subfigure}[]{\\includegraphics[width=3.2cm]{Drawings\/sel_net_dpdt_eo.pdf}}\n\t\t\\end{subfigure}\\hspace{0.4cm}\n\t\t\\begin{subfigure}[]{\\includegraphics[width=3.2cm]{Drawings\/sel_net_dpdt_adjacent.pdf}}\n\t\t\\end{subfigure}\n\t\t\\caption{Current flow paths between (a) the $2^{nd}$ and the $7^{th}$ cell, (b) the $4^{th}$ and the $5^{th}$ cell in the proposed low-frequency selection network.}\n\t\t\\label{sel_net_8bat}\n\t\\end{figure}\n\t\n\t\n\t\n\t\n\t\n\t\\subsection{Proposed Cell Selection Network}\n\tFig.\\,\\ref{dpdt}(b) shows the proposed low frequency cell-to-cell selection network, which uses One SPST switch and $n$ DPDT switches to select two cells at a time.\n\tOne DPDT switch is used here to connect each common node between two adjacent cells to a voltage rail. However, each voltage rail is switched to a second rail if an upstream DPDT switch is turned on. Thus, each common node can effectively be connected to two voltage rails. In case the polarities of the voltage rails do not match with the converter port polarities, two DPDT relays $S_{pol1}$ and $S_{pol2}$ are used for polarity reversal. In this case, a polarity reversal is required for each pair of voltage rails when the corresponding cell is even-numbered.     \n\tIn the case of two adjacent cells, the common node is connected to both converter ports by an SPST switch $S_{short}$. Thus, the proposed selection network requires ($n+2$) DPDT and $2$ SPST switches, and the selection switch count is reduced to almost half compared to the existing LFSSCC equalizers for a large number of cells.\n\t\n\t\n\tLets consider that the selected cells are the $k^{th}$ and the $l^{th}$ cell, where $k>l$. Then, the $k^{th}$ cell is connected to port 1 and the $l^{th}$ cell is connected to port 2 with following steps,         \n\t\\begin{itemize}\n\t\t\\item Turn on $S_{pol1}$ if $k$ is even. \n\t\t\\item Turn on $S_{pol2}$ if $l$ is even. \n\t\t\\item Turn on $S_{short}$ if the $k^{th}$ and the $l^{th}$ cells are adjacent.\n\t\t\\item Turn on $S_k$, $S_{k-1}$, $S_{l}$, and $S_{l-1}$.\n\t\\end{itemize}\n\n\n\t\n\t\n\t\n\t\n\t\n\t\n\tThe selection network's operation is explained here for eight cells in two different equalization situations, as shown in Fig.\\,\\ref{sel_net_8bat}. The red lines indicate the current flow paths.\n\t\n\t\\subsubsection{Two non-adjacent cells} Fig.\\,\\ref{sel_net_8bat}(a) shows the current flow path between two non-adjacent cells, cell 2 and cell 7. The switches  $S_{1}$, $S_{2}$, $S_{6}$, and $S_{7}$ are turned on to connect the cells to the converter ports.  The polarity reversal switch $S_{pol2}$ is turned on to ensure correct polarity connection.\n\t\n\t\\subsubsection{Two adjacent cells} Fig.\\,\\ref{sel_net_8bat}(b) shows the current flow path for two adjacent cells, cell 4 and cell 5. The switches  $S_{3}$, $S_{4}$, $S_{5}$, and $S_{pol2}$ are turned on to connect cell 5 to port 1 and cell 4 to port 2 with the correct polarity. As the cell 4 and cell 5 are adjacent, their common node should be connected to the negative terminal of port 1 and positive terminal of port 2. This is achieved by turning on the shorting switch $S_{short}$.\n\t\n\t\n\t\n\t\\subsection{Implementation of Selection Switches}\n\tThe low-frequency selection switches can be implemented with MOSFETs or relays as discussed below,\n\t\\subsubsection{DPDT switch for cell selection}\n\tFig.\\,\\ref{dpdt_mos}(a) and (b) show the off and on states of the DPDT switches $S_1$ to $S_n$. \t\n\tEach switch blocks the voltages $V_{P1\\_T1b}$, $V_{P2\\_T2b}$ in on-state, and $V_{P2\\_T2a}$ in off-state. Fig.\\,\\ref{dpdt} and Fig.\\,\\ref{sel_net_8bat} show that $V_{P1\\_T1b}$ is bipolar and $V_{P2\\_T2b}$, $V_{P2\\_T2a}$ are unipolar. Hence, the DPDT switch can be implemented with four MOSFETs, as shown in Fig.\\,\\ref{dpdt_mos}(c). Alternatively, it can also be implemented with a DPDT relay.\n\t\\begin{figure}[h!]\n\t\t\\centering\n\t\t\\begin{subfigure}[]{\\includegraphics[width=2.7cm]{Drawings\/dpdt_mos_a.pdf}}\n\t\t\\end{subfigure}\\hspace{0.0cm}\n\t\t\\begin{subfigure}[]{\\includegraphics[width=2.7cm]{Drawings\/dpdt_mos_b.pdf}}\n\t\t\\end{subfigure}     \\hspace{0.0cm}     \n\t\t\\begin{subfigure}[]{\\includegraphics[width=2.7cm]{Drawings\/dpdt_mos_c.pdf}}\n\t\t\\end{subfigure}\n\t\t\\caption{DPDT switches for cell connection, $S_1$ to $S_n$: (a) off state, (b) on state, (c) a MOSFET based implementation.}\n\t\t\\label{dpdt_mos}\n\t\\end{figure}\n\t\\subsubsection{DPDT switch for polarity reversal}\n\tFig.\\,\\ref{dpdt_pol_mos}(a) and (b) show the off and on state of the polarity reversal DPDT switches $S_{pol1}$ and $S_{pol2}$. It can be observed that the voltage between each pole to each of its throw is positive or zero in off-state. The voltage between a throw and corresponding pole is positive or zero in on-state. The current in the DPDT switch is bi-directional. Hence, the DPDT switch can be implemented with a DPDT relay or four MOSFETs as shown in Fig.\\,\\ref{dpdt_pol_mos}(c).\t\n\t\\begin{figure}[h!]\n\t\t\\centering\n\t\t\\begin{subfigure}[]{\\includegraphics[width=2.7cm]{Drawings\/dpdt_pol_mos_a.pdf}}\n\t\t\\end{subfigure}\\hspace{0.0cm}\n\t\t\\begin{subfigure}[]{\\includegraphics[width=2.7cm]{Drawings\/dpdt_pol_mos_b.pdf}}\n\t\t\\end{subfigure}     \\hspace{0.0cm}     \n\t\t\\begin{subfigure}[]{\\includegraphics[width=2.7cm]{Drawings\/dpdt_pol_mos_c.pdf}}\n\t\t\\end{subfigure}\n\t\t\\caption{DPDT switch for polarity reversal, $S_{pol1}$ and $S_{pol2}$: (a) off state, (b) on state, (c) a MOSFET based implementation.}\n\t\t\\label{dpdt_pol_mos}\n\t\\end{figure}\n\t\n\t\\subsubsection{SPST switches}\n\tThe SPST switches block unipolar voltages, and each of them can be implemented with an SPST relay or a MOSFET.\t\n\t\n\t\n\tThus, the proposed selection switch network can be implemented with $(n+2)$ DPDT and $2$ SPST relays or with $(4n+10)$ MOSFETs.\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\\section{Cell Voltage Recovery Compensation}\\label{sec_volt_comp}\n\tWhen the discharging of a cell is stopped, the cell voltage $V_b$ recovers to a higher voltage after some time. Similarly, when the cell's charging is stopped, $V_b$ falls and settles to a lower voltage. This recovery in $V_b$ is often modeled as a voltage drop across the cell impedance. Ideally, when the cell's equalization is complete, $V_b$ should settle within the acceptable voltage band. However, if the cell's equalization is stopped when $V_b$ reaches within the acceptable voltage band, $V_b$ settles outside of the band due to voltage recovery. Thus, further rounds of equalization for the same cell are necessary, resulting in a higher number of switching of the selection switches. \n\t\n\t\n\t\n\t\n\tThe control algorithm for a low-frequency cell selection network provides a small amount of time-gap, usually 10s to 30s for Li-ion cell, between two switching transitions for allowing the cell voltages to settle before selecting the next pair of cells for equalization. This time-gap helps to avoid high-frequency switching of the selection switches during transients in cell voltages. The measured cell voltages in low-frequency cell-to-stack\\cite{kim_auto,lin} and cell-to-cell\\cite{shang_cell} equalizers show the voltage recovery effect and resulting high number of charge or discharge rounds of each cell. Due to slow switching transitions and time-gap between transitions, a higher number of switchings leads to a longer equalization time. \n\t\n\tSeveral attempts have been made to consider the effect of the cell voltage recovery within the equalization algorithm to avoid a higher number of switchings of selection switches. Estimation of the impedance drop\\cite{hannan,pham} and self-learning fuzzy logic based method\\cite{zhang_interleaved} have been employed for this purpose. The impedance drop estimation methods are computationally intensive. The self-learning fuzzy logic algorithm requires additional computation resources and prior knowledge of charge-discharge characteristics.\n\n\tA cell voltage recovery compensation is proposed here to reduce the number of switchings of the selection switches. The proposed method is computationally simpler than the impedance drop estimation methods and the fuzzy logic method. It does not require any prior knowledge of the charge-discharge characteristics of the cells.\n\t\n\t\n\t\\begin{figure}[h!]\n\t\t\\centering\n\t\t\\includegraphics[width=8cm]{Drawings\/volt_comp.pdf}\n\t\t\\caption{Cell voltage behavioral response when a step charging of the cell is initiated or terminated.}\n\t\t\\label{volt_comp}\n\t\\end{figure}\n\t\n\t\n\t\n\tThe cell voltage recovery effect is explained here with an example of an under-charged cell. The equalization strategy is to charge the cell so that its voltage becomes equal to the average voltage of all the cells. The equalizer charges the under-charged cell with current $I$ from time $t_1$ to $t_2$. Fig.\\,\\ref{volt_comp} shows the cell voltage and current. \n\tAt the beginning of charging at $t_1$, $V_b$ rises from $V_1$ and mostly settles within a short time $\\triangle t$.\n\tHowever, the cell voltage continues to increase slowly after time $(t_1+\\triangle t)$ due to the charging of the cell. The charging is stopped at $t_2$ when the cell voltage is $(V_2+V_{rcv})$. The cell voltage $V_b$ then settles to voltage $V_2$. Thus, if the cell charging is stopped when $V_b$ reaches the average voltage $V_{avg}$, then $V_b$ settles to ($V_{avg}-V_{rcv}$).\n\t\n\t\n\tHence, the equalizer should charge an under-charged cell until $V_b$ reaches $(V_{avg}+V_{rcv})$, so that $V_b$ settles to $V_{avg}$ after charging is stopped. Similarly, it should should discharge an over-charged cell until $V_b$ reduces to $(V_{avg}-V_{rcv})$. However, the estimation of $V_{rcv}$ with internal battery parameters requires significant computation efforts and often suffers from estimation error. In this work, the change in $V_b$ in time $\\triangle t$ from the start of the charging is measured and stored in memory as $V_{imp}$. The voltage $V_{imp}$ is used as an estimate of $V_{rcv}$. However, $V_{rcv}$ is a function of cell condition and, hence, the use of $V_{imp}$ as an estimate of $V_{rcv}$ will have a lower error when the time duration $(t_2-t_1)$ is small.\n\t\n\tThe cell voltage recovery compensation based algorithm in this work charges an under-charged cell till its voltage reaches $(V_{avg}+V_{imp})$ and discharges an over-charged cell till its voltage reduces to $(V_{avg}-V_{imp})$. \n\tIf the initial cell voltage $V_1$ is not close to the final cell voltage $V_2$, then $(t_2-t_1)$ is large and $V_{imp}$ is not a good estimate of $V_{rcv}$.  However, $V_b$ comes close to the acceptable voltage range after the first round of equalization and requires further charging or discharging for a shorter duration. In the second round of equalization, $(t_2-t_1)$ is small and $V_{imp}$ is a good approximation of $V_{rcv}$. Thus, $V_b$ settles within the acceptable voltage range within a few rounds of equalization. \n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\\begin{table*}[h]\n\t\t\\centering\n\t\t\\small\n\t\t\\caption{Comparison of the proposed equalizer with the existing low-frequency selection switch cell-to-cell (LFSSCC) equalizers. \\vspace{0.0cm}}\n\t\n\t\t\\renewcommand{\\arraystretch}{1.2}\n\t\t\\renewcommand{\\tabcolsep}{4pt}\n\t\t\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}\n\t\t\t\\hline\n\t\t\t\\multirow{3}{*}{\\parbox{1.5cm}{\\centering{Topology}}} &  \\multirow{3}{*}{\\parbox{2cm}{\\centering{Type of selection switch}}} &  \\multicolumn{8}{c|}{\\centering{Number of components}}  & \\multirow{3}{*}{\\parbox{1.5cm}{\\centering{Efficiency\\\\ (\\%)}}} \\\\\n\t\t\t\\cline{3-10}\n\t\t\t\n\t\t\t& & \\multicolumn{3}{c|}{Selection switch network} & \\multicolumn{5}{c|}{Dc-dc converter} & \\\\\n\t\t\t\\cline{3-10}\n\t\t\t\n\t\t\t& & MOSFET & DPDT relay & SPST relay & MOSFET & Capacitor & Inductor & Transformer & Diode & \\\\\n\t\t\t\\hline\n\t\t\t\n\t\t\t\\multirow{2}{*}{Ref.\\cite{xiong}}  & \\parbox{1.8cm}{\\centering{MOSFET}} & 8$n$ & 0 & 0 & \\multirow{2}{*}{1} & \\multirow{2}{*}{2} & \\multirow{2}{*}{0} & \\multirow{2}{*}{1} & \\multirow{2}{*}{1} & \\multirow{2}{*}{59.4}\\\\\n\t\t\t\\cline{2-5} \n\t\t\t& \\parbox{1.8cm}{\\centering{Relay}} & 0 & 2$n$ & 0 &  & & & & & \\\\\n\t\t\t\\hline\n\t\t\t\n\t\t\t\\multirow{2}{*}{Ref.\\cite{pham}}  & \\parbox{1.8cm}{\\centering{MOSFET}} & 8$n$ & 0 & 0 & \\multirow{2}{*}{2} & \\multirow{2}{*}{2} & \\multirow{2}{*}{0} & \\multirow{2}{*}{1} & \\multirow{2}{*}{2} & \\multirow{2}{*}{\\parbox{1.5cm}{\\centering{85.3 to 89.5}}}\\\\\n\t\t\t\\cline{2-5} \n\t\t\t& \\parbox{1.8cm}{\\centering{Relay}} & 0 & 2$n$ & 0 & & & & & & \\\\\n\t\t\t\\hline\n\t\t\t\n\t\t\t\\multirow{2}{*}{Ref.\\cite{shang_cell}}  & \\parbox{1.8cm}{\\centering{MOSFET}} & 8$n$ & 0 & 0 & \\multirow{2}{*}{5} & \\multirow{2}{*}{2} & \\multirow{2}{*}{2} & \\multirow{2}{*}{0} & \\multirow{2}{*}{5} & \\multirow{2}{*}{\\parbox{1.5cm}{\\centering{98.6 to 99.5}}}\\\\\n\t\t\t\\cline{2-5} \n\t\t\t& \\parbox{1.8cm}{\\centering{Relay}} & 0 & 2$n$ & 0 & & & & & & \\\\\n\t\t\t\\hline\n\t\t\t\n\t\t\t\\multirow{2}{*}{\\parbox{1.5cm}{\\centering{Proposed equalizer}}}  & \\parbox{1.8cm}{\\centering{MOSFET}} & 4$n$+10  & 0 & 0 & \\multirow{2}{*}{2} & \\multirow{2}{*}{2} & \\multirow{2}{*}{2} & \\multirow{2}{*}{0} & \\multirow{2}{*}{0} & \\multirow{2}{*}{\\parbox{1.5cm}{\\centering{90.1 to 92.9}}}\\\\\n\t\t\t\\cline{2-5} \n\t\t\t& \\parbox{1.8cm}{\\centering{Relay}} & 0 & $n$+2 & 2 & & & & & & \\\\\n\t\t\t\\hline\n\t\t\t\n\t\t\t\n\t\t\n\t\t\n\t\t\n\t\t\\end{tabular}\n\t\t\\label{tab_comp_low_freq}\n\t\\end{table*}\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\\begin{table*}[h]\n\t\t\\centering\n\t\t\\small\n\t\t\\caption{Comparison of the proposed low-frequency selection switch cell-to-cell (LFSSCC) equalizer with other types of equalizers. \\vspace{0.0cm}}\n\t\n\t\t\\renewcommand{\\arraystretch}{1.2}\n\t\t\\renewcommand{\\tabcolsep}{2.5pt}\n\t\t\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}\n\t\t\t\\hline\n\t\t\t\\multirow{3}{*}{\\parbox{1.2cm}{\\centering{Topology}}} &  \\multirow{3}{*}{\\parbox{3.2cm}{\\centering{Type of Equalizer}}} &  \\multicolumn{7}{c|}{\\centering{Number of components}} & \\multicolumn{2}{c|}{\\centering{Number of drivers}} & \\multirow{3}{*}{\\parbox{0.7cm}{\\centering{Effici-ency (\\%)}}} &  \\multirow{3}{*}{\\parbox{1.0cm}{\\centering{Equaliza- tion speed}}} & \\multirow{3}{*}{\\parbox{1.5cm}{\\centering{Voltage difference dependent}}} \\\\\n\t\t\t\\cline{3-11}\n\t\t\t\n\t\t\t& & \\multicolumn{2}{c|}{Semiconductor} & \\multicolumn{2}{c|}{Relay} & \\multicolumn{3}{c|}{Passives} &\\multirow{2}{*}{\\parbox{1.2cm}{\\centering{High frequency}}} & \\multirow{2}{*}{\\parbox{1.2cm}{\\centering{Low frequency}}} & & &\\\\\n\t\t\t\\cline{3-9}\n\t\t\t\n\t\t\t& & MOSFET & Diode & DPDT & SPST & Cap. & Ind. & Trans. & & & & &  \\\\\n\t\t\t\\hline\n\t\t\t\n\t\t\tRef\\cite{ye_zero}  &  \\parbox{3cm}{\\vspace{0.1 cm}\\centering{Adjacent cell resonant-tank}\\vspace{0.1 cm}} & 2$n$ & 0 & 0 & 0 & 2$n$-1 & $n$-1 & 0 & 2$n$ & 0 & 98.2 & Low  & Yes\\\\\t\t\t\t\n\t\t\t\\hline\n\t\t\t\n\t\t\tRef\\cite{li}  &  \\parbox{3cm}{\\vspace{0.1 cm}\\centering{Multi-cell to stack multi-winding trans.}\\vspace{0.1 cm}} & $n$+1 & 0 & 0 & 0 & $n$ & 0 & \\parbox{0.7cm}{\\centering{$n$+1 wind.}} & $n$+1 & 0 & 84.8 & Moderate  & Yes\\\\\t\t\t\t\n\t\t\t\\hline\n\t\t\t\n\t\t\tRef\\cite{ye_star}  &  \\parbox{3cm}{\\vspace{0.1 cm}\\centering{Multi-cell to multi-cell switched capacitor}\\vspace{0.1 cm}} & 2$n$ & 0 & 0 & 0 & 2$n$ & 0 & 0 & 2$n$ & 0 & - & Good  & Yes\\\\\t\t\t\t\n\t\t\t\\hline\n\t\t\t\n\t\t\tRef\\cite{wang}  &  \\parbox{3cm}{\\vspace{0.1 cm}\\centering{Multi-cell to multi-cell dual-active bridge}\\vspace{0.1 cm}} & 3$n$ & 0 & 0 & 0 & $n$ & 0 & $n$\/2 & 3$n$ & 0 & 84.5 & Excellent  & No\\\\\t\t\t\t\n\t\t\t\\hline\n\t\t\t\n\t\t\tRef\\cite{hannan}  &  \\parbox{3cm}{\\vspace{0.1 cm}\\centering{Cell-to-stack MOSFET based cell-selection}\\vspace{0.1 cm}} & 4$n$+2 & 2 & 0 & 0 & 2 & 0 & 2 & 2 & 4$n$ & 92.0 & Moderate  &  No\\\\\t\t\t\t\n\t\t\t\\hline\n\t\t\t\n\t\t\tRef\\cite{lin}  &  \\parbox{3cm}{\\vspace{0.1 cm}\\centering{Cell-to-stack Relay based cell-selection}\\vspace{0.1 cm}} & 1 & 1 & 0 & 2$n$ & 2 & 2 & 0 & 1 & 2$n$ & - & Moderate  & No\\\\\t\t\t\t\n\t\t\t\\hline\n\t\t\t\n\t\t\t\\multirow{2}{*}{Proposed}  &  \\parbox{3.2cm}{\\vspace{0.1 cm}\\centering{MOSFET based cell-to-cell}\\vspace{0.1 cm}} & 4$n$+12 & 0 & 0 & 0 & 2 & 2 & 0 & 2 & 4$n$+10 & \\multirow{2}{*}{\\parbox{0.7cm}{\\centering{90.1-92.9}}} & \\multirow{2}{*}{Good}  & \\multirow{2}{*}{No}\\\\\n\t\t\t\\cline{2-11}\n\t\t\t& \\parbox{3.2cm}{\\vspace{0.1 cm}\\centering{Relay based cell-to-cell}\\vspace{0.1 cm}} & 2 & 0 & $n$+2 & 2 & 2 & 2 & 0 & 2 & $n$+2 &  &  &\\\\\t\t\t\t\n\t\t\t\\hline\n\t\t\t\n\t\t\t\\multicolumn{14}{l}{{$^*Note$: Cap.: Capacitor, Ind.: Inductor, Trans.: Transformer, wind.: winding, $n$ represents the number of cells.}}\n\t\t\t\n\t\t\\end{tabular}\n\t\t\\label{tab_comp_others}\n\t\\end{table*}\n\t\n\t\n\t\n\t\n\t\\section{Comparison of Cell-to-Cell Selection Networks}\\label{sec_comp}\n\tA comparison of the component count and efficiency of the proposed equalizer with the existing LFSSCC equalizers, presented in Table \\ref{tab_comp_low_freq}, shows that it can work with a lower number of selection switches compared to other LFSSCC equalizers. It can also be observed that the proposed equalizer achieves higher efficiency than \\cite{xiong,pham} with similar complexity of the dc-dc converter, which is significantly simpler than \\cite{shang_cell}. Thus, the proposed method achieves above 90\\% efficiency with simple circuit implementation.\n\t\n\tTable \\ref{tab_comp_others} compares the proposed LFSSCC equalizer with different types of existing equalizers. The followings can be observed from the comparison of component counts, driver requirements, efficiency, equalization speed, and voltage difference dependence,\n\t\\begin{enumerate}\n\t\t\\item Although the adjacent cell and multi-cell equalizers can work with a lower number of MOSFETs, they require a large number of passive components and high-frequency isolated gate drivers. \n\t\t\\item The equalization current in the adjacent cell and multi-cell equalizers, except \\cite{wang}, is proportional to the cell voltage differences. Thus, these equalizers become less effective when the voltage difference is not large, especially in Li-ion cells, where even a large difference in SOC results in a small voltage difference. The work in \\cite{wang} achieves voltage difference independence by controlling each of the cell currents simultaneously, leading to higher sensor and computation cost.\n\t\t\\item The proposed equalizer offers similar component counts, driver requirements, and efficiencies compared to low-frequency cell-to-stack equalizers, but achieves twice as fast equalization.\n\t\t\\item A relay-based implementation offers the lowest component count and driver requirements.\n\t\\end{enumerate}\n\t\n\t\n\t\n\t\n\t\n\t\n\t\\begin{figure}[h!]\n\t\t\\centering\n\t\t\n\t\t\\includegraphics[width=8.5cm]{Results\/equalizer_image.pdf}\n\t\t\n\t\t\n\t\t\\caption{Image of the developed 8-cell equalizer prototype.}\n\t\t\\label{setup_image}\n\t\\end{figure}\n\t\n\t\n\t\n\t\n\t\\begin{table}[h]\n\t\t\\centering\n\t\t\\small\n\t\t\\caption{Circuit components and parameters of the proposed equalizer prototypes based on relays. \\vspace{0.0cm}}\n\t\n\t\t\\renewcommand{\\arraystretch}{1.5}\n\t\t\\renewcommand{\\tabcolsep}{3pt}\n\t\t\\begin{tabular}{|c|c|}\n\t\t\t\\hline\n\t\t\tComponent\/parameter   &    Ratings\/part no.    \\\\    \\hline\n\t\t\tSwitching frequency & $30$ kHz\\\\    \\hline\n\t\t\t\\parbox{5cm}{\\centering{\\vspace{0.0cm}  Inductances in port 1: $L_1$, $L_{1'}$, $M_1$}}  &  $62.5$ $\\mu$H, $0.7$ A  \\\\    \\hline\n\t\t\t\\parbox{5cm}{\\centering{\\vspace{0.0cm}  Inductances in port 2: $L_2$, $L_{2'}$, $M_2$}}  &  $62.5$ $\\mu$H, $0.7$ A  \\\\    \\hline\n\t\t\tCapacitors: $C_1$, $C_2$ &  $50$ $\\mu$F, $50$ V  \\\\    \\hline\n\t\t\tMOSFETs: $Q_1$, $Q_2$   &   BSC009NE2LS5I \\\\    \\hline\n\t\t\tSPST relay  &    OJE-SH-124LMH    \\\\    \\hline\n\t\t\tDPDT relay  &    RT424024    \\\\    \\hline\n\t\t\tWait time for $V_{imp}$ measurement, $\\triangle t$ & $20$ s\\\\ \\hline \n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\\end{tabular}\n\t\t\\label{tab_comp_ratings}\n\t\\end{table}\t\n\t\n\t\n\t\n\t\\section{Experimental Results}\\label{experiment}\n\tAn 8-cell prototype is developed to validate the proposed low-frequency selection switch cell-to-cell (LFSSCC) equalizer using DPDT relays as the selection switches. Table \\ref{tab_comp_ratings} provides the ratings of the equalizer components, and Fig.\\,\\ref{setup_image} shows the image of the developed prototype. The prototype is tested with eight $3.6$ V, $2.6$ Ah Li-ion cells\\cite{li_cell} to verify the converter operation, equalizer efficiency, and the control algorithm. The current of the cell connected to port $1$ of the converter is controlled to $0.5$ A, and the equalization algorithm decides its direction based on cell voltages.\n\t\n\t\n\t\\subsection{Operation of Cuk Converter} \n\tFig.\\ref{dpdt_wf} shows the measured current and voltage waveforms of the capacitively level-shifted Cuk converter in the prototype to verify the converter operation. The measured current waveforms show that the port 1 current is $0.5$ A, and the peak-to-peak ripple present in each of the port currents is $100$ mA. \n\tFig.\\,\\ref{dpdt_wf}(c) shows the measured capacitor voltages for the case when cell $1$ and cell $4$ are selected for equalization. It can be observed that the capacitor $C_1$ blocks the total voltage of $4$ cells, and $C_2$ blocks that total voltage of $2$ cells, as expected theoretically in (\\ref{vc1}). The measured voltage ripples in $V_{C1}$ and $V_{C2}$ in Fig.\\,\\ref{dpdt_wf}(d) show peak-to-peak ripples of $20$ mV for both of the capacitors. \n\t\n\t\n\t\n\t\n\t\n\t\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\t\n\t\\begin{figure}[h!]\n\t\t\\centering\n\t\t\\begin{subfigure}[]{\\includegraphics[width=4cm]{Results\/DPDT_Iout.pdf}}\n\t\t\\end{subfigure}\\hspace{0.0cm}\n\t\t\\begin{subfigure}[]{\\includegraphics[width=4cm]{Results\/DPDT_Iin.pdf}}\n\t\t\\end{subfigure}\n\t\t\\begin{subfigure}[]{\\includegraphics[width=4.25cm]{Results\/DPDT_vc.pdf}}\n\t\t\\end{subfigure}\\hspace{0.0cm}\n\t\t\\begin{subfigure}[]{\\includegraphics[width=4.25cm]{Results\/DPDT_vc_ripple.pdf}}\n\t\t\\end{subfigure}\n\t\t\n\t\t\\caption{Experimental waveforms of (a) port 1 current, (b) port 2 current, (c) capacitor voltages, and (d) capacitor voltage ripples of the Cuk converter.}\n\t\t\\label{dpdt_wf}\n\t\\end{figure}\n\t\n\t\n\t\n\t\n\t\n\t\n\t\\begin{figure}[h!]\n\t\t\\centering\n\t\t\n\t\t\\includegraphics[width=8.5cm]{Plots\/eff_8cell.eps}\n\t\t\n\t\t\n\t\t\\caption{Plot of measured efficiency of the prototype with its output power.}\n\t\t\\label{eff_plot}\n\t\\end{figure}\n\t\n\t\n\tThe efficiency of the prototype is measured for different output powers up to $2$ W and is plotted in Fig.\\,\\ref{eff_plot}. Fig.\\,\\ref{eff_plot} shows that the prototype has a maximum efficiency of $92.9\\%$ and an efficiency of $90.1\\%$ at the rated power. \n\t\n\t\n\t\n\t\\begin{figure}[h!]\n\t\t\\centering\n\t\t\n\t\t\\begin{subfigure}[]{\\includegraphics[width=8.8cm]{Results\/8cell_rest_volt_comp.eps}}\n\t\t\\end{subfigure}\\hspace{0.0cm}\n\t\t\\begin{subfigure}[]{\\includegraphics[width=8.8cm]{Results\/8cell_rest_wo_volt_comp.eps}}\n\t\t\\end{subfigure}\n\t\t\n\t\t\\caption{Plot of measured cell voltages during voltage equalization (a) with, and (b) without cell voltage recovery compensation.}\n\t\t\\label{spst_volt_conv}\n\t\\end{figure}\n\t\n\t\\subsection{Voltage Convergence Test}\n\tThe performance of the equalizer and its control algorithm with the proposed cell voltage recovery compensation is tested on an 8-cell stack under rest condition, and Fig.\\,\\ref{spst_volt_conv}(a) shows the measured cell voltages. The maximum voltage difference is $200$ mV at the beginning. All the cell voltages converge within an acceptable voltage band of $20$ mV in $100$ minutes. \n\t\n\tThe same test is performed without the cell voltage recovery compensation to verify the necessity of this compensation, and Fig.\\,\\ref{spst_volt_conv}(b) shows the measured cell voltages. The initial cell voltages are close to those of the previous test for a proper comparison. The equalizer takes about $130$ minutes in this case to converge the cell voltages within a band of $30$ mV. Thus, the voltage convergence time is $30$\\% longer when the cell voltage recovery compensation is not used. It can also be observed from Fig.\\,\\ref{spst_volt_conv}(a) and (b) that the number of switchings of the relays is significantly lower when cell voltage recovery compensation is employed. For example, the highest switched relay's number of switching transitions is reduced from $166$ to only $18$ using the compensation. The reduction of the number of switchings significantly improves the life and reliability of the relays. \n\t\n\t\n\t\n\t\\begin{figure}[h!]\n\t\t\\centering\n\t\t\n\t\t\\includegraphics[width=8.8cm]{Results\/8cell_ch_dis.pdf}\n\t\t\n\t\t\n\t\t\\caption{Plot of measured cell voltages over a charge-discharge cycle.}\n\t\t\\label{volt_plot_ch_dis}\n\t\\end{figure}\n\t\n\t\\begin{figure}[h!]\n\t\t\\centering\n\t\t\n\t\t\\includegraphics[width=8.8cm]{Results\/8cell_load.pdf}\n\t\t\n\t\t\n\t\t\\caption{Plot of measured cell voltages with time during changes in load current $I_b$ of the cell stack.}\n\t\t\\label{volt_plot_load}\n\t\\end{figure}\n\t\n\t\\subsection{Dynamic Performance of the Equalizer}\n\tThe equalizer's performance is tested during the charge-discharge cycle of the eight-cell Li-ion stack, and Fig.\\,\\ref{volt_plot_ch_dis} shows the cell voltages. The cell voltages are initially unbalanced, with the maximum difference among them being $160$ mV. The cell stack is discharged from this condition at $0.5$ C-rate until the cell voltages reach $3$ V. The cell stack is then subjected to constant current (CC) charging at $0.5$ C-rate and constant voltage (CV) charging at $4$ V. Once, the CV charging is finished the stack is discharged once again at $0.5$ C-rate. It can be observed from the voltage plot that the equalizer equalizes the initially unbalanced cell voltages during the first discharge and maintains these voltages within $20$ mV voltage band. Once the cell voltages are equalized, the equalizer is disabled. However, on the course of charging and discharging, one or two cell voltages occasionally gets out of the $20$ mV voltage band the equalizer is enabled again, as observed near $115$, $170$, $180$, and $190$ minutes of Fig.\\,\\ref{volt_plot_ch_dis}. Fig.\\,\\ref{volt_plot_ch_dis} also shows that a very few switchings of the relays are required to maintain the cell voltages within the  $20$ mV band after the initial voltage difference is mitigated.  \n\t\n\tThe performance of the equalizer is also verified under variable load conditions. Fig.\\,\\ref{volt_plot_load} shows the load current variation and the corresponding cell voltages.  It can be observed that a sudden change in current often produces unbalance in cell voltages, and the equalizer is activated to eliminate this unbalance. The equalizer is also activated when any cell voltage is out of the  $20$ mV voltage band during constant current discharging between two step-changes in current, as observed near $65$ and $70$ minutes of Fig.\\,\\ref{volt_plot_load}. Thus, the proposed equalizer meets performance requirements under different practical operating conditions.\n\n\t\n\t\n\t\\section{Conclusion}\n\tA cell-to-cell voltage equalizer with low-frequency selection switches is proposed based on a capacitively level-shifted Cuk converter. The avoidance of an isolation transformer and diodes for the proposed equalizer's operation helps achieve high efficiency.    \n\tThe use of low-frequency selection switches with simple drive circuits in the proposed equalizer leads to lower component count and cost.\n\tA low-frequency cell-to-cell selection network is proposed with bipolar voltage buses. This reduces the number of selection switches to almost half compared to the existing low-frequency selection networks for a large number of cells. \n\tA comparison of the proposed equalizer with the existing LFSSCC equalizers shows its advantages in terms of switch count, drive circuit requirements, and efficiency. \n\tAn 8-cell prototype of the proposed equalizer is implemented with relays, and the operation of the capacitively level-shifted Cuk converter is experimentally verified. The developed prototype shows an efficiency above $90$\\% over its entire power range and a peak efficiency of $92.9\\%$. The equalizer is shown to successfully converge the voltages of eight Li-ion cells within a voltage band of $20$ mV from an initial voltage imbalance of $200$ mV. The prototype shows good voltage balancing performance under different conditions such as constant current charging, constant voltage charging, constant current discharge, and varying load.  \n\tA cell voltage recovery compensation scheme is proposed that estimates the voltage recovery due to cell impedance to reduce the number of switchings and the total equalization time. Experimental results show that the proposed compensation leads to the voltage convergence in a shorter time with about one order of magnitude reduction in the number of relay switchings. Thus, the proposed equalizer offers cell-to-cell voltage equalization with lower circuit complexity, component count, and good performance.\n\t\n\t\n\t\n\t\n\n\t\n\t\n\n\n\t\n\t\\bibliographystyle{IEEEtranTIE}\n\t","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe interaction of the $\\Sigma$ hyperon with nuclear matter may be\nrepresented by the complex single particle (s.p.) optical model\npotential $U_\\Sigma=V_\\Sigma-iW_\\Sigma$. In this paper we present\nour attempts to determine $V_\\Sigma$ and $W_\\Sigma$. We also point\nout the most realistic two-body $\\Sigma N$ interaction among the\navailable OBE models of the baryon-baryon interaction.\n\nIn the present paper we discuss the\nfollowing sources of information on $U_\\Sigma$:\n$\\Sigma N$ scattering data in Sec.2, $\\Sigma^-$ atoms\nin Sec.3, associated production reactions in Sec.4,\nand strangeness exchange reactions in Sec.5. Our conclusions\nare presented in Sec.6.\n\n\\section{$\\Sigma N$ scattering}\n\nThe way from the $\\Sigma N$ scattering data to $U_\\Sigma$ consists\nof two steps: first, we determine the two-body $\\Sigma N$ interaction\n${\\cal V}_{\\Sigma N}$, and second, with this ${\\cal V}_{\\Sigma N}$\nwe calculate $U_\\Sigma$. The scarcity of the two-body $\\Sigma N$\ndata makes the first step\nvery difficult. A way of overcoming these difficulties was followed\nby de Swart and his collaborators in Nijmegen: they assumed the\nmechanism of one-boson exchange (OBE) and the SU(3) symmetry\nwhich enabled them to employ the numerous $NN$ data in determining\nthe parameters of their two-body interaction. In this way they\nproduced a number of the Nijmegen models of the baryon-baryon\ninteraction: models D \\cite{D}, F \\cite{F}, soft core (SC) model\n\\cite{SC}, and the new soft-core (NSC) model \\cite{NSC}.\n\\begin{figure}[h]\n\\vspace{1mm}\n\\begin{minipage}[t]{0.475\\linewidth}\n\\centering \\vspace{-8mm} {\\psfig{file=fig1cc.eps,width=7cm}}\n\\caption{The isoscalar potential $V_\\Sigma$ as a function of the\nnucleon density $\\rho$ at $k_\\Sigma=0$ for the indicated Nijmegen\nmodels of the $\\Sigma N$ interaction.}\n\\end{minipage}\n\\vspace{20mm} \\hspace{1mm}\n\\begin{minipage}[t]{0.475\\linewidth}\n\\centering \\vspace{-8mm}\n\\includegraphics[height=5.5cm,width=8.cm]{fig2cc.eps}\n\\vspace{-6mm} \\caption{The component $W_c,  W_\\e$, and $W_t$ of\nthe $\\Sigma$ absorptive potential in nuclear matter of density\n$\\rho_0$ as functions of $k_\\Sigma$.}\n\\end{minipage}\n\\vspace{-16mm}\n\\end{figure}\n\\vspace{-8mm}\n\\subsection{The real potential $V_\\Sigma$}\n\nIn calculating $V_\\Sigma$ we use the real part of the effective\n$\\Sigma N$ interaction YNG \\cite{YM} in nuclear matter. The YNG interaction\nis the configuration space representation of the G matrix calculated\nin the low order Brueckner approximation with the Nijmegen\nmodels of the baryon-baryon interaction. Our results obtained for\n$V_\\Sigma$ as function of the nucleon density\n$\\rho$ are shown in Fig. 1. As the dependence of\n$V_\\Sigma$ on the $\\Sigma$ momentum $k_\\Sigma$ is not very strong\nin the relevant interval of $k_\\Sigma$ \\cite{JJDD}, we use for\n$V_\\Sigma$ its value calculated at $k_\\Sigma=0$.\nWe see that all the Nijmegen interaction models, except for model\nF, lead to pure attractive $V_\\Sigma$ which implies the existence of\nbound states of $\\Sigma$ hyperons in the nuclear core, \\ie,\n$\\Sigma$ hypernuclei. Since no $\\Sigma$ hypernuclei have been\nobserved,\\footnote{The observed bound state of $^4_\\Sigma$He\n\\cite{He} is an exception. In the theoretical description\nof this state, Harada and his collaborators \\cite{Ha} apply\nphenomenological $\\Sigma N$ interactions, in particular, the\ninteraction SAP-F simulating at low energies the Nijmegen\nmodel F interaction. They show that essential for the existence\nof the bound state of $^4_\\Sigma$He is a strong Lane component\n$V_\\tau$ in $V_\\Sigma$, and among the Nijmegen models the\nstrongest $V_\\tau$ is implied by model F.\\cite{JDJD}}\nwe conclude that among the Nijmegen interaction models\nmodel F is the only realistic representation of the $\\Sigma N$\ninteraction.\n\n\\subsection{The absorptive potential $W_\\Sigma$}\n\nAs pointed out in \\cite{YM}, the imaginary part of the YNG\ninteraction is very sensitive to the choice of the intermediate\nstate energies in the $G$ matrix equation. In this situation\nwe decided to use for $W_\\Sigma$ the semi-classical\nexpression in terms of the total cross sections (modified by\nthe exclusion principle) for $\\Sigma N$ scattering,\ndescribed in \\cite{JDPR}. We denote by $W_c$ the contribution to the\nabsorptive potential of\nthe $\\Sigma \\Lambda$ conversion process $\\Sigma N\n\\rightarrow \\Lambda N^\\prime$ and by $W_e$ the contribution\nof the $\\Sigma N$ elastic scattering, and have\n$W_\\Sigma = W_t = W_c + W_e$.\n\\footnote{\nNotice that in the case of the nucleon optical potential\nin nuclear matter (for nucleon energies below the threshold\nfor pion production),\n$V_N-iW_N$, only the elastic $NN$ scattering contributes to\n$W_N$, and the situation is similar as in the case of the\ncontribution $W_e$ to $W_\\Sigma$.}\n\n\nOur results obtained for $W_c, W_e, W_t$ for nuclear matter (with\nN=Z) at equilibrium density $\\rho = \\rho_0 =$ 0.166 fm$^{-3}$ are shown in\nFig. 2.\nWith increasing momentum $k_\\Sigma$ the $\\Sigma\\Lambda$ conversion\ncross section decreases, on the other hand the suppression of $W_c$ by the\nexclusion principle weakens. As the net result\n$W_c$ does not change very much with $k_\\Sigma$. The same two mechanisms\nact in the case of $W_e$. Here, however, the action of the exclusion\nprinciple is much more pronounced: at $k_\\Sigma=0$ the suppression of\n$W_e$ is complete. At higher momenta, where the Pauli blocking is not\nimportant, the total elastic cross section is much bigger than the\nconversion cross section, and we have $W_e>>W_c$, and consequently\n$W_\\Sigma>>W_c$.\n\n\\section{$\\Sigma^-$ atoms}\n\nThe available data on strong interaction effects in $\\Sigma^-$ atoms\nconsist of 23 data points: strong interaction shifts $\\epsilon$ and\nwidths $\\Gamma$ of the observed levels. These shifts and widths can\nbe measured directly only in the lowest $\\Sigma^-$ atomic levels.\nThe widths of the next to the last level can be obtained indirectly\nfrom measurements of the relative yields of X-rays.\n\nIn \\cite{JRA}, we have estimated the 23 values of $\\epsilon$ and\n$\\Gamma$ from the difference between the eigenvalues of the\nSchr\\\"{o}dinger equation of $\\Sigma^-$ in $\\Sigma^-$ atoms with\nthe strong $\\Sigma^-$-atomic nucleus interaction and without\nthis interaction. To obtain this strong interaction, we\napplied the local density approximation, and used our\noptical model of Sec. 2. The agreement of our results,\ncalculated with the optical potentials (obtained with the\n4 Nijmegen $\\Sigma N$ interaction models)\nwith the 23 empirical data points is characterized by the\nfollowing values of $\\chi^2$: $\\chi^2$(model D) $>$ 130,\n$\\chi^2$(model F) = 38.1, $\\chi^2$(model SC) = 55.0,\n $\\chi^2$(model NSC) $>$ 904, and we conclude that the $\\Sigma^-$\natomic data point out at model F as the best representation\nof the $\\Sigma N$ interaction.\\footnote{Notice that the positive\nsign of the measured values of $\\epsilon$ requires an attractive\n$\\Sigma$ potential at the nuclear surface, \\ie at low densities.}\n\n\\section{The associated production reactions}\n\nThe first associated $\\Sigma$ production reaction $(\\pi^-,K^+)$\nwas observed at KEK on $^{28}$Si target at pion momentum of\n1.2 GeV\/c  (\\cite{anna1},\\cite{anna3}), and this reaction is\nthe subject of the present analysis. We consider the reaction\n$(\\pi^-,K^+)$ in which the pion $\\pi^-$ with momentum\n${\\bf k}_\\pi$ hits a proton in the $^{28}$Si target in the state $\\psi_P$\nand emerges in the final state as kaon $K^+$ moving in the direction\n$\\hat{k}_K$ with energy $E_K$, whereas the hit proton emerges in the final\nstate as a $\\Sigma^-$ hyperon with momentum $\\bf{k}_\\Sigma$.\nWe apply the simple\nimpulse approximation described in \\cite{I2}, with $K^+$ and $\\pi^-$ plane\nwaves, and obtain:\n\\begin{equation}\n\\label{ia}\nd^3\\sigma\/d\\hat{k}_\\Sigma d\\hat{k}_K dE_K\\sim|\\int d{\\bf r}\\exp(-i{\\bf qr})\n\\psi_{\\Sigma,{\\bf k}_\\Sigma}({\\bf r})^{(-)*}\\psi_P({\\bf r})|^2,\n\\end{equation}\nwhere the momentum transfer ${\\bf q}={\\bf k}_K-{\\bf k}_\\pi$, and\n$\\psi_{\\Sigma,{\\bf k}_\\Sigma}({\\bf r})^{(-)}$ is the $\\Sigma$ scattering\nwave function which is the solution of the s,p. Schr\\\"{o}dinger equation\nwith the s.p. potential\n\\begin{equation}\n\\label{us}\nU_\\Sigma(r)=(V_\\Sigma-iW_\\Sigma)\\theta(R-r),\n\\end{equation}\nwhere for $V_\\Sigma$ and $W_\\Sigma$ we use the nuclear matter results\ndiscussed in Section 2, calculated at $\\rho=n\/[(4\\pi\/3)R^3]$, where\n$n$=27 is the number of nucleons in the final state.\n\nFor the $^{28}$Si target nucleus we assume a simple shell model\nwith a square well s.p, potential $V_P(r)$ (which determines $\\psi_P$)\nwith the radius $R_P$ (and with a spin-orbit term).\nThe parameters of $V_P(r)$ are adjusted to the proton separation\nenergies (in particular $R_P=3.756$fm). For $R$ we make the simple\nand plausible assumption: $R=R_P$.\n\nIn the inclusive KEK experiments \\cite{anna1}-\\cite{anna3} only the energy\nspectrum of kaons at fixed $\\hat{k}_\\Sigma$ was measured.\nTo obtain this\nenergy spectrum, we have to integrate the cross section (\\ref{ia}) over\n$\\hat{k}_\\Sigma$.\n\n\\begin{figure}[h]\n\\vspace{0mm}\n\\begin{minipage}[t]{0.475\\linewidth}\n\\centering \\vspace{-8mm} {\\psfig{file=fig3cc.eps,width=7.cm}}\n\\caption{Kaon spectrum from $(\\pi^-,K^+)$ reaction on $^{28}$Si at\n$\\theta_K=6^\\circ$ at $p_\\pi=1.2$ GeV\/c obtained with $V_\\Sigma$\ndetermined by models F and D of the $\\Sigma N$ interaction. Curves\ndenoted by $c(t)$ were obtained with $W_\\Sigma = W_c(W_t)$. Data\npoints are taken from \\cite{anna3}.}\n\\end{minipage}\n\\vspace{20mm} \\hspace{1mm}\n\\begin{minipage}[t]{0.475\\linewidth}\n\\centering \\vspace{-1mm}\n\\includegraphics[height=4.9cm,width=5.6cm]{fig4cc.eps}\n\\vspace{-3mm} \\caption{Pion spectrum from $(K^-,\\pi^+)$ reaction\non $^9$Be at $\\theta_\\pi=4^\\circ$ at $p_K=0.6$ GeV\/c obtained with\n$V_\\Sigma$ determined by models F and D of the $\\Sigma N$\ninteraction. Curves denoted by $c(t)$ were obtained with $W_\\Sigma\n= W_c(W_t)$. Data points are taken from \\cite{bart}.}\n\\end{minipage}\n\\vspace{-16mm}\n\\end{figure}\nWe present our results for the inclusive cross\nsection as a function of $B_\\Sigma$, the separation (binding)\nenergy of $\\Sigma$ from the hypernuclear system produced.\nOur model F and D results\n\\footnote{The remaining models SC and NSC are\nsimilar to model D: they all lead to attractive $V_\\Sigma$ in\ncontradistinction to model F leading to repulsive $V_\\Sigma$ (at\ndensities inside nulei - see Fig. 1). Consequently, the results\nfor the kaon spectrum for models SC and NSC are expected to be\nsimilar as in case of model D.}\nfor kaon spectrum from $(\\pi^-,K^+)$ reaction on $^{28}$Si\nat $\\theta_K=6^o$ at $p_\\pi= 1.2$ GeV\/c\nare shown in Fig. 3.\nWe see that the best fit to the data points is obtained for\n$V_\\Sigma$ derived from model F and with $W_\\Sigma=W_t=W_c+W_e$.\nThe fit would improve if we considered the distortion of kaon and\nespecially of pion waves\n(it was noticed already in Ref. \\cite{anna1} that this\ndistortion pushes the kaon spectrum down). Inclusion into the\nabsorptive potential of the contribution $W_e$ of the elastic\n$\\Sigma N$ scattering is essential for obtaining this result with\n$V_\\Sigma$(model F) = 17.25 MeV. Earlier estimates of the kaon\nspectrum without this contribution suggested a repulsive\n$V_\\Sigma$ with an unexpected strength of about 100 MeV. Notice\nthat the action of the absorptive potential $W_\\Sigma$ on the\n$\\Sigma$ wave function (decrease of this wave function) is similar\nas the action of a repulsive $V_\\Sigma$. Therefore we achieve with\nstrong absorption the same final effect with a relatively weaker\nrepulsion.\n\n\n\n\\section{The strangeness exchange reactions}\n\nFirst observations of the strangeness exchange $(K^-,\\pi)$ reactions\nwith a reliable accuracy were performed at BNL. Here,\nwe shall discuss the $(K-,\\pi^+)$ reaction observed at BNL on\nBe$^9$ target with 600 MeV\/c kaons.\\cite{bart} Proceeding similarly\nas in the case of the associated production described in Sec.4, we\nget the results shown in Fig. 4. We see that similarly as in Sect. 4\nthe fit to the data points obtained for $V_\\Sigma$\nderived from model F is much better than the fit obtained with\nmodel D.\n\n\\newpage\n\n\\section{Conclusions}\n\n$\\bullet$ The real part $V_\\Sigma$ of the $\\Sigma$ optical potential\nis repulsive inside the nucleus and has a shallow attractive pocket\nat the nuclear surface.\n\n$\\bullet$ Among the Nijmegen models of the baryon-baryon interaction\nonly model F leads to this form of $V_\\Sigma$.\n\n$\\bullet$ The contribution of the elastic $\\Sigma N$ scattering to the\nabsorptive part $W_\\Sigma$ of the $\\Sigma$ optical potential is\nessential in the analysis of $\\Sigma$ production processes.\n\\vspace{0.6cm}\n\nThis research was partly supported by the Polish Ministry of Science\nand Higher Education under Research Project No. N N202 046237.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Introduction} \n\nThis paper is part of a long term project concerned with a \nsystematic approach to unitary representations \nof Banach--Lie groups in terms of conditions \non spectra in the derived representation. A unitary \nrepresentation $\\pi \\: G \\to \\U(\\cH)$ \nis said to be {\\it smooth} if the subspace \n$\\cH^\\infty$ of smooth vectors is dense. \nThis is automatic for continuous \nrepresentations of finite dimensional groups but not in general \n(cf.\\ \\cite{Ne10a}). For any smooth \nunitary representation, the {\\it derived representation} \n\\[ \\dd\\pi \\: \\g = \\L(G)\\to \\End(\\cH^\\infty), \\quad \n\\dd\\pi(x)v := \\derat0 \\pi(\\exp tx)v\\]  \ncarries significant information in the sense that the closure of the \noperator $\\dd\\pi(x)$ coincides with the infinitesimal generator of the \nunitary one-parameter group $\\pi(\\exp tx)$. We call $(\\pi, \\cH)$ \n{\\it semibounded} if the function \n\\[ s_\\pi \\: \\g \\to \\R \\cup \\{ \\infty\\},\\ \\ \ns_\\pi(x) \n:= \\sup\\big(\\Spec(i\\dd\\pi(x))\\big) \n= \\sup\\{ \\la i\\dd\\pi(x)v,v\\ra \\: v \\in \\cH^\\infty, \\|v\\|=1\\}\\] \nis bounded on the neighborhood of some point in $\\g$ \n(cf.\\ \\cite[Lemma~5.7]{Ne08}). \nThen the set $W_\\pi$ of all such points \nis an open invariant convex cone in the Lie algebra $\\g$. \nWe call $\\pi$ {\\it bounded} if $s_\\pi$ is bounded on some $0$-neighborhood, \nwhich is equivalent to $\\pi$ being norm continuous \n(cf.\\ Proposition~\\ref{prop:contin}). \nAll finite dimensional unitary representations are bounded \nand many of the unitary representations appearing in physics are semibounded \n(cf. \\cite{Ne10c}). \n\nOne of our goals is a classification of the irreducible semibounded \nrepresentations and the development of \ntools to obtain direct integral decompositions \nof semibounded representations. To this end, realizations \nof unitary representations in spaces of holomorphic sections of \nvector bundles turn out to be extremely helpful. \nClearly, the bundles in question should be allowed to have \nfibers which are infinite dimensional Hilbert spaces, and to \ntreat infinite dimensional groups, we also have to admit \ninfinite dimensional base manifolds. \nThe main point of the present paper is to provide \neffective methods to treat unitary representations \nof Banach--Lie groups in spaces of holomorphic sections of homogeneous \nHilbert bundles. \n\nIn Section~\\ref{sec:1} we explain how to parametrize \nthe holomorphic structures on Banach vector bundles \n$\\bV = G \\times_H V$ over a Banach homogeneous space $M = G\/H$ \nassociated to a norm continuous representation \n$(\\rho, V)$ of the isotropy group~$H$. The main result \nof Section~\\ref{sec:1} is \nTheorem~\\ref{thm:a.2} which generalizes the corresponding \nresults for the finite dimensional case by Tirao and Wolf \n(\\cite{TW70}). As in  finite dimensions, the complex bundle \nstructures are specified by  \n``extensions'' $\\beta \\: \\fq \\to \\gl(V)$ of the differential \n$\\dd \\rho \\: \\fh \\to \\gl(V)$ to a representation of the complex subalgebra \n$\\fq \\subeq \\g_\\C$ specifying the complex structure on~$M$ in the sense that \n$T_H(M) \\cong \\g_\\C\/\\fq$ \n(cf.~\\cite{Bel05}). The main point in \\cite{TW70} is that the homogeneous space \n$G\/H$ need not be realized as an open $G$-orbit in a complex \nhomogeneous space of a complexification $G_\\C$, which is impossible \nif the subgroup of $G_\\C$ generated by $\\exp \\fq$ is not closed. \nIn the Banach context, two additional difficulties appear: \nThe Lie algebra $\\g_\\C$ may not be integrable in the sense that \nit does not belong to any Banach--Lie group (\\cite{GN03}) and, even if $G_\\C$ \nexists and the subgroup $Q := \\la \\exp \\fq \\ra$ is closed, it \nneed not be a Lie subgroup so that there is no natural construction \nof a manifold structure on the quotient space~$G\/Q$. \nAs a consequence, the strategy of the proof \nin \\cite{TW70} can not be used for Banach Lie groups. Another \ndifficulty of the infinite dimensional context is that there is \nno general existence theory for solutions of $\\oline\\partial$-equations \n(see in particular \\cite{Le99}). \n\nThe next step, carried out in Section~\\ref{sec:2},  \nis to analyze Hilbert subspaces of the space $\\Gamma(\\bV)$ \nof holomorphic sections of $\\bV$ \non which $G$ acts unitarily. In this context \n$(\\rho, V)$ is a bounded unitary representation. The most regular \nHilbert spaces with this property are those that we call \n{\\it holomorphically induced from $(\\rho, \\beta)$}. \nThey contain $(\\rho, V)$ as an $H$-subrepresentation \nsatisfying a compatibility condition with respect to $\\beta$. \nHere we show that if \nthe subspace $V \\subeq \\cH$ is invariant under the commutant \n$B_G(\\cH) = \\pi(G)'$, then restriction to $V$ yields an isomorphism \nof the von Neumann algebra $B_G(\\cH)$ with a suitably defined commutant \n$B_{H,\\fq}(V)$ of $(\\rho,\\beta)$ (Theorem~\\ref{thm:5.5}). \nThis has remarkable \nconsequences. One is that the representation of $G$ on $\\cH$ \nis irreducible (multiplicity free, discrete, type I) if and only \nif the representation of $(H,\\fq)$ on $V$ has this property. \nThe second main result in Section~\\ref{sec:2} is a \ncriterion for a unitary representation \n$(\\pi, \\cH)$ of $G$ to be holomorphically induced \n(Theorem~\\ref{thm:a.3}). \n\nSection~\\ref{sec:3} is devoted to a description \nof environments in which Theorem~\\ref{thm:a.3} applies naturally. \nHere we consider an element $d \\in \\g$ which is {\\it elliptic} in \nthe sense that the one-parameter group $e^{\\R \\ad d}$ of automorphisms \nof $\\g$ is bounded. This is equivalent to the existence of an \ninvariant compatible norm. Suppose that $0$ is isolated in $\\Spec(\\ad d)$. \nThen the subgroup $H := Z_G(d) = \\{ g \\in G \\: \\Ad(g)d = d \\}$ \nis a Lie subgroup and the homogeneous space $G\/H$ carries a natural \ncomplex manifold structure. A smooth unitary representation \n$(\\pi, \\cH)$ of $G$ is said to be of \n{\\it positive energy} if the selfadjoint operator \n$-i \\dd\\pi(d)$ is bounded from below. Note that this is in particular \nthe case if $\\pi$ is semibounded with $d \\in W_\\pi$. \nAny positive energy representation \nis generated as a $G$-representation by the closed subspace \n$V := \\oline{(\\cH^\\infty)^{\\fp^-}}$, where \n$\\fq = \\fp^+ \\rtimes \\fh_\\C \\subeq \\g_\\C$ is the complex subalgebra \ndefining the complex structure on $G\/H$ and \n$\\fp^- := \\oline{\\fp^+}$. \nIf the $H$-representation $(\\rho, V)$ is bounded, then \n$(\\pi, \\cH)$ is holomorphically induced \nby $(\\rho,\\beta)$, where $\\beta$ is determined by  $\\beta(\\fp^+) = \\{0\\}$ \n(Theorem~\\ref{thm:6.2}). In Theorem~\\ref{thm:3.15} we further \nshow that, under these assumptions, $\\pi$ is semibounded with $d \\in W_\\pi$. \nThese results are rounded off by Theorem~\\ref{thm:6.2b} which shows \nthat, if $\\pi$ is semibounded with $d \\in W_\\pi$, then \n$\\pi$ is a direct sum of holomorphically induced representations \nand if, in addition, $\\pi$ is irreducible, then $V$ coincides \nwith the minimal eigenspace of $-i\\dd\\pi(d)$ and the representation \nof $H$ on this space is automatically bounded. \nFor all this we use refined analytic tools based on the fact that \nthe space \n$\\cH^\\infty$ of smooth vectors is a Fr\\'echet space on which \n$G$ acts smoothly (\\cite{Ne10a}) and the $\\R$-action on \n$\\cH^\\infty$ defined by $\\pi_d(t) := \\pi(\\exp td)$ is equicontinuous. \nThese properties permit us to use a \nsuitable generalization of Arveson's spectral theory, developed\n in Appendix~\\ref{app:1}. \n\nIn \\cite{Ne11} we shall use the techniques developed \nin the present paper to obtain a complete descriptions of semibounded \nrepresentations for the class of hermitian Lie groups. \nOne would certainly like to extend the tools developed here to \nFr\\'echet--Lie groups such as diffeomorphism groups and groups of \nsmooth maps. Here a serious problem is the construction of \nholomorphic vector bundle structures on associated bundles, \nand we do not know how to extend this beyond the Banach context, \nespecially because of the non-existing solution theory for \n$\\oline\\partial$-equations (cf.\\ \\cite{Le99}). \n\nSeveral of our results have natural predecessors in more restricted\n contexts. In \\cite{BR07} Belti\\c{t}\\u{a} and Ratiu \nstudy holomorphic Hilbert bundles over Banach manifolds $M$ \nand consider the endomorphism bundle $B(\\bV)$  over $M \\times \\oline M$ \n(using a different terminology). They also relate Hilbert subspaces \n$\\cH$ of $\\Gamma(\\bV)$ to \nreproducing kernels, which in this context are \nsections of $B(\\bV)$ satisfying a certain holomorphy condition \nwhich under the assumption of local boundedness of the kernel \nis equivalent to \nholomorphy as a section of $B(\\bV)$ (\\cite[Thm.~4.2]{BR07}; \nsee also \\cite[Thm.~1.4]{BH98} and \\cite{MPW97} for the case of \nfinite dimensional \nbundles over finite dimensional manifolds and \\cite{Od92} for the \ncase of line bundles). \nBelti\\c{t}\\u{a} and Ratiu \nuse this setup to realize certain representations  \nof a $C^*$-algebra $\\cA$, which define bounded unitary representations \nof the unitary group $G := \\U(\\cA)$, in spaces of holomorphic sections of a \nbundle over a homogeneous space of the unit group $\\cA^\\times$ on \nwhich $\\U(\\cA)$ acts transitively (\\cite[Thm.~5.4]{BR07}). \nThese homogeneous spaces are of the form \n$G\/H$, where $H$ is the centralizer of some hermitian element \n$a \\in \\cA$ with finite spectrum, so that the realization \nof these representations by holomorphic sections could also be derived \nfrom our Corollary~\\ref{cor:6.2}. This work has been continued by \nBelti\\c{t}\\u{a} with Gal\\'e in a different direction, focusing \non complexifications of real homogeneous spaces \ninstead of invariant complex structures on the real spaces (\\cite{BG08}). \n\nIn \\cite{Bo80} Boyer constructs irreducible unitary representations \nof the Hilbert--Lie group \n$\\U_2(\\cH) = \\U(\\cH) \\cap (\\1 + B_2(\\cH))$ via holomorphic induction \nfrom characters of the diagonal subgroup. They live in spaces of \nholomorphic sections on homogeneous spaces of the complexified \ngroup $\\GL_2(\\cH) = \\GL(\\cH) \\cap (\\1 + B_2(\\cH))$, \nwhich are restricted versions of flag manifolds carrying \nstrong K\\\"ahler structures. \n\n\nIn the present paper we only deal with representations \nassociated to homogeneous vector bundles. To understand branching laws \nfor restrictions of representations to subgroups, one should \nalso study situations where the group $G$ does not act transitively \non the base manifold. For finite dimensional holomorphic vector \nbundles, this has been done extensively by \nT.~Kobayashi who obtained powerful criteria for representations \nin Hilbert spaces of holomorphic sections to be multiplicity \nfree (cf.\\ \\cite{KoT05}, \\cite{KoT06}). It would be interesting to explore \nthe extent to which \nKobayashi's technique of visible actions on complex manifolds \ncan be extended to Banach manifolds. \\\\\n\n\n{\\bf Notation:} For a group $G$ we write $\\1$ for the neutral element \nand $\\lambda_g(x) = gx$, resp., $\\rho_g(x) = xg$ \nfor left multiplications, resp., right multiplications. \nWe write $\\g_\\C$ for the complexification of a real Lie algebra $\\g$ and \n$\\oline{x + iy} := x - iy$ for the complex conjugation on $\\g_\\C$, which is an \nantilinear Lie algebra automorphism. \nFor two Hilbert spaces $\\cH_1, \\cH_2$, we write \n$B(\\cH_1, \\cH_2)$ for the space of continuous (=bounded) linear operators \n$\\cH_1 \\to \\cH_2$ and $B_p(\\cH)$, $1 \\leq p < \\infty$, \nfor the space of Schatten class operators \n$A \\: \\cH \\to \\cH$ of order $p$, i.e., $A$ is compact with \n$\\tr((A^*A)^{p\/2}) < \\infty$. \\\\\n\n{\\bf Acknowledgment:} We thank Daniel Belti\\c{t}\\u{a} for a careful \nreading of earlier versions of this paper, for pointing out references \nand for various interesting discussions on its subject matter. \n\n\\tableofcontents\n\n\n\\section{Holomorphic Banach bundles} \\mlabel{sec:1}\n\nTo realize unitary representations \nof Banach--Lie groups in spaces of holomorphic sections of \nHilbert bundles, we first need a \nparametrization of holomorphic bundle structures on \ngiven homogeneous vector bundles for real Banach--Lie groups. \nAs we shall see in Theorem~\\ref{thm:a.2} below, \nformulated appropriately, the corresponding results\nfrom the finite dimensional case (cf.\\ \\cite{TW70}) \ncan be generalized to the Banach context. \n\nThe following observation provides some information \non the assumptions required on the isotropy representation \nof a Hilbert bundle. It is a slight modification \nof results from \\cite[Sect.~3]{Ne09}. \n\n\\begin{prop}\\label{prop:contin} Let $(\\pi,\\cH)$ be a \nunitary representation of the Banach--Lie group $G$ \nfor which all vectors are smooth. \nThen $\\pi \\: G \\to \\U(\\cH)$ is a morphism of \nLie groups, hence in particular norm continuous, i.e., a \nbounded representation. \n\\end{prop}\n\n\\begin{prf} Our assumption implies that, for each $x\\in \\g$, \nthe infinitesimal generator $\\dd\\pi(x)$ of the unitary \none-parameter group $\\pi_x(t) := \\pi(\\exp_G(tx))$ \nis everywhere defined, hence a bounded operator because its \ngraph is closed. \n\nTherefore the derived representation leads to a morphism of \nBanach--Lie algebras \n$\\dd\\pi \\: \\g \\to B(\\cH).$\nSince the function $s_\\pi$ \nis a sup of a set of continuous linear functionals, it is \nlower semi-continuous. Hence the function \n\\[ x \\mapsto \\|\\dd\\pi(x)\\| \n= \\max(s_\\pi(x), s_\\pi(-x)) \\] \nis a lower semi-continuous seminorm \n and therefore continuous because $\\g$ is barreled \n(cf.\\ \\cite[\\S III.4.1]{Bou07}).\nWe conclude that the set of all linear functional \n$\\la \\dd\\pi(\\cdot)v,v\\ra$, $v \\in \\cH^\\infty$ a unit vector, \n is equicontinuous in  $\\g'$, \nso that the assertion follows from \\cite[Thm.~3.1]{Ne09}. \n\\end{prf}\n\nWe now turn to the case where $M$ is a Banach homogeneous space. \nLet $G$ be a Banach--Lie group with Lie algebra $\\g$ and \n$H \\subeq G$ be a split Lie subgroup, i.e., \nthe Lie algebra $\\fh$ of $H$ has a closed complement in $\\g$, \n for which the coset space \n$M := G\/H$ carries the structure of a complex manifold such that \nthe projection $q_M\\: G\\to G\/H$ is a smooth $H$-principal bundle and \n$G$ acts on $M$ by holomorphic maps. \nLet $m_0 = q_M(\\1) \\in M$ be the canonical base point and \n$\\fq \\subeq \\g_\\C$ be the kernel of the complex linear extension \nof the map $\\g \\to T_{m_0}(G\/H)$ to $\\g_\\C$, so that $\\fq$ is a closed \nsubalgebra of~$\\g_\\C$ invariant under $\\Ad(H)$ \n(cf.\\ \\cite[Thm.~15]{Bel05}). We call $\\fq$ the \n{\\it subalgebra defining the complex structure on $M = G\/H$} \nbecause specifying $\\fq$ means to identify \n$T_{m_0}(G\/H)\\cong \\g\/\\fh$ with the complex Banach space $\\g_\\C\/\\fq$ \nand thus specifying the complex structure on~$M$. \n\n\n\\begin{rem} \\mlabel{rem:norm-cont} \nIf the Banach--Lie group $G$ acts smoothly by isometric bundle automorphisms \non the holomorphic Hilbert bundle $\\bV$ over $M = G\/H$, then \nthe action of the stabilizer group $H$ on $V := \\bV_{m_0}$ \nis smooth, so that Proposition~\\ref{prop:contin} shows that it \ndefines a bounded unitary representation \n$\\rho \\: H \\to \\U(V)$. \n\nIf $\\sigma_M \\: G \\times M \\to M$ denotes the corresponding action on \n$M$ and $\\dot\\sigma_M \\: \\g \\to \\cV(M)$ the derived action, \nthen, for the closed subalgebra \n\\[ \\fq := \\{ x \\in \\g_\\C \\: \\dot\\sigma_M(x)(m_0) = 0\\}, \\] \nthe representation $\\beta \\: \\fq \\to B(V)$ \nis given by a continuous bilinear map \n\\[ \\hat\\beta \\: \\fq \\times V \\to V, \\quad \n(x,v) \\mapsto \\beta(x)v.\\] \nThis means that $\\beta$ is a continuous morphism \nof Banach--Lie algebras. \n\\end{rem}\n\nThis observation leads us to the following structures. \n\n\n\\begin{defn}\\mlabel{def:a.1} Let $H \\subeq G$ be a \nLie subgroup and  $\\fq \\subeq \\g_\\C$ be a closed subalgebra containing \n$\\fh_\\C$. \nIf  $\\rho \\: H \\to \\GL(V)$ is a norm continuous representation \non the Banach space $V$, then a morphism \n$\\beta \\: \\fq \\to \\gl(V)$ of complex Banach--Lie algebras \nis said to be an {\\it extension of $\\rho$} if \n\\begin{equation}\\label{eq:comprel} \n\\dd\\rho = \\beta\\res_\\fh \\quad \\mbox{ and } \\quad \n\\beta(\\Ad(h)x) = \\rho(h)\\beta(x)\\rho(h)^{-1} \\quad \\mbox{ for } \\quad \nh \\in H, x \\in \\fq. \n\\end{equation}\n\\end{defn}\n\n\\begin{defn} (a) \nIf $q \\: \\bV = G \\times_H V \\to M$ is a homogeneous vector bundle \ndefined by the norm continuous representation $\\rho \\: H \\to \\GL(V)$, \nwe associate to each section $s \\: M \\to \\bV$ the function \n$\\hat s \\: G \\to V$ specified by $s(gH) = [g, \\hat s(g)]$. \nA function $f \\: G \\to V$ is of the form $\\hat s$ for a \nsection of $\\bV$ if and only if \n\\begin{equation} \\label{eq:equiv-sec} \nf(gh) = \\rho(h)^{-1} f(g) \\quad \\mbox{ for } \\quad \ng \\in G, h \\in H. \n\\end{equation}\nWe write $C(G,V)_\\rho$, resp., $C^\\infty(G,V)_\\rho$ for the \ncontinuous, resp., smooth functions satisfying \n\\eqref{eq:equiv-sec}.  \n\n(b) We associate to each \n$x \\in \\g_\\C$ the left invariant \ndifferential operator on $C^\\infty(G,V)$ defined by \n\\[ (L_x f)(g) := \\derat0 f(g\\exp(tx)) \\quad \\mbox{ for } \\quad \nx \\in \\g. \\]\nBy complex linear extension, we define the operators \n\\[L_{x+ iy} := L_x + i L_y \\quad \\mbox{ for } \\quad z = x + iy \\in \\g_\\C, \nx,y \\in \\g. \\]\n\n(c) For any extension $\\beta$ of $\\rho$, \nwe write $C^\\infty(G,V)_{\\rho,\\beta}$ for the \nsubspace of those elements $f \\in C^\\infty(G,V)_\\rho$ \nsatisfying, in addition,  \n\\begin{equation} \\label{eq:inf-equiv}\nL_x f = - \\beta(x) f \\quad \\mbox{ for } \\quad x \\in \\fq. \n\\end{equation}\n\\end{defn}\n\n\\begin{rem}\nFor $x \\in \\g$, $h \\in H$ and any smooth function \n$\\phi$ defined on an open right $H$-invariant subset of \n$G$, we have \n\\begin{eqnarray*}\n(L_x\\phi)(gh) \n&=& \\derat0 \\phi\\big(g\\exp(t\\Ad(h)x)h\\big)=\n (L_{\\Ad(h)x}(\\phi \\circ \\rho_h))(g), \n\\end{eqnarray*}\nso that we obtain for each $x \\in \\g_\\C$ and $h \\in H$ the \nrelation \n\\begin{eqnarray}\\label{eq:liederrel}\n(L_x\\phi)\\circ \\rho_h = L_{\\Ad(h)x}(\\phi \\circ \\rho_h). \n\\end{eqnarray}\n\\end{rem}\n\n\nThe proof of the following theorem is very much inspired by \\cite{TW70}.\n\n\n\\begin{thm} \\mlabel{thm:a.2} Let $M = G\/H$, $V$ be a complex Banach space  \nand $\\rho \\: H \\to \\GL(V)$ be a norm continuous representation.  \nThen, for any extension $\\beta \\: \\fq \\to \\gl(V)$ of $\\rho$, \nthe associated bundle $\\bV := G \\times_H V$ carries \na unique structure of a holomorphic vector bundle over $M$, \nwhich is determined by the \ncharacterization of the holomorphic sections $s \\: M \\to \\bV$ \nas those for which $\\hat s \\in C^\\infty(G,V)_{\\rho,\\beta}$. \nAny such holomorphic bundle structure is $G$-invariant in the sense that \n$G$ acts on $\\bV$ by holomorphic bundle automorphism. \nConversely, every $G$-invariant holomorphic vector bundle \nstructure on $\\bV$ is obtained from this construction.\n\\end{thm}\n\n\\begin{prf} {\\bf Step 1:} Let $\\beta$ be an extension \nof $\\rho$ and  $E \\subeq \\g$ be a closed subspace \ncomplementing $\\fh$. Then $E \\cong \\g\/\\fh \\cong T_{m_0}(M)$ \nimplies the existence \nof a complex structure $I_E$ on $E$ for which $E \\to \\g\/\\fh$ is an isomorphism \nof complex Banach spaces. Therefore \nthe $I_E$-eigenspace decomposition \n\\[ E_\\C = E_+ \\oplus E_-, \\quad E_\\pm = \\ker(I_E \\mp i\\1),\\] \nis a direct decomposition into closed subspaces. The \nquotient map $\\g_\\C \\to \\g_\\C\/\\fq \\cong \\g\/\\fh$ \nis surjective on $E_+$ and annihilates $E_-$. Now \n$\\fr := E_+$ is a closed complex complement of $\\fq \n= \\fh_\\C \\oplus E_-$ in $\\g_\\C$ (\\cite[Thm.~15]{Bel05}). \n\nPick open convex $0$-neighborhoods \n$U_\\fr \\subeq \\fr$ and $U_\\fq \\subeq \\fq$ such that the BCH multiplication \n$*$ defines a biholomorphic map \n$\\mu \\: U_\\fr \\times U_\\fq \\to U, (x,y) \\mapsto x * y$, \nonto the open $0$-neighborhood $U \\subeq \\g_\\C$ \nand that $*$ defines an associative multiplication \ndefined on all triples of elements of $U$. \n\n{\\bf Step 2:} On the $0$-neighborhood  $U \\subeq \\g_\\C$, we consider the \nholomorphic function \n$$ F \\: U \\to \\GL(V), \\quad \nF(x * y) := e^{-\\beta(y)}. $$\nLet $U_\\g \\subeq U$ be an open $0$-neighborhood which is \nmapped by $\\exp_G$ diffeomorphically onto an open $\\1$-neighborhood \n$U_G$ of $G$. Then we consider the smooth function \n$$ f \\: U_G \\to \\GL(V), \\quad \\exp_G z \\mapsto F(z). $$ \n\nFor $w \\in \\g$ and $z \\in U_\\g$, the BCH product $z * tw \\in \\g_\\C$ \nis defined if \n$t$ is small enough, and we have \n\\begin{equation}\\label{eq:1.4} \n(L_w f)(\\exp_G z) = \\derat0 F(z * tw) \n= \\dd F(z) \\dd\\lambda_z^*(0)w, \n\\end{equation}\nwhere $\\dd\\lambda_z^*(0) \\: \\g_\\C \\to \\g_\\C$ is the differential \nof the multiplication map $\\lambda_z^*(x) = z * x$ in $0$.\nAs \\eqref{eq:1.4} is complex linear in $w \\in\\g_\\C$, it follows that \n$$ (L_w f)(\\exp_G z) = \\dd F(z) \\dd\\lambda_z^*(0)w $$  \nhold for every $w \\in \\g_\\C$, hence in particular for $w \\in \\fq$. \nFor $w \\in \\fq$ and $z = \\mu(x,y)$, we thus obtain \n\\begin{align*}\n\\dd F(z) \\dd\\lambda_z^*(0)w \n&= \\derat0 F(x * y * tw)\n= \\derat0 e^{-\\beta(y * tw)} \n= \\derat0  e^{-t \\beta(w)} e^{-\\beta(y)}\\\\\n&=  -\\beta(w) e^{-\\beta(y)}. \n\\end{align*}\nWe conclude that \n$$ (L_w f)(g) = -\\beta(w) f(g) \\quad \\mbox{ for } \\quad g \\in U_G. $$ \nIn particular, we obtain \n$f(gh) = \\rho(h)^{-1} f(g)$ for $g \\in U_G, h \\in H_0.$ \n\n{\\bf Step 3:} \nSince $H$ is a complemented Lie subgroup, there \nexists a connected submanifold $Z \\subeq G$ containing $\\1$ for which \nthe multiplication map \n$Z\\times H \\to G, (x,h) \\mapsto xh$ is a diffeomorphism onto \nan open subset of $G$. Shrinking $U_G$, we may therefore \nassume that $U_G = U_Z U_H$ holds for a connected open $\\1$-neighborhood \n$U_Z$ in $Z$ and a connected open $\\1$-neighborhood \n$U_H$ in $H$. Then \n\\[ \\tilde f(z h) := \\rho(h)^{-1} f(z) \n\\quad \\mbox{ for } \\quad z \\in U_Z, h \\in H, \\] \ndefines a smooth function $\\tilde f \\: U_Z H \\to \\GL(V)$. \nThat it extends $f$ follows from the fact that \n$u = zh \\in U_G$ with $z \\in U_Z$ and $h \\in U_H$ implies \n$h \\in H_0$, so that \n$$ \\tilde f(u) \n= \\rho(h)^{-1} f(z)\n= f(zh) = f(u). $$\n\nFor $w \\in \\fq$, formula \\eqref{eq:liederrel}\nleads to \n\\begin{align*}\n(L_w \\tilde f)(zh) \n&= (L_{\\Ad(h)w}(\\tilde f \\circ \\rho_h))(z)\n= L_{\\Ad(h)w}(\\rho(h)^{-1}\\tilde f)(z)\n= L_{\\Ad(h)w}(\\rho(h)^{-1}f)(z)\\\\\n&= - \\rho(h)^{-1}\\beta(\\Ad(h)w)f(z) \n= - \\beta(w)\\rho(h)^{-1}f(z) \n= - \\beta(w)\\tilde f(zh) . \n\\end{align*}\nTherefore $\\tilde f$ satisfies \n\\begin{equation}\\label{eq:comprel2}\nL_w \\tilde f = - \\beta(w) \\tilde f \\quad \\mbox{ for } \\quad w \\in \\fq.\n\\end{equation}\n\n{\\bf Step 4:} For $m \\in M$ we choose an element $g_m \\in G$ with $g_m.m_0 = m$ \nand put $U_m := g q_M(U_Z)$, so that \n$$ G^{U_m} := q_M^{-1}(U_m) = g_m U_Z H. $$ \nOn this open subset of $G$, the function \n$$ F_m \\: G^{U_m} \\to \\GL(V), \\quad \nF_m(g) := \\tilde f(g_m^{-1}g) $$ \nsatisfies \n\\begin{description}\n\\item[\\rm(a)] $F_m(gh) = \\rho(h)^{-1} F_m(g)$ for $g \\in G^{U_m}$, $h \\in H$.\n\\item[\\rm(b)] $L_w F_m = -\\beta(w) F_m$ for $w \\in \\fq$. \n\\item[\\rm(c)] $F_m(g_m) = \\1 = \\id_V$. \n\\end{description}\n\nNext we note that the function \n$F_m$ defines a smooth trivialization \n$$ \\phi_m \\: U_m \\times V \\to \\bV\\res_{U_m} = G^{U^m} \\times_H V, \\quad \n(gH, v) \\mapsto [g, F_m(g)v]. $$\nThe corresponding transition functions are given by \n$$ \\phi_{m,n} \\: U_{m,n} := U_m \\cap U_n \\to \\GL(V), \\quad \ngH \\mapsto \\tilde\\phi_{m,n}(g) := F_m(g)^{-1} F_n(g). $$ \n\nTo verify that these transition functions are holomorphic, \nwe have to show that the functions \n$\\tilde\\phi_{m,n} \\: G^{U_{m,n}} \\to B(V)$ \nare annihilated by the differential operators $L_w$, $w \\in \\fq$. \nThis follows easily from the product rule and (b): \n\\begin{align*}\nL_w(F_m^{-1} F_n) \n&= L_w(F_m^{-1}) F_n + F_m^{-1} L_w(F_n)\n= - F_m^{-1} L_w(F_m) F_m^{-1} F_n + F_m^{-1} L_w(F_n)\\\\\n&= F_m^{-1}(\\beta(w) F_m F_m^{-1}- \\beta(w)) F_n \n= F_m^{-1}(\\beta(w) - \\beta(w)) F_n = 0.\n\\end{align*}\nWe conclude that the transition functions $(\\phi_{m,n})_{m,n \\in M}$ \ndefine a holomorphic vector bundle atlas on $\\bV$. \n\n{\\bf Step 5:} To see \nthat this holomorphic structure is determined uniquely \nby $\\beta$, pick an open connected subset $U \\subeq M$ containing \n$m_0$ on which the bundle is holomorphically trivial and \nthe trivialization is specified by a \nsmooth function $F \\: G^U \\to \\GL(V)$. \nThen $L_w F = -\\beta(w)F$ implies that \n$\\beta(w) = - (L_wF)(\\1) F(\\1)^{-1}$ \nis determined uniquely by the  holomorphic structure on $\\bV$. \n\n{\\bf Step 6:} The above construction also shows that, for \nany holomorphic vector bundle structure on $\\bV$, for which \n$G$ acts by holomorphic bundle automorphisms, we may consider \n\\begin{equation}\n  \\label{eq:beta}\n\\beta \\: \\fq \\to B(V), \\quad \\beta(w) := - (L_wF)(\\1) F(\\1)^{-1} \n\\end{equation}\nfor a local trivialization given by $F \\: G^U \\to \\GL(V)$, where \n$m_0 \\in U$. Then \n$\\beta$ is a continuous linear map. \nTo see that it does not depend on the choice of $F$, we note that, \nfor any other trivialization $\\tilde F \\: G^U \\to \\GL(V)$, \nthe function $F^{-1}\\cdot \\tilde F$ factors through a holomorphic \nfunction on $U$, so that \n$$ 0 = L_w(F^{-1}\\tilde F)(\\1) \n= F(\\1)^{-1}\\Big(- (L_w F)(\\1)F(\\1)^{-1} + (L_w \\tilde F)(\\1)\\tilde F(\\1)^{-1}\\Big)\\tilde F(\\1). $$\nWe conclude that the right hand side of \\eqref{eq:beta} \ndoes not depend on the choice of $F$. \nApplying this to functions of the form \n$F_g(g') := F(gg')$, defined on $g^{-1}G^U$, we obtain in particular \n\\[ -\\beta(w) = -(L_w f)(\\1)F(\\1)^{-1} = (L_w F_g)(\\1) F_g(\\1)^{-1}\n= (L_w F)(g) F(g)^{-1}, \\]\nso that $L_w F = -\\beta(w)F$. \n\nFor $g= h \\in H$, we obtain  with \\eqref{eq:liederrel} \n\\begin{align*}\n-\\beta(w) &= (L_w F_h)(\\1) F(h)^{-1}\n= (L_w F)(h) F(\\1)^{-1}\\rho(h)\\\\\n&= \\rho(h)^{-1}(L_{\\Ad(h)w}F)(\\1) F(\\1)^{-1}\\rho(h)\n= -\\rho(h)^{-1}\\beta(\\Ad(h)w)\\rho(h).\n\\end{align*}\nFor $w \\in \\fh$, the relation $\\beta(w) = \\dd\\rho(w)$ \nfollows immediately from the $H$-equivariance of~$F$. \nTo see that $\\beta$ is an extension of $\\rho$, it now remains \nto verify that it is a homomorphism of Lie algebras. \nFor $w_1, w_2 \\in \\fq$, we have \n\\begin{align*}\n\\beta([w_1, w_2])F \n&= - L_{[w_1, w_2]}F\n= L_{w_2} L_{w_1} F - L_{w_1} L_{w_2} F \\\\\n&= -L_{w_2} \\beta(w_1) F + L_{w_1} \\beta(w_2)F \n= \\beta(w_1)  \\beta(w_2)F - \\beta(w_2)\\beta(w_1)F,\n\\end{align*}\nwhich shows that $\\beta$ is a homomorphism of Lie algebras. \n\\end{prf} \n\n\\begin{ex}  We consider the special case  \nwhere $G$ is contained as a Lie subgroup in \na complex Banach--Lie group $G_\\C$ with Lie algebra $\\g_\\C$ \nand where $Q := \\la \\exp \\fq \\ra$ is a Lie subgroup of $G_\\C$ \nwith $Q \\cap G = H$. Then the orbit mapping $G \\to G_{\\C}\/Q, \ng \\mapsto g Q$, induces an open \nembedding of $M = G\/H$ as an open $G$-orbit in the complex manifold\n$G_{\\C}\/Q$.\n\nIn this case every holomorphic representation \n$\\pi \\: Q \\to \\GL(V)$ defines an associated holomorphic Banach \nbundle $\\bV := G_\\C \\times_Q V$ over the complex manifold \n$G_\\C\/Q \\cong G\/H$. \n\nSince the Lie algebra $\\g_\\C$ need not be integrable in the sense \nthat it is the Lie algebra of a Banach--Lie group \n(cf.~\\cite[Sect.~6]{GN03}), \nthe assumption $G \\subeq G_\\C$ is not general enough to \ncover every situation. One therefore needs the general Theorem~\\ref{thm:a.2}.\n\nIn \\cite[Sect.~6]{GN03} one finds examples \nof simply connected Banach--Lie groups \n$G$ for which $\\g_\\C$ is not integrable. Here \n$G$ is a quotient of a simply connected Lie group \n$\\hat G$ by a central subgroup $Z \\cong \\R$, \n$\\hat G$ has a simply connected universal complexification \n$\\eta_{\\hat G} \\: {\\hat G} \\to {\\hat G}_\\C$, and the subgroup \n$\\exp_{{\\hat G}_\\C}(\\fz_\\C)$ is not closed; its closure is a $2$-dimensional \ntorus. \n\nThen the real Banach--Lie group $G$ acts smoothly and faithfully \nby holomorphic maps on a complex Banach \nmanifold $M$ for which $\\g_\\C$ is not integrable. \nIt suffices to pick a suitable tubular neighborhood \n${\\hat G} \\times U \\cong {\\hat G} \\exp(U) \\subeq {\\hat G}_\\C$, \nwhere \n$U \\subeq i\\hat\\fg$ is a convex $0$-neighborhood. \nThen the quotient \n$M := {\\hat G}\/Z \\times (U + i\\z)\/i\\z$ is a complex manifold on which \n$G \\cong {\\hat G}\/Z$ acts faithfully, but $\\g_\\C$ is not integrable. \n\\end{ex}\n\n\\section{Hilbert spaces of holomorphic sections} \\mlabel{sec:2}\n\nIn this section we take a closer look at \nHilbert spaces of holomorphic sections in $\\Gamma(\\bV)$ for \nholomorphic Hilbert bundles $\\bV$ constructed with the methods from \nSection~\\ref{sec:1}. Here we are interested in Hilbert spaces with \ncontinuous point evaluations (to be defined below) \non which $G$ acts unitarily. This leads to the concept of a \nholomorphically induced representation, which for infinite dimensional \nfibers is a little more subtle than in the finite dimensional case. \nThe first key result of this section is \nTheorem~\\ref{thm:5.5} relating the commutant of a \nholomorphically induced unitary $G$-representation \n$(\\pi, \\cH)$ to the commutant of the representation \n$(\\rho,\\beta)$ of $(H,\\fq)$ on the fiber~$V$.\nThis result is complemented by Theorem~\\ref{thm:a.3} which is a \nrecognition devise for holomorphically induced representations. \n \n\n\\begin{defn} Let $q \\: \\bV \\to M$ be a holomorphic Hilbert bundle \non the complex manifold $M$. We write $\\Gamma(\\bV)$ for the space of \nholomorphic sections of $\\bV$. A Hilbert subspace \n$\\cH \\subeq \\Gamma(\\bV)$ is said to have {\\it \ncontinuous point evaluations} if all the evaluation maps \n\\[ \\ev_m \\: \\cH \\to \\bV_m, \\quad s \\mapsto s(m)\\] \nare continuous and the function $m \\mapsto \\|\\ev_m\\|$ is locally bounded. \n\n\nIf $G$ is a group acting on $\\bV$ by holomorphic \nbundle automorphisms, $G$ acts on $\\Gamma(\\bV)$ by \n\\begin{equation}\n  \\label{eq:g-act}\n(g.s)(m) := g.s(g^{-1}m).   \n\\end{equation}\nWe call a Hilbert subspace $\\cH \\subeq \\Gamma(\\bV)$ with continuous \npoint evaluations {\\it $G$-invariant} if \n$\\cH$ is invariant under the action defined by \\eqref{eq:g-act} \nand the so obtained representation of $G$ on $\\cH$ is \nunitary. \n\\end{defn}\n\n\\begin{rem} \\mlabel{rem:2.2} \nLet $q \\: \\bV \\to M$ be a holomorphic Hilbert bundle \nover $M$. Then we can represent holomorphic sections of this \nbundle by holomorphic functions on the dual bundle $\\bV^*$ \nwhose fiber $(\\bV^*)_m$ is the dual space  \n$\\bV_m^*$ of $\\bV_m$.  We thus obtain an embedding \n\\begin{equation}\n  \\label{eq:funcdual}\n\\Psi \\: \\Gamma(\\bV) \\to \\cO(\\bV^*), \\quad \n\\Psi(s)(\\alpha_m) = \\alpha_m(s(m)),  \n\\end{equation}\nwhose image consists of \nholomorphic functions on $\\bV^*$ which are fiberwise linear. \n\nIf $G$ is a group acting on $\\bV$ by holomorphic \nbundle automorphisms, then $G$ also acts naturally \nby holomorphic maps on $\\bV^*$ via \n$(g.\\alpha_m)(v_{g.m}) := \\alpha_m(g^{-1}.v_{g.m})$ \nfor $\\alpha_m \\in \\bV^*_m$. \nTherefore \n\\[ \\Psi(g.s)(\\alpha_m) \n= \\alpha_m(g.s(g^{-1}.m)) \n= (g^{-1}.\\alpha_m)(s(g^{-1}.m)) \n= \\Psi(s)(g^{-1}.\\alpha_m) \\] \nimplies that $\\Psi$ is equivariant with respect to the natural \n$G$-actions on $\\Gamma(\\bV)$ and $\\cO(\\bV^*)$. \n\\end{rem}\n\n\n\\subsection{Existence of analytic vectors}  \n\n\n\\begin{lem}\n\\mlabel{lem:3.1} \nIf $M = G\/H$ is a Banach homogeneous space with a $G$-invariant \ncomplex structure and $\\bV = G\\times_H V$ a $G$-equivariant \nholomorphic vector bundle over $M$ defined by \na pair $(\\rho,\\beta)$ as in {\\rm Theorem~\\ref{thm:a.2}}, \nthen the $G$-action on $\\bV$ is analytic.  \n\\end{lem}\n\n\\begin{prf} The manifold \n$\\bV$ is a $G$-equivariant quotient of the product \nmanifold $G \\times V$ on which $G$ acts analytically by left \nmultiplications in the left factor. Since the quotient map \n$q \\: G \\times V \\to \\bV$ is a real analytic submersion, the \naction of $G$ on $\\bV$ is also analytic. \n\\end{prf}\n\n\\begin{defn} Let $M$ be a complex manifold (modeled on a locally \nconvex space) and $\\cO(M)$ the space of holomorphic complex-valued \nfunctions on $M$. We write $\\oline M$ for the conjugate complex manifold. \nA holomorphic function \n$Q \\: M \\times \\oline M \\to \\C$\nis said to be a {\\it reproducing kernel} of a Hilbert subspace \n$\\cH \\subeq \\cO(M)$ if for each $w\\in M$ the function \n$Q_w(z) := Q(z,w)$ is contained in $\\cH$ and satisfies \n$$ \\la f, Q_z \\ra = f(z) \\quad \\mbox{ for } \\quad z \\in M, f \\in \\cH. $$\nThen $\\cH$ is called a {\\it reproducing kernel Hilbert space} \nand since it is determined uniquely by the kernel $Q$, it is \ndenoted $\\cH_Q$ (cf.\\ \\cite[Sect.~I.1]{Ne00}). \n\nNow let $G$ be a group and \n$\\sigma \\: G \\times M  \\to M, (g,m) \\mapsto g.m$ \nbe a smooth right action of $G$ on $M$ by holomorphic maps. Then \n $(g.f)(m) := f(g^{-1}.m)$ defines a unitary representation \nof $G$ on a reproducing kernel Hilbert space $\\cH_Q \\subeq \\cO(M)$ \nif and only if the kernel $Q$ is {\\it invariant}: \n$$ Q(g.z, g.w) = Q(z,w) \\quad \\mbox{ for } \\quad z,w \\in M, g \\in G $$ \n(\\cite[Rem.~II.4.5]{Ne00}). In this case we call \n${\\cal H}_Q$ a $G$-invariant reproducing kernel Hilbert space. \n\\end{defn} \n\n\n\\begin{lem} \\mlabel{lem:2.2} \nLet $G$ be a Banach--Lie group \nacting analytically via \n\\[\\sigma \\: G \\times M \\to M, \\quad \n(g,m) \\mapsto \\sigma_g(m) \\] \nby holomorphic maps on the complex manifold~$M$. \n\n{\\rm(a)} Let $\\cH \\subeq \\cO(M)$ be a reproducing kernel Hilbert \nspace whose kernel $Q$ is a $G$-invariant holomorphic function on \n$M \\times \\oline M$. \nThen the elements $(Q_m)_{m \\in M}$ representing the evaluation \nfunctionals in $\\cH$ are analytic \nvectors for the representation of $G$, defined by \n$\\pi(g)f := f \\circ \\sigma_g^{-1}$. \n\n{\\rm(b)} Let $\\bV \\to M$ be a holomorphic $G$-homogeneous \nHilbert bundle and $\\cH \\subeq \\Gamma(\\bV)$ be a $G$-invariant \nHilbert space with continuous point evaluations. \nThen every vector of the form \n$\\ev_m^*v$, $m \\in M$, $v \\in \\bV_m$, is analytic for the \n$G$-action on $\\cH$. \n\\end{lem}\n\n\\begin{prf} (a) Since $Q$ is holomorphic on \n$M \\times \\oline M$, it is in particular real analytic. \nFor $m \\in M$, we have \n\\[ \\la \\pi(g)Q_m, Q_m \\ra \n= (\\pi(g)Q_m)(m) = Q_m(g^{-1}m) = Q(g^{-1}m,m), \\]\nand since $Q$ and the $G$-action are real analytic, \nthis function is analytic on $M$. Now \\cite[Thm.~5.2]{Ne10b} \nimplies that $Q_m$ is an analytic vector. \n\n(b) As in Remark~\\ref{rem:2.2}, we \nrealize $\\Gamma(\\bV)$ by holomorphic functions on the dual \nbundle $\\bV^*$. We thus obtain a \nreproducing kernel Hilbert space \n$\\cH_Q := \\Psi(\\cH) \\subeq \\cO(\\bV^*)$, \nand since $\\Psi$ is $G$-equivariant, the reproducing \nkernel $Q$ is $G$-invariant. \nFor $v \\in \\bV_m$, evaluation in the corresponding element \n$\\alpha_m := \\la \\cdot, v \\ra \\in \\bV^*_m$ is given  by \n\\[ s \\mapsto \\la s(m), v \\ra = \\la \\ev_m(s), v\\ra \n= \\la s, \\ev_m^*v\\ra.\\] \nFor the corresponding $G$-invariant kernel $Q$ on $\\bV^*$ \nthis means that $Q_{\\alpha_m} = \\ev_m^*v$, so that the assertion \nfollows from (a). \n\\end{prf}\n\n\n\\subsection{The endomorphism bundle and commutants} \n\nThe goal of this section is Theorem~\\ref{thm:5.5} which connects \nthe commutant of the $G$-representation on a Hilbert space $\\cH_V$ \nof a holomorphically induced representation with the \ncorresponding representation $(\\rho, \\beta)$ \nof $(H,\\fq)$ on $V$. The remarkable \npoint is that, under natural assumptions, these commutants are isomorphic, \nso that both representations have the same decomposition theory. \nThis generalizes an important \nresult of S.~Kobayashi concerning irreducibility \ncriteria for the $G$-representation on $\\cH_V$ (\\cite{Ko68}, \n\\cite[Thm.~2.5]{BH98}). \n\nLet $\\bV = G \\times_H V \\to M$ be a $G$-homogeneous holomorphic \nHilbert bundle as in Theorem~\\ref{thm:a.2}. Then \nthe complex manifold $M \\times \\oline M$ is a complex homogeneous \nspace $(G \\times G)\/(H \\times H)$, \nwhere the complex structure is defined by the closed subalgebra \n$\\fq \\oplus \\oline\\fq$ of $\\g_\\C \\oplus \\g_\\C$. \nOn the Banach space $B(V)$ we consider the norm continuous \nrepresentation of $H \\times H$ by \n\\[ \\tilde\\rho(h_1, h_2)A = \\rho(h_1)A\\rho(h_2)^* \\] \nand the the corresponding extension \n$\\tilde\\beta \\: \\fq \\oplus \\oline \\fq \\to \\gl(B(V))$ by \n\\[ \\tilde\\beta(x_1,x_2)A := \\beta(x_1)A + A \\beta(\\oline{x_2})^*.\\]\nWe write ${\\mathbb L} := (G \\times G) \\times_{H \\times H} B(V)$ for the \ncorresponding holomorphic Banach bundle over \n$M \\times \\oline M$ (Theorem~\\ref{thm:a.2}). \n\n\\begin{rem} For every $g \\in G$, we have isomorphisms \n$V \\to \\bV_{gH} = [g,V], v \\mapsto [g,v]$. \nAccordingly, we have for every pair $(g_1, g_2) \\in G \\times G$ an \nisomorphism \n\\[ \\nu \\: B(V) \\to B(\\bV_{g_2 H}, \\bV_{g_1H}), \\quad \n\\nu(A)[g_2,v] \\mapsto [g_1,Av]. \\] \nThis defines a map \n\\[ \\gamma \\: G \\times G \\times B(V) \\to \nB(\\bV) := \\bigcup_{m,n \\in M} B(\\bV_m,\\bV_n), \\quad \n\\gamma(g_1, g_2,A)[g_2,v] = [g_1, Av]. \\] \nFor $h_1, h_2 \\in H$, we then have \n\\begin{align*}\n&\\gamma(g_1h_1, g_2h_2,\\tilde\\rho(h_1, h_2)^{-1}A)[g_2,v] \n= \\gamma(g_1h_1, g_2h_2,\\tilde\\rho(h_1, h_2)^{-1}A)[g_2h_2,\\rho(h_2)^{-1}v] \\\\\n&= [g_1 h_1, \\rho(h_1)^{-1} A v] \n= [g_1, A v] = \\gamma(g_1, g_2, A)[g_2,v], \n\\end{align*}\nso that $\\gamma$ factors through a bijection \n$\\oline\\gamma \\: {\\mathbb L} \\to B(\\bV).$ \nThis provides an interpretation of the bundle ${\\mathbb L}$ as the \n{\\it endomorphism bundle of $\\bV$} (cf.\\ \\cite{BR07}). \n\\end{rem}\n\nSince the group $G$ acts (diagonally) on the bundle \n${\\mathbb L}$, it makes sense to consider $G$-invariant holomorphic \nsections. \n\n\\begin{lem} \\mlabel{lem:2.7} \nThe space $\\Gamma({\\mathbb L})^G$ of $G$-invariant holomorphic \nsections of ${\\mathbb L}$ has the property that the evaluation map \n\\[ \\ev \\: \\Gamma({\\mathbb L})^G \\cong C^\\infty(G \\times G, B(V))_{\\tilde \\rho, \n\\tilde\\beta} \\to B_H(V), \\quad s \\mapsto \\hat s(\\1,\\1) \\] \nis injective. \n\\end{lem}\n\n\\begin{prf} If $\\hat s(\\1,\\1) = 0$, then \nthe corresponding holomorphic section $s \\in \\Gamma({\\mathbb L})$ vanishes on the \ntotally real submanifold \n\\[ \\Delta_M := \\{ (m,m) \\: m \\in M \\} = G(m_0, m_0)\\subeq \nM \\times \\oline M,\\] \nand hence on all of $M \\times \\oline M$. \nFor the function $\\hat s \\: G \\times G \\to B(V)$, we \nhave for $h \\in H$ \n\\[ \\hat s(\\1,\\1) = (h^{-1}.\\hat s)(\\1,\\1) \n= \\hat s(h, h) = \\rho(h)^{-1} \\hat s(\\1,\\1) \\rho(h), \\] \nshowing that $\\hat s(\\1,\\1)$ lies in $B_H(V)$. \n\\end{prf}\n\n\\begin{rem} In general, the holomorphic bundle \n${\\mathbb L}$ does not have any non-zero holomorphic section, although \nits restriction to the diagonal $\\Delta_M$ \nhas the trivial section $R$ given by $R_{(m,m)} = \\id_{\\bV_m}$ \nfor every $m\\in M$. \n\nIf $R \\: M \\times \\oline M \\to {\\mathbb L}$ is a holomorphic section, \nthen we obtain for every $v \\in V$ and $n \\in M$ a holomorphic section \n$R_{n,v} \\: M \\to \\bV, m \\mapsto R(m,n)v$    \nwhich is non-zero in $m$ if $R_{m,m}v \\not=0$. \nTherefore $\\bV$ has nonzero holomorphic sections if ${\\mathbb L}$ does, and this \nis not always the case. \n\\end{rem}\n\n\\begin{ex} \\mlabel{ex:2.11}  \n(a) Let $(\\pi, \\cH)$ be a unitary representation of \n$G$ and $\\Psi \\: \\cH \\to \\Gamma(\\bV)$ be $G$-equivariant \nsuch that the evaluation maps $\\ev_m \\circ \\Psi \\: \\cH \\to \\bV_m$ \nare continuous and $m \\mapsto \\|\\ev_m\\|$ is locally bounded. \nThen \n\\[ R(m,n) := (\\ev_m \\Psi) (\\ev_n \\Psi)^* \\in B(\\bV_n, \\bV_m)\\]  \ndefines a holomorphic section of $B(\\bV)$. \nSince this  assertion is local, it follows from \nthe corresponding assertion for \ntrivial bundles treated in \\cite[Lemma~A.III.9(iii)]{Ne00}. \nThe corresponding smooth function \n\\[ \\hat R  \\: G \\times G \\to B(V), \\quad \n\\hat R(g_1, g_2) = (\\ev_{g_1} \\Psi) (\\ev_{g_2} \\Psi)^*\\] \nis a $G$-invariant $B(V)$-valued \nkernel on $G \\times G$ because \n\\[ \\ev_{gh} \\Psi = \\ev_h \\Psi \\circ \\pi(g)^{-1} \n\\quad \\mbox{ for } \\quad g,h \\in G. \\] \nWe conclude that $\\hat R \\in \\Gamma({\\mathbb L})^G$, so that it is completely \ndetermined by \n\\[ \\hat R(\\1,\\1) = (\\ev_\\1 \\Psi) (\\ev_\\1 \\Psi)^*\\] \n(Lemma~\\ref{lem:2.7}). \nIn particular, $\\Psi$ is completely determined by $\\ev_\\1 \\circ \\Psi$. \nThis follows also from its equivariance, which leads to \n$\\Psi(v)(g) = \\ev_\\1(g^{-1}.\\Psi(v)) \n= \\ev_\\1\\Psi(\\pi(g)^{-1}v).$ \n\n(b) For $A, B \\in B(V)$ commuting \nwith $\\rho(H)$ and $\\beta(\\fq)$ and $\\hat R \\in C^\\infty(G \\times G, \nB(V))_{\\tilde\\rho, \\tilde\\beta}$, the function \n\\[ \\hat R^{A,B} \\: G \\times G \\to B(V), \\quad \n\\hat R^{A,B}(g_1, g_2) = A R(g_1, g_2) B\\] \nis also contained in $C^\\infty(G \\times G, B(V))_{\\tilde\\rho, \\tilde\\beta}$, \nhence defines a holomorphic section of the bundle~${\\mathbb L}$. \n\\end{ex} \n\n\\begin{defn} \\mlabel{def:2.10} \n(a) For $(\\rho, \\beta)$ as in Theorem~\\ref{thm:a.2} \nand the corresponding associated bundle $\\bV = G \\times_H V$, \na unitary representation $(\\pi, \\cH)$ of $G$ is said to be \n{\\it holomorphically induced from $(\\rho,\\beta)$} \nif there exists a realization \n$\\Psi \\: \\cH \\to \\Gamma(\\bV)$ as an invariant Hilbert \nspace with continuous point evaluations whose kernel \n$R \\in \\Gamma({\\mathbb L})^G$ satisfies $\\hat R(\\1,\\1) = \\id_V$. \n\n(b) Since this condition determines $R$ by Lemma~\\ref{lem:2.7}, \nit also determines the reproducing kernel space $\\Psi(\\cH)$ \nand its norm. \nWe conclude that, for every pair $(\\rho,\\beta)$, \nthere is at most one holomorphically induced unitary representation \nof $G$ up to unitary equivalence. \nAccordingly, we call $(\\rho, \\beta, V)$ {\\it inducible} \nif there exists a corresponding holomorphically induced unitary \nrepresentation of~$G$. \n\nIf this is the case, then we use the isometric embedding \n$\\ev_\\1^* \\: V  \\to \\cH$ to identify $V$ with a subspace of \n$\\cH$ and note that \nthe evaluation map $\\ev_\\1 \\: \\cH \\to V$ corresponds to the\northogonal projection $p_V \\: \\cH \\to V$. \n\\end{defn}\n\n\\begin{rem} \\mlabel{rem:2.13} (a) If $\\cH_V  \\into \\Gamma(\\bV)$ \nis a $G$-invariant Hilbert \nspace on which the representation is holomorphically induced, \nthen we obtain a $G$-invariant element $Q \\in B(\\bV)$ by  \n\\[ Q(m,n) := \\ev_m \\ev_n^* \\in B(\\bV_n, \\bV_m),\\] \nand the relation $\\ev_g = \\ev_\\1 \\circ \\pi(g)^{-1} \n= p_V \\circ \\pi(g)^{-1}$ yields \n\\[ \\hat Q(g_1, g_2) = \\ev_{g_1} \\ev_{g_2}^*  \n= p_V \\pi(g_1)^{-1} \\pi(g_2)p_V = p_V \\pi(g_1^{-1}g_2)p_V \\] \nand in particular $\\hat Q(\\1,\\1) = \\id_V.$\n\n(b) Suppose that $\\cH \\subeq \\Gamma(\\bV)$ is holomorphically \ninduced. Since $\\ev_\\1$ is $H$-equivariant, the closed subspace \n$V \\subeq \\cH$ is $H$-invariant and the $H$-representation \non this space is equivalent to $(\\rho,V)$, hence in particular \nbounded. \n\nMoreover, $V \\subeq \\cH^\\omega$ follows from \nLemma~\\ref{lem:2.2}(b). Since the evaluation maps \\break $\\ev_g \\: \\cH \\to V$ \nseparate the points, the analyticity of the elements \n$\\ev_g^*v$ even implies that $\\cH^\\omega$ is dense in $\\cH$. \n\nFor $x \\in \\fq$  and $s \\in \\cH^\\infty$ we have \n\\[ \\ev_\\1\\dd\\pi(x) s \n= (\\dd\\pi(x)s)\\,\\hat{}(\\1) \n= - L_x \\hat  s(\\1) \n=  \\beta(x) \\hat s(\\1) \n=  \\beta(x) \\ev_\\1 s.\\]\nTherefore $\\dd\\pi(\\fq)$ preserves the subspace \n$\\cH^\\infty \\cap V^\\bot = \\cH^\\infty \\cap \\ker(\\ev_\\1)$. \nFrom $V \\subeq \\cH^\\infty$, we derive \n\\[ \\cH^\\infty = V \\oplus (V^\\bot \\cap \\cH^\\infty), \\] \nso that the density of $\\cH^\\infty$ in $\\cH$ implies that \n$V = (V^\\bot \\cap \\cH^\\infty)^\\bot$, and hence that \nthis space is invariant under the restriction \n$\\dd\\pi(\\oline x)$ of the adjoint $- \\dd\\pi(x)^*$. \nFor $s_j = \\ev_\\1^*v_j \\in V$, $j =1,2$,  we further obtain \n\\begin{align*}\n\\la \\dd\\pi(\\oline x)s_1, s_2 \\ra \n&=  -\\la \\ev_\\1^* v_1, \\dd\\pi(x) s_2 \\ra \n=  -\\la v_1, \\ev_\\1\\dd\\pi(x) s_2 \\ra \n=  -\\la v_1, \\beta(x) \\ev_\\1 s_2 \\ra \\\\\n&=  -\\la v_1, \\beta(x) v_2 \\ra \n=  -\\la \\beta(x)^*v_1,v_2 \\ra,\n\\end{align*}\nso that \n\\begin{equation}\\label{eq:a3} \n\\dd\\pi(\\oline x)\\res_V = - \\beta(x)^*, \\quad x \\in \\fq.\n\\end{equation}\n\nFinally we observe that \n\\[ (\\pi(G)V)^\\bot = \\{ s \\in \\cH \\: (\\forall g \\in G)\\, \\hat s(g) = 0\\}\n= \\{0\\}\\] \n implies that $\\cH = \\oline{\\Spann(\\pi(G)V)}$. \n\\end{rem}\n\nIf $V$ is of the form $\\oline{(\\cH^\\infty)^\\fn}$ for a subalgebra \n$\\fn \\subeq \\g_\\C$, then it is invariant under the commutant $B_G(V)$, \nbut we do not know if this is always true for holomorphically \ninduced representation. \nTo make the following proposition as flexible as possible, we \nassume this naturality condition of~$V$ (cf.\\ Remark~\\ref{rem:2.14} below).\n\n\\begin{thm} \\mlabel{thm:5.5} \nSuppose that $(\\pi, \\cH_V)$ is holomorphically induced from \nthe representation $(\\rho,\\beta)$ of $(H,\\fq)$ on~$V$ \nand that \n\\[ B_{H,\\fq}(V) := \n\\{ A \\in B_H(V) \\: (\\forall x \\in \\fq)\\, \nA \\beta(x) = \\beta(x) A, A^* \\beta(x) = \\beta(x) A^* \\} \\] \nis the involutive commutant of $\\rho(H)$ and $\\beta(\\fq)$. \nIf $V$ is invariant under $B_G(V)$, then the map \n\\[ R \\: B_G(\\cH_V) \\to B_{H,\\fq}(V), \\quad  \nA \\mapsto A\\res_V \\]\nis an isomorphism of von Neumann algebras. \n\\end{thm}\n\n\\begin{prf} By assumption, every $A \\in B_G(\\cH_V)$ preserves \n$V$, so  that \n$A\\res_V$ can be identified with the operator $p_V A p_V  \\in B(V)$, \nwhere $p_V \\: \\cH \\to V$ is the orthogonal projection. \nClearly $A\\res_V$ commutes with each $\\rho(h) = \\pi(h)\\res_V$. \nIt also preserves the subspace $\\cH^\\infty$ on which it satisfies \n$\\pi(x)A = A \\pi(x)$ for every $x \\in \\g_\\C$. Therefore \n$A\\res_V$ commutes with $\\dd\\pi(\\oline\\fq)$ and hence with \n$\\beta(\\fq)$ (cf.~\\eqref{eq:a3} in Remark~\\ref{rem:2.13}). \n\nTherefore $R$ defines a homomorphism of von Neumann algebras. \nIf $R(A) = 0$, then $A V = 0$ implies that \n$A \\pi(G)V = \\{0\\}$, which leads to $A = 0$. \nHence $R$ is injective. \n\nSince each von Neumann algebra is generated by orthogonal projections \n(\\cite[Chap.~1, \\S 1.2]{Dix96}) \nand images of von Neumann algebras under restriction maps \nare von Neumann algebras \n(\\cite[Chap.~1, \\S 2.1, Prop.~1]{Dix96}), \nwe are done if we can show that every orthogonal \nprojection in $B_{H,\\fq}(V)$ is contained in the image of \n$R$. So let $P \\in B_{H,\\fq}(V)$. \nThen $V_1 := P(V)$ and $V_2 := (\\1-P)(V)$  yields an \n$(H,\\fq)$-invariant orthogonal decomposition $V = V_1 \\oplus V_2$. \n\nLet $\\hat Q \\: G \\times G \\to B(V), (g_1, g_2) \\mapsto \np_V \\pi(g_1^{-1}g_2) p_V$ be the natural kernel function \ndefining the inclusion $\\cH_V \\into \\Gamma(\\bV)$ and consider the \n$G$-invariant kernel \n\\[  \\hat R(g_1, g_2) := P \\hat Q(g_1, g_2) (\\1 - P). \\]\nAccording to Example~\\ref{ex:2.11}(b), \nit is contained in $C^\\infty(G \\times G, B(V))_{\\tilde\\rho,\\tilde\\beta}$, \nhence defines an element $R \\in \\Gamma({\\mathbb L})^G$. \nIn view of \n\\[  \\hat R(\\1,\\1) = P Q(\\1,\\1) (\\1 - P) = P(\\1 - P) = 0, \\]\nthis section vanishes in the base point, hence on all of \n$M \\times \\oline M$ (Lemma~\\ref{lem:2.7}). We conclude that \n\\[ 0 = \\hat R(g_1, g_2) \n= P  p_V \\pi(g_1^{-1}g_2) p_V (\\1-P) \n= P \\pi(g_1^{-1}g_2) (\\1 - P), \\] \nso that, for every $g \\in G$, we have \n$P \\pi(g)(\\1-P) = 0.$ \nThis leads to $P  \\cH_{V_2} = \\{0\\}$, and hence to \n$\\cH_{V_1} \\bot \\cH_{V_2}$. We derive that \n$\\cH_V = \\cH_{V_1} \\oplus \\cH_{V_2}$ is an orthogonal \ndirect sum. Therefore the \northogonal projection $\\tilde P \\in B_G(\\cH_V)$ onto $\\cH_{V_1}$  \nleaves $V$ invariant and satisfies \n$\\tilde P\\res_V = P$. This proves that $R$ is surjective. \n\\end{prf}\n\n\\begin{rem} The preceding proof shows that, under the assumptions of \nTheorem~\\ref{thm:5.5}, the range of the injective map \n\\[ \\ev \\: \\Gamma({\\mathbb L})^G \\to B_H(V), \\quad R \\mapsto \\hat R(\\1,\\1)\\] \ncontains $B_{H,\\fq}(V)$. If $\\beta(\\fq) = \\beta(\\fh_\\C)$, then \n$B_{H,\\fq}(V) = B_H(V)$, so that we obtain a linear isomorphism \n$\\Gamma({\\mathbb L})^G\\cong B_H(V)$. \n\\end{rem}\n\nThe preceding theorem has quite remarkable \nconsequences because it implies that the representations \n$(\\pi, \\cH_V)$ and $(\\rho,V)$ decompose in the same way. \n\n\\begin{cor} \\mlabel{cor:commutant} \nSuppose that $(\\pi, \\cH_V)$ is holomorphically induced from \n$(\\rho,\\beta)$, that $V$ is $B_G(V)$-invariant,  \nand that $\\beta(\\fq) = \\beta(\\fh_\\C)$, so that \n$B_H(V) = B_{H,\\fq}(V)$. \nThen the $G$-representation $(\\pi, \\cH_V)$ \nhas any of the following properties if and only if \nthe $H$-representation $(\\rho,V)$ does. \n\\begin{description}\n\\item[\\rm(i)] Irreducibility. \n\\item[\\rm(ii)] Multiplicity freeness. \n\\item[\\rm(iii)]  Type $I$, $II$ or $III$. \n\\item[\\rm(iv)] Discreteness, i.e., being a direct \nsum of irreducible representations. \n\\end{description}\n\\end{cor}\n\n\\begin{prf} (i) follows from the fact that, according to Schur's Lemma, \nirreducibility means that the commutant equals $\\C \\1$. \n  \n(ii) is clear because multiplicity freeness means that the commutant \nis commutative. \n\n(iii) is clear because the type of a representation is defined as \nthe type of its commutant as a von Neumann algebra. \n\n(iv) That a unitary representation decomposes discretely \nmeans that its commutant is an $\\ell^\\infty$-direct sum of \nfactors of type $I$. Hence the $G$ representation on $\\cH_V$ has\nthis property if only if the $H$-representation on $V$~does.\n\\end{prf}\n\nCorollary~\\ref{cor:commutant}(i) is a version of \nS.~Kobayashi's Theorem in the Banach context \n(cf.\\ \\cite{Ko68}). \n\n\\begin{prob} Theorem~\\ref{thm:5.5} should \nalso be useful to derive direct integral decompositions \nof the $G$-representation on $\\cH_V$ from direct integral \ndecompositions of the $H$-representation on $V$. \n\nSuppose that the bounded unitary representation \n$(\\rho, V)$ of $H$ is of type I and holomorphically inducible. \nThen it is a direct integral of irreducible representations \n$(\\rho_j, V_j)$. Are these irreducible representations of \n$H$ also inducible?   \n\\end{prob}\n\n\n\\begin{cor} \\mlabel{cor:2.15} Suppose that the $G$-representations \n$(\\pi_1, \\cH_{V_1})$, resp., $(\\pi_2, \\cH_{V_2})$ \nare holomorphically induced from the $(H,\\fq)$-representations \n$(\\rho_1, \\beta_1, V_1)$, resp., $(\\rho_2, \\beta_2, V_2)$. \nThen any unitary isomorphism \n$\\gamma \\: V_1 \\to V_2$ of $(H,\\fq)$-modules extends uniquely to a \nunitary equivalence $\\tilde\\gamma \\: \\cH_{V_1} \\to \\cH_{V_2}$. \n\\end{cor}\n\n\\begin{prf} Since $\\gamma$ is $(H,\\fq)$-equivariant, we have a \nwell-defined $G$-equivariant bijection \n\\[ \\tilde \\gamma \\: \\Gamma(\\bV_1) \\cong \nC^\\infty(G,V_1)_{\\rho_1, \\beta_1} \\to \nC^\\infty(G,V_2)_{\\rho_2, \\beta_2}, \\quad \nf \\mapsto \\gamma \\circ f \\] \nobtained from a corresponding isomorphism \n$[(g,v)] \\mapsto [(g,\\gamma(v))]$ of holomorphic $G$-bundles. \nTherefore $\\tilde\\gamma(\\cH_{V_1})$ is an invariant Hilbert space \nwith continuous point evaluations. The corresponding \n$G$-invariant kernel $\\hat Q \\in C^\\infty(G \\times G, B(V_2))$ \nsatisfies \n\\[ \\hat Q(\\1,\\1) = \\gamma \\circ \\id_{V_1} \\circ \\gamma^* \n= \\gamma \\gamma^* = \\id_{V_2}.\\] \nThis implies that $\\tilde\\gamma(\\cH_{V_1}) \n= \\cH_{V_2}$ and that the map \n$\\tilde \\gamma \\: \\cH_{V_1} \\to \\cH_{V_2}$ is unitary \nbecause both spaces have the same reproducing kernels \n(cf.\\ Definition~\\ref{def:2.10}). \n\nFor $v \\in V_1$, we further note that \n$(\\tilde\\gamma \\ev_\\1^*v)(\\1) \n= \\gamma \\ev_\\1 \\ev_\\1^*v = \\gamma v$, so that \n$\\tilde\\gamma$ extends $\\gamma$, when considered as a map \non the subspace $V_1 \\cong \\ev_\\1^*V_1 \\subeq \\cH_1$. \n\\end{prf}\n\n\n\\subsection{Realizing unitary representations by holomorphic sections} \n\nWe conclude this section with a result that helps to realize \ncertain subrepresentations of unitary representations in \nspaces of holomorphic sections. We continue in the setting \nof Section~\\ref{sec:1}, where $M = G\/H$ is a Banach homogeneous \nspace with a complex structure defined by the subalgebra \n$\\fq \\subeq \\g_\\C$. \n\nFrom the discussion in Remark~\\ref{rem:2.13}, we know that \nevery holomorphically induced representation \nsatisfies the assumptions (A1\/2) in the theorem below, \nwhich is our main tool to prove that a given unitary representation \nis holomorphically induced. \n\n\\begin{thm}\\mlabel{thm:a.3} \nLet $(\\pi, \\cH)$ be a continuous unitary representation of $G$ \nand $V \\subeq \\cH$ be a closed subspace satisfying the \nfollowing conditions: \n\\begin{description}\n\\item[\\rm(A1)] $V$ is $H$-invariant and the representation \n$\\rho$ of $H$ on $V$ is bounded. In particular,  \n$\\dd\\pi \\res_{\\fh}\\: \\fh \\to \\gl(V)$ \ndefines a continuous homomorphism \nof Banach--Lie algebras. \n\\item[\\rm(A2)] The exists a subspace $\\cD_V \\subeq V \\cap \\cH^\\infty$ dense \nin $V$ which is invariant under $\\dd\\pi(\\oline\\fq)$, \nthe operators $\\dd\\pi(\\oline\\fq)\\res_{\\cD_V}$ are bounded, \n and the so obtained representation of \n$\\oline\\fq$ on $V$ defines a continuous morphism \nof Banach--Lie algebras \n$$\\beta \\: \\fq \\to \\gl(V), \\quad x \\mapsto \n-(\\dd\\pi(\\oline x)\\res_V)^*.$$ \n\\end{description}\nThen the following assertions hold: \n\\begin{description}\n\\item[\\rm(i)] $\\beta$ is an extension of $\\rho$ defining on \n$\\bV := G \\times_H V$ the structure of a complex \nHilbert bundle. \n\\item[\\rm(ii)] If $p_V \\: \\cH \\to \\cH$ denotes \nthe orthogonal projection to~$V$, then \n$$ \\Phi \\: \\cH \\to C(G,V)_\\rho, \\quad \\Phi(v)(g) := p_V(\\pi(g)^{-1}v) $$ \nmaps $\\cH$ into $C^\\infty(G,V)_{\\rho,\\beta} \\cong \\Gamma(\\bV)$, \nand we thus obtain a $G$-equivariant unitary isomorphism \nof the closed subspace \n$\\cH_V := \\oline{\\Spann \\pi(G)V}$ \nwith the representation holomorphically induced from $(\\rho, \\beta)$. \n\\item[\\rm(iii)] $V \\subeq \\cH^\\omega$. \n\\end{description}\n\\end{thm}\n\n\\begin{prf} (i) For $x \\in \\fh$, we have \n$\\beta(x) = -\\dd\\rho(x)^*\\ = \\dd\\rho(x),$ \nand it is also easy to see that \n$\\beta(\\Ad(h)x) = \\pi(h) \\beta(x)\\pi(h)^{-1}$ for \n$h \\in H$ and $x \\in \\fq$. Therefore $\\beta$ is an extension \nof $\\rho$, and we can use \nTheorem~\\ref{thm:a.2} to see that \n$\\beta$ defines the structure of a holomorphic Hilbert \nbundle on $\\bV$. \n\n(ii) Clearly, $\\Phi(\\cH^\\infty) \\subeq C^\\infty(G,V)_\\rho$. \nFor $v \\in \\cD_V$, $w \\in \\cH^\\infty$ and $x \\in \\fq$, we further \nderive from (A2) that \n\\begin{align*}\n\\la p_V(\\dd\\pi(x)w), v \\ra \n&= \\la \\dd\\pi(x)w, v \\ra = \\la w, \\dd\\pi(-\\oline x)v \\ra \n= \\la p_V(w), \\dd\\pi(-\\oline x)v \\ra \n= \\la \\beta(x)p_V(w),v \\ra, \n\\end{align*}\nso that the density of $\\cD_V$ in $V$ implies \n$$ p_V \\circ \\dd\\pi(x) =  \\beta(x) \\circ p_V\\res_{\\cH^\\infty} \n\\quad \\mbox{ for } \n\\quad x \\in \\fq. $$\nFor $v \\in \\cH^\\infty$ and $x \\in \\fq$, we now obtain \n\\begin{equation}\n  \\label{eq:lx-rel}\n\\big(L_x \\Phi(v)\\big)(g) \n= -p_V(\\dd\\pi(x)\\pi(g)^{-1}v) \n= -\\beta(x)p_V(\\pi(g)^{-1}v)\n= -\\beta(x)\\Phi(v)(g). \n\\end{equation}\nThis means that \n$\\Phi(\\cH^\\infty) \\subeq C^\\infty(G,V)_{\\rho,\\beta}$. \nWriting $\\Gamma_c(\\bV)$ for the space of continuous sections of \n$\\bV$, we also obtain a map \n$\\tilde\\Phi \\: \\cH \\to \\Gamma_c(\\bV)$ which is continuous if \n$\\Gamma_c(\\bV)$ is endowed with the compact open topology. \nAs $\\Gamma(\\bV)$ is closed in $\\Gamma_c(\\bV)$ with respect to this topology \n(\\cite[Cor.~III.12]{Ne01}), \n$\\tilde\\Phi(\\oline{\\cH^\\infty}) \\subeq \\Gamma(\\bV)$, \nresp.,  $\\Phi(\\oline{\\cH^\\infty}) \\subeq C^\\infty(G,V)_{\\rho,\\beta}$. \n\nClearly $\\Phi(v) = 0$ is equivalent to \n$v \\bot \\pi(G)V$, so that $(\\ker \\Phi)^\\bot = \\cH_V$. \nSince (A2) implies that $V \\subeq \\oline{\\cH^\\infty}$, the same \nholds for $\\Spann(\\pi(G)V)$. This shows that \n\\[ \\Phi(\\cH) = \\Phi(\\cH_V) \n= \\Phi(\\oline{\\cH^\\infty}) \\subeq C^\\infty(G,V)_{\\rho,\\beta}.\\] \nThe corresponding kernel $\\hat Q \\: G \\times G \\to B(V)$ is given by \n\\[ \\hat Q(g_1, g_2) = \\ev_{g_1} \\ev_{g_2}^* \n= p_V \\pi(g_1)^{-1} (p_V \\circ \\pi(g_2)^{-1})^* \n= p_V \\pi(g_1^{-1}g_2) p_V,\\] \nso that we have in particular $\\hat Q(\\1,\\1) = \\id_V$. \n\n(iii) To see that $V$ consists of analytic vectors, \nwe may w.l.o.g.\\ assume that $\\cH = \\cH_V$ and hence that \n$\\cH \\subeq \\Gamma(\\bV)$ is holomorphically induced and that \n$\\Phi = \\id_\\cH$. \nLet $\\ev_{\\1 H} \\: \\cH \\to \\bV_{\\1 H} \\cong V$ be the evaluation map. \nThe corresponding map \n\\[ \\ev_\\1 \\: C^\\infty(G,V)_{\\rho,\\beta} \\to V \\] \nis simply given by evaluation in $\\1 \\in G$. \nNow $\\ev_\\1\\res_V \\: V \\to V$ is the identity, so that \n$\\ev_\\1^* \\: V\\to \\cH$ is simply the isometric inclusion. Hence \nthe analyticity of $v = \\ev_\\1^*v = \\ev_{\\1 H}^*v$ \nfollows from Lemma~\\ref{lem:2.2}. \n\\end{prf} \n\n\\begin{rem} \\mlabel{rem:2.14} \nSuppose that there exist subalgebras $\\fp^\\pm \\subeq \\g_\\C$ with \n$\\fq = \\fh_\\C \\rtimes \\fp^+$ and $\\Ad(H)\\fp^\\pm = \\fp^\\pm$. \nThen, for every unitary representation \n$(\\pi, \\cH)$ of $G$, the closed subspace \n$V := \\oline{(\\cH^\\infty)^{\\fp^-}}$ \nis invariant under $H$ and $B_G(\\cH)$. \nIf the representation $(\\rho, H)$ on $V$ is bounded, \nthen the dense subspace $\\cD_V := (\\cH^\\infty)^{\\fp^-}$ \nsatisfies (A2) if we put $\\beta(\\fp^+) = \\{0\\}$. \nTheorem~\\ref{thm:a.3} now implies that \n$V \\subeq \\cH^\\omega$, so that we see in particular that \n$V = (\\cH^\\infty)^{\\fp^-}$ \nis a closed subspace of $\\cH$. \nMoreover, Corollary~\\ref{cor:commutant} applies. \n\\end{rem}\n\n\nThe following remark sheds some extra light on condition (A2). \n\n\\begin{rem}\nLet $\\cH \\subeq \\Gamma(\\bV)$ be a $G$-invariant Hilbert subspace \nwith continuous point evaluations, $m \\in M$ and \n$V := \\oline{\\im(\\ev_{m}^*)} \\subeq \\cH$. \nThen $V$ is a $G_m$-invariant closed subspace of $\\cH$ and \n\\[ V^\\bot = \\im(\\ev_m^*)^\\bot = \\ker(\\ev_m) \n= \\{ s \\in \\cH \\: s(m) =0\\}. \\] \n\nThe action of $G$ on $\\bV$ by holomorphic bundle automorphisms \nleads to a homomorphism $\\dot\\sigma_{\\bV} \\: \\g_\\C \\to \\cV(\\bV)$ \nand each $x \\in \\fq_m$ thus leads to a linear vector field \n$-\\beta_m(x)$ on the fiber $\\bV_m$. Passing to derivatives in the formula \n$(g.s)(m) := g.s(g^{-1}m)$, we obtain for $x \\in \\fq_m$ \n\\[ (x.s)(m) = -\\beta_m(x)\\cdot s(m).\\] \nIn particular, $V^\\bot \\cap \\cH^\\infty$ is invariant under \nthe derived action of $\\fq_m$, so that one can expect that the adjoint \noperators coming from $\\oline{\\fq_m}$ act on $\\bV_m$\n\nAs we have seen in Lemma~\\ref{lem:2.2}(b), the \nsubspace $\\im(\\ev_m^*)$ of $\\cH$ consists of smooth vectors, so that \n$V \\cap \\cH^\\infty$ is dense in $V$. \n\\end{rem}\n\n\n\\begin{ex} \\mlabel{ex:grass} \nWe consider the identical representation of \n$G = \\U(\\cH)$ on the complex Hilbert space $\\cH$. \nLet $\\cK$ be a closed subspace of $\\cH$. Then \nthe subgroup $Q := \\{ g \\in \\GL(\\cH) \\: g\\cK = \\cK\\}$ \nis a complex Lie subgroup of $\\GL(\\cH)$ and the Gra\\ss{}mannian \n$\\Gr_\\cK(\\cH) := \\GL(\\cH)\\cK \\cong \\GL(\\cH)\/Q$ carries the \nstructure of a complex homogeneous space on which the unitary \ngroup $G = \\U(\\cH)$ acts transitively and which is isomorphic \nto $G\/H$ for $H := \\U(\\cH)_\\cK \\cong \\U(\\cK) \\oplus \\U(\\cK^\\bot)$. \n\nWriting elements of $B(\\cH)$ as $(2 \\times 2)$-matrices according to \nthe decomposition $\\cH = \\cK \\oplus \\cK^\\bot$, we have \n\\[ \\fq = \\Big\\{ \\pmat{ a & b \\\\ 0 & d} \\: a \\in B(\\cK), b \\in B(\\cK^\\bot, \\cK), \nd \\in B(\\cK^\\bot)\\Big\\},\\] \nand $\\gl(\\cH) = \\fq \\oplus \\fp^-$ holds for \n$\\fp^- = \\Big\\{ \\pmat{ 0 & 0 \\\\ c & 0} \\: c \\in B(\\cK, \\cK^\\bot)\\Big\\}.$ \n\nThe representation of $\\U(\\cH)$ on $\\cH$ is bounded with \n$V := \\cH^{\\fp^-} = \\cK^\\bot$, and the representation of \n$H \\cong \\U(\\cK) \\oplus \\U(\\cK^\\bot)$ on this space is bounded. \nIn view of Theorem~\\ref{thm:a.3}, the \ncanonical extension $\\beta \\: \\fq \\to \\gl(V), \\beta(x) := (x^*\\res_V)^*$ \nnow leads to a holomorphic vector bundle \n$\\bV := \\GL(\\cH) \\times_Q V \\cong G \\times_H V$ and a $G$-equivariant \nrealization $\\cH \\into \\Gamma(\\bV)$. \n\nIn this sense every Hilbert space can be realized as a space of \nholomorphic sections of a holomorphic vector bundle over any Gra\\ss{}mannian \nassociated to~$\\cH$. Note that $\\U(\\cH)_\\cK = \\U(\\cH)_V$ shows that \n$\\Gr_\\cK(\\cH) \\cong G\/H$ can be identified in a natural way with\n$\\Gr_V(\\cH)$. \n\\end{ex} \n\n\\begin{rem} Let $(\\pi, \\cH)$ be a smooth unitary representation \nof the Lie group $G$ and $V \\subeq \\cH$ be a closed $H$-invariant subspace. \nWe then obtain a natural $G$-equivariant map \n$\\eta \\: G\/H \\to \\Gr_V(\\cH), gH \\mapsto \\pi(g)V$. \nIf this map is holomorphic, then we can pull back the natural \nbundle $\\bV \\to \\Gr_V(\\cH)$ from Example~\\ref{ex:grass} and \nobtain a realization of $\\cH$ in $\\Gamma(\\eta^*\\bV)$. This works \nvery well if the representation \n$(\\pi, \\cH)$ is bounded because in this case $\\pi \\: G \\to \\U(\\cH)$\nis a morphism of Banach--Lie groups, but if \n$\\pi$ is unbounded, then it seems difficult to verify \nthat $\\eta$ is smooth, resp., holomorphic. \n\nIf (A1\/2) in Theorem~\\ref{thm:a.3} are satisfied, then \n$V \\subeq \\cH^\\omega \\subeq \\cH^\\infty$ implies that, \nthe operators $\\dd\\pi(x)$, $x \\in \\g$, are defined on $V$. \nSince they are closable, the graph of these restrictions is closed, \nwhich implies that the restrictions $\\dd\\pi(x)\\res_V \\: V \\to \\cH$ \nare continuous linear operators. We thus obtain a \nnatural candidate for a tangent map \n\\[ T_H(\\eta) \\: T_H(G\/H) \\cong \\g\/\\fh \\to \nT_V(\\Gr_V(\\cH)) \\cong B(V,V^\\bot), \\quad \nx \\mapsto (\\1-p_V)\\dd\\pi(x) p_V.\\] \n\nFor the special case where $\\dim V = 1$, we have \n$V= \\C v_0$ for a smooth vector $v_0$, and since the projective orbit map \n$G \\to \\bP(\\cH) \\cong \\Gr_V(\\cH), g \\mapsto [\\pi(g)v_0]$ is smooth, \nthe induced map $\\eta \\:  G\/H \\to \\bP(\\cH)$ is smooth as well.\nThis construction is the key idea \nbehind the theory of coherent state representations \n(\\cite{Od88}, \\cite{Od92}, \\cite{Li91}, \\cite{Ne00}, \\cite{Ne01}), \nwhere one uses a holomorphic map $\\eta \\: G\/H \\to \\bP(\\cH^*)$ of a \ncomplex homogeneous space $G\/H$ to realize a unitary representation \n$(\\pi, \\cH)$ of $G$ in the space of holomorphic sections \nof the line bundle $\\eta^*{\\mathbb L}$, where \n${\\mathbb L} \\to \\bP(\\cH^*)$ is the canonical bundle on the dual projective\nspace with $\\Gamma({\\mathbb L}) \\cong \\cH$. \n\\end{rem}\n\n\n\\section{Realizing positive energy representations}  \\mlabel{sec:3}\n\nIn this section we fix an element $d \\in \\g = \\L(G)$ for which \nthe one-parameter group $e^{\\R \\ad d} \\subeq \\Aut(\\g)$ is bounded, \ni.e., preserves an equivalent norm. We call such elements \n{\\it elliptic}. Then $H := Z_G(d)$ is a Lie subgroup and if \n$0$ is isolated in $\\Spec(D)$, then $G\/H$ carries a natural complex \nstructure. The class of representations which one may expect \nto be realized by holomorphic sections of Hilbert bundles \n$\\bV$ over $G\/H$ is the class of {\\it positive energy representations}, \nwhich is defined by the condition that the selfadjoint operator \n$-i\\dd\\pi(d)$ is bounded below. \n\n\\subsection{The splitting condition} \n\nLet $d \\in \\g$ be an elliptic element. Then \n\\[ H = Z_G(\\exp \\R d) = Z_G(d) \n= \\{  g \\in G \\: \\Ad(g)d = d \\} \\] \nis a closed subgroup of $G$, not necessarily connected, \n with Lie algebra $\\fh = \\z_\\g(d) = \\ker(\\ad d)$. \nSince $\\g$ contains arbitrarily small $e^{\\R \\ad d}$-invariant \n$0$-neighborhoods $U$, there exists such an open $0$-neighborhood \nwith $\\exp_G(U) \\cap H = \\exp_G(U \\cap \\L(H)).$ \nTherefore $H$ is a Lie subgroup of $G$, i.e., a Banach--Lie group \nfor which the inclusion $H \\into G$ is a topological embedding. \n\nOur assumption implies that $\\alpha_t := e^{t\\ad d}$ defines an \nequicontinuous one-paramter group of automorphisms of the \ncomplex Banach--Lie algebra $\\g_\\C$. For \n$\\delta > 0$, we consider the Arveson spectral subspace \n$\\fp^+ := \\g_\\C([\\delta,\\infty[)$ \n(cf.\\ Definition~\\ref{def:arv}). \nApplying Proposition~\\ref{prop:spec-add} to the Lie bracket \n$\\g_\\C \\times \\g_\\C \\to \\g_\\C$, we see that \n$\\fp^+$ is a closed complex subalgebra. \nFor $f \\in L^1(\\R)$, $\\alpha(f) := \\int_\\R f(t) \\alpha_t\\, dt$ \nand $x \\in \\g_\\C$, \nthe relations $\\oline{\\alpha(f)x} = \\alpha(\\oline f)\\oline x$ and \n$\\hat{\\oline f}(\\xi) = \\oline{\\hat f(-\\xi)}$ imply that \n$\\fp^- := \\oline{\\fp^+} = \\g_\\C(]-\\infty, -\\delta])$. \nTo make the following constructions work, we assume the \n{\\it splitting condition:} \n\\begin{equation}\n  \\label{eq:splitcond}\n\\g_\\C = \\fp^+ \\oplus \\h_\\C \\oplus \\fp^-.\\tag{SC}\n\\end{equation}\nIn view of Lemma~\\ref{lem:a.17}, it is satisfied for some \n$\\delta > 0$ if and only if $0$ is isolated in $\\Spec(\\ad d)$. \n\nSince $\\Ad(H)$ commutes with $e^{\\R \\ad d}$, \nthe closed subalgebras $\\fp^\\pm \\subeq \\g_\\C$ are invariant \nunder $\\Ad(H)$ and $e^{\\R \\ad d}$. \nNow $\\g \\cap (\\fp^+ \\oplus \\fp^-)$ is a closed complement for \n$\\fh$ in $\\g$, so that $M := G\/H$ carries the structure of a \nBanach homogeneous space and \n$\\fq := \\fh_\\C + \\fp^+ \\cong \\fp^+ \\rtimes \\fh_\\C$ \ndefines a $G$-invariant complex manifold \nstructure on $M$ (cf.\\ Section~\\ref{sec:1}). \n\n\\begin{rem} (a) For bounded derivations of compact \n$L^*$-algebras similar splitting conditions have been used \nby Belti\\c{t}\\u{a} in \\cite{Bel03} to obtain K\\\"ahler polarizations \nof coadjoint orbits. In \\cite{Bel04} this is extended to \nbounded normal derivations of a complex Banach Lie algebra. \n\n(b) If $\\g$ is a real Hilbert--Lie algebra, then one can use\n spectral measures to obtain natural \ncomplex structures on $G\/H$ even if the splitting condition is \nnot satisfied, i.e., $0$ need not be isolated in the spectrum \nof $\\ad d$ (\\cite[Prop.~5.4]{BRT07}). \n\\end{rem}\n\n\n\\begin{ex} If $\\alpha$ factors through an action of \nthe circle group $\\T = \\R\/2\\pi\\Z$, then the Peter--Weyl Theorem \nimplies that the sum $\\sum_{n \\in \\Z} \\g_\\C^n$ of the corresponding \neigenspaces $\\g_\\C^n := \\ker(\\ad d - in \\1)$ \nis dense in $\\g_\\C$ with $\\fh_\\C = \\g_\\C^0$. Since the operator \n$\\ad d$ is bounded, only finitely many $\\g_\\C^n$ are non-zero, so that \nwe actually have $\\g_\\C = \\sum_{n \\in \\Z} \\g_\\C^n$. \nFrom $[\\g_\\C^n, \\g_\\C^m] \\subeq \\g_\\C^{n+m}$ it follows \nthat $\\fp^\\pm := {\\sum_{\\pm n > 0} \\g_\\C^n}$ are closed subalgebras \nfor which $\\g_\\C = \\fp^+  \\oplus \\fh_\\C \\oplus \\fp^-$ is direct. \nIn this case the splitting condition is always satisfied \nand $\\Spec(D) \\subeq i\\Z$. \n\nIf, conversely, $d \\in \\g$ is an element for which the complex \nlinear extension of $\\ad d$ to $\\g_\\C$ is diagonalizable \nwith finitely many eigenvalues in $i\\Z$, then $e^{\\R \\ad d} \n\\subeq \\Aut(\\g_\\C)$ is compact, hence preserves a compatible norm. \nAn important special situation, where we have all this \nstructure are hermitian Lie groups (cf.\\ \\cite{Ne11}). \nIn this case we simply have $\\fp^\\pm = \\g_\\C^{\\pm 1}$. \n\\end{ex}\n\n\n\\subsection{Positive energy representations} \n\n\\begin{lem} \\mlabel{lem:c.1} \nLet $\\gamma \\: \\R \\to \\U(\\cH)$ be a strongly continuous \nunitary representation and $A = A^* = -i\\gamma'(0)$ be its selfadjoint \ngenerator, so that $\\gamma(t) = e^{itA}$ in terms of measurable functional \ncalculus. Then the following assertions hold: \n\\begin{description}\n\\item[\\rm(i)]  For each $f \\in L^1(\\R)$, we have \n$\\gamma(f) = \\hat f(A),$\nwhere $\\hat f(x) := \\int_\\R e^{ixy} f(y)\\, dy$ is the Fourier transform \nof $f$. \n\\item[\\rm(ii)]  Let $P \\: {\\mathfrak B}(\\R) \\to B(\\cH)$ be the unique \nspectral measure with $A = P(\\id_\\R)$. \nThen, for every closed subset $E \\subeq \\R$, the \nrange $P(E)\\cH$ coincides with the Arveson spectral subspace \n$\\cH(E)$. \n\\end{description}\n\\end{lem}\n\n\\begin{prf} Since the unitary representation $(\\gamma,\\cH)$ is a direct sum \nof cyclic representation, it suffices to prove the assertions for \ncyclic representations. Every cyclic representation of \n$\\R$ is equivalent to the representation on \nsome space $\\cH = L^2(\\R,\\mu)$, where $\\mu$ is a Borel probability \nmeasure on $\\R$ and $(\\gamma(t)\\xi)(x) = e^{itx}\\xi(x)$ \n(see \\cite[Thm.~VI.1.11]{Ne00}). \n\n(i) We have $(A\\xi)(x) = x\\xi(x)$, so that \n$(\\hat f(A)\\xi)(x) = \\hat f(x)\\xi(x)$. On the other hand, we have for \n$f \\in L^1(\\R)$ in the space $\\cH = L^2(\\R,\\mu)$ the relation \n\\[ (\\gamma(f)\\xi)(x) \n= \\int_\\R f(t) e^{itx}\\xi(x)\\, dt = \\hat f(x)\\xi(x). \\] \n\n(ii) see \\cite[p.~226]{Ar74}. \n\\end{prf}\n\n\n\\begin{prop} \\mlabel{prop:c.3} \nLet $(\\pi, \\cH)$ be a smooth unitary representation \nof the Banach--Lie group $G$, $d \\in \\g$ be elliptic, \nand $P \\: {\\mathfrak B}(\\R) \\to \\cL(\\cH)$ be the spectral measure \nof the unitary one-parameter group \n$\\pi_d(t) := \\pi(\\exp_G td)$. \nThen the following assertions hold: \n\\begin{description}\n\\item[\\rm(i)] $\\cH^\\infty$ carries a Fr\\'echet structure for which \n$\\pi_d(t)_{t \\in \\R}$ defines a continuous equicontinuous action of \n$\\R$ on $\\cH^\\infty$. In particular, $\\cH^\\infty$ is invariant under \n$\\pi_d(f)$ for every $f \\in L^1(\\R)$. \n\\item[\\rm(ii)] For every closed subset $E \\subeq \\R$, we have \n$\\cH^\\infty(E) = (P(E)\\cH) \\cap \\cH^\\infty$ for the corresponding \nspectral subspace. \n\\item[\\rm(iii)]  For every open subset $E \\subeq \\R$, \n$(P(E) \\cH) \\cap \\cH^\\infty$ is dense in $P(E)\\cH^\\infty$. \nMore precisely, there exists a sequence $(f_n)_{n \\in \\N}$ \nin $L^1(\\R)$ for which $\\pi_d(f_n) \\to P(E)$ in the \nstrong operator topology and $\\supp(\\hat f_n) \\subeq E$, so that \n$\\pi_d(f_n)v \\in \\cH^\\infty \\cap P(E)\\cH^\\infty$ for every $v \\in \\cH^\\infty$. \n\\item[\\rm(iv)] \nFor closed subsets $E, F \\subeq \\R$, \nthe Arveson spectral subspaces $\\g_\\C(F)$ satisfies \n\\begin{equation}\n  \\label{eq:shift}\n\\dd\\pi(\\g_\\C(F))\\big(\\cH^\\infty \\cap P(E)\\cH\\big) \\subeq P(\\oline{E+ F})\\cH. \n\\end{equation}\n\\end{description}\n\\end{prop}  \n\n\\begin{prf} (i) We may w.l.o.g.\\ assume that the norm on \n$\\g$ is invariant under $e^{\\R \\ad d}$. \nOn $\\cH^\\infty$ we consider the Fr\\'echet topology \ndefined by the seminorms \n$$ p_n(v) := \\sup \\{ \\|\\dd\\pi(x_1)\\cdots \\dd\\pi(x_n)v\\| \\: \nx_i \\in \\g, \\|x_i\\| \\leq 1\\} $$\nwith respect to which the action of $G$ on $\\cH^\\infty$ is smooth  \n(cf.~\\cite[Thm.~4.4]{Ne10a}). In particular, the bilinear map \n\\begin{equation}\n  \\label{eq:applic}\n\\g_\\C \\times \\cH^\\infty \\to \\cH^\\infty, \\quad \n(x,v) \\mapsto \\dd\\pi(x) v\n\\end{equation}\nis continuous because it can be obtained as a \nrestriction of the tangent map of the $G$-action. \n \nIn view of the relation \n$\\pi_d(t) \\dd\\pi(x) \\pi_d(t)^{-1} = \\dd\\pi(e^{t \\ad d}x)$ \nfor $t \\in \\R, x \\in \\g,$ \nthe isometry of $e^{t\\ad d}$ on $\\g$ implies that \nthe seminorms $p_n$ on $\\cH^\\infty$ \nare invariant under $\\pi_d(\\R)$. Since the $\\R$-action \non $\\cH^\\infty$ defined by the operators $\\pi_d(t)$ is smooth, \nhence in particular continuous, we obtain with Definition~\\ref{def:arv} \nan algebra homomorphism \n$$ \\pi_d \\: (L^1(\\R), *) \\to \\End(\\cH^\\infty), \\quad \nf \\mapsto \\int_\\R f(t)\\pi_d(t)\\, dt $$  \n(cf.\\ Definition~\\ref{def:arv} below), and this implies (i). \n\n(ii) Since $\\cH^\\infty(E) = \\cH(E) \\cap \\cH^\\infty$ follows immediately \nfrom the definition of spectral subspaces (Remark~\\ref{rem:a.6}), \nthis assertion is a consequence of Lemma~\\ref{lem:c.1}(ii). \n\n(iii) We write the open set $E$ as an increasing \nunion of compact subsets $E_n$ \nand observe that $\\bigcup_n P(E_n) \\cH$ is dense in $P(E)\\cH$.  \nFor every $n$, there exists a compactly supported function \n$h_n \\in C^\\infty_c(\\mathbb R,\\mathbb R)$ such that \n\\[ \\supp(h_n) \\subseteq E, \\quad 0 \\leq h_n \\leq 1, \\quad \n\\mbox{ and } \\quad h_n\\big|_{E_n} = 1.\\] \n Let $f_n \\in \\cS(\\mathbb R)$ with \n$\\hat f_n = h_n$. Then \n$\\pi_d(f_n) = \\hat f_n(-i\\gamma'(0)) = h_n(-i\\gamma'(0))$ \n(Lemma~\\ref{lem:c.1}(i)) and consequently\n\\[ P(E_n)\\cH \\subseteq \\pi_d(f_n)\\cH \\subseteq \nP(E)\\cH. \\]\nTherefore the subspace \n$\\pi_d(f_n)\\cH^\\infty$ of $\\cH^\\infty$ is contained \nin $P(E)\\cH$. \nIf $w = P(E)v$ for some $v \\in \\cH^\\infty$\nthen \n\\[ \\pi_d(f_n)w  = \\pi_d(f_n)P(E)v = \\pi_d(f_n)v\n\\in \\cH^\\infty\\] \nand \n\\[ \\|\\pi_d(f_n)w-w\\|^2 \n=  \\|h_n(-i\\pi_d'(0))w -w\\|^2 \\leq \\|P(E\\backslash E_n)w\\|^2 \\to 0\\] \nfrom which it follows that $\\pi_d(f_n)w \\to w$. \n\n(iv) This follows  from the continuity of \n\\eqref{eq:applic}, Proposition~\\ref{prop:spec-add} and (ii). \n\\end{prf}\n\n\nResults of a similar type as Proposition~\\ref{prop:c.3}(iv)  \nand the more universal Proposition~\\ref{prop:spec-add} in the \nappendix are well known in the context of bounded operators \n(cf.\\ \\cite{FV70}, \\cite{Ra85}, \\cite[Prop.~1.1, Cor.~1.2]{Bel04}). \nArveson also obtains variants for automorphism groups of operator algebras \n(\\cite[Thm.~2.3]{Ar74}). \n\n\\begin{rem} Combining Lemma~\\ref{lem:c.1}(i) with \nProposition~\\ref{prop:c.3}(i), we derive that \nthe subspace $\\cH^\\infty$ of $\\cH$ is invariant under all operators \n$P(\\hat f) = \\hat f(-i\\dd\\pi(d))$ for $f \\in L^1(\\R)$. This implies \nin particular to the operators $P(h)$, $h \\in \\cS(\\R)$, but not \nto the spectral projections $P(E)$.   If $E \\subeq \\R$ is open, \nProposition~\\ref{prop:c.3}(iii) provides a suitable approximate \ninvariance. \n\\end{rem}\n\nThe following proposition is of key importance for the following. \nIt contains the main consequences of Arveson's spectral theory  \nfor the actions on $\\g_\\C$ and $\\cH^\\infty$. \n\n\\begin{prop} \\mlabel{prop:6.2} If $d \\in \\g$ is elliptic with \n$0$ isolated in $\\Spec(\\ad d)$, then for any \nsmooth positive energy representation \n $(\\pi, \\cH)$ of $G$,  the $H$-invariant subspace \n$V := \\oline{(\\cH^\\infty)^{\\fp^-}}$ satisfies \n$\\cH = \\oline{\\Spann(\\pi(G)V)}$. \n\\end{prop} \n\n\\begin{prf} First we show that $V \\not=\\{0\\}$ whenever \n$\\cH\\not=\\{0\\}$. Let \n\\[ s := \\inf(\\Spec(-i\\dd\\pi(d))) > -\\infty . \\]\nFor some $\\eps \\in ]0,\\delta[$, we consider the closed subspace \n\\begin{equation}\n  \\label{eq:vdef}\nW := P([s,s+ \\eps[) \\cH = P(]s-\\eps, s+ \\eps[) \\cH, \n\\end{equation}\nwhere $P \\: {\\mathfrak B}(\\R) \\to B(\\cH)$ is the spectral measure of~$\\pi_d$. \nThen Proposition~\\ref{prop:c.3} implies that  \n$W^\\infty :=  W\\cap \\cH^\\infty$ is dense in $W$ \nand that \n\\[  \\dd\\pi(\\fp^-) W^\\infty \n\\subeq P(]-\\infty, s+ \\eps - \\delta])\\cH = \\{0\\},\\]\nwhich leads to $\\{0\\}\\not= W \\subeq V$. \n\nApplying the preceding argument to  the positive energy \nrepresentation on the orthogonal complement of \n$\\cH_V := \\oline{\\Spann \\pi(G)V}$, the relation \n$V \\cap \\cH_V^\\bot=\\{0\\}$ implies that \n$\\cH_V^\\bot=\\{0\\}$, and hence that $\\cH = \\cH_V$. \n\\end{prf}\n\n\\begin{thm} \\mlabel{thm:6.2} If $d \\in \\g$ is elliptic with \n$0$ isolated in $\\Spec(\\ad d)$ and $(\\pi,\\cH)$ \nis a smooth positive energy representation \nfor which the $H$-representation $\\rho(h) := \\pi(h)\\res_V$ \non $V := \\oline{(\\cH^\\infty)^{\\fp^-}}$  \nis bounded, then $(\\pi, \\cH)$ is holomorphically induced from \nthe representation $(\\rho,\\beta)$ of $(H,\\fq)$ on \n$V$ defined by $\\beta(\\fp^+) = \\{0\\}$. In particular, \n$V$ consists of analytic vectors. \n\\end{thm} \n\n\\begin{prf} Since $\\pi(G)V$ spans a dense subspace of $\\cH$, i.e., \n$\\cH = \\cH_V$ (Proposition~\\ref{prop:6.2}), \nthe assertion follows from Remark~\\ref{rem:2.14}. \n\\end{prf}\n\n\\begin{rem}\nSince $\\cH_V \\cong \\cH_{V'}$ as $G$-representations \nif and only if $V \\cong V'$ as $H$-representations \n(cf.\\ Corollary~\\ref{cor:2.15}), the description of \nall $G$-representations of  positive energy \nfor which the $H$-representation $(\\rho, V)$ is bounded  \nis equivalent to the determination of all bounded $H$-representations \n$(\\rho, V)$ for which $(\\rho, \\beta, V)$ is \ninducible if we put $\\beta(\\fp^+) = \\{0\\}$. \n\\end{rem}\n\n\\begin{cor} \\mlabel{cor:6.2} If \n$d \\in \\g$ is elliptic with $0$ isolated in $\\Spec(\\ad d)$, \nthen every bounded representation of $G$  is holomorphically induced from \nthe representation $(\\rho,\\beta)$ of $(H,\\fq)$ on \n$V := \\oline{(\\cH^\\infty)^{\\fp^-}}$ defined by $\\beta(\\fp^+) = \\{0\\}$. \n\\end{cor} \n\nFrom Corollary~\\ref{cor:commutant} we obtain in particular:\n\n\\begin{cor} A positive energy representation \n$(\\pi, \\cH)$ of $G$ for which the representation $(\\rho,V)$ of $H$ \nis bounded is a direct sum of irreducible ones if and only \nif $(\\rho, V)$ has this property. \n\\end{cor}\n\n\n\\begin{ex}  The complex Banach--Lie algebra \n$\\g$ is called {\\it weakly root graded} if there exists \na finite reduced root system $\\Delta$ such that $\\g$ \ncontains the corresponding \nfinite dimensional semisimple Lie algebra $\\g_\\Delta$ and for some \nCartan subalgebra $\\fh \\subeq \\g_\\Delta$, the Lie algebra $\\g$ is a direct \nsum of finitely many $\\ad \\fh$-eigenspaces. \n\nNow suppose that $\\g$ is a real Banach--Lie algebra for which \n$\\g_\\C$ is weakly root graded, that \n$\\fh$ is invariant under conjugation and, for every \n$x \\in \\fh \\cap i\\g$, the derivation $\\ad x$ has real spectrum.\nThen the realization results for bounded unitary representations \nof $G$ which follows from \n\\cite[Thm.~5.1]{MNS09}, applied to their holomorphic \nextensions $G_\\C \\to \\GL(\\cH)$, can be derived \neasily from Corollary~\\ref{cor:6.2}. \n\\end{ex}\n\nThe following theorem shows that, assuming that $(\\pi, \\cH)$ is semibounded \nwith $d \\in W_\\pi$ permits us to get rid of the quite implicit assumption \nthat the $H$-representation on $V$ is bounded. It is an important \ngeneralization of Corollary~\\ref{cor:6.2} to semibounded representations. \n\n\\begin{thm} \\mlabel{thm:6.2b} Let $(\\pi, \\cH)$ be a semibounded \nunitary representation of the Banach--Lie group $G$ and \n$d \\in W_\\pi$ be an elliptic element for which $0$ \nis isolated in $\\Spec(\\ad d)$. \nWe write $P \\: {\\mathfrak B}(\\R) \\to B(\\cH)$ for the spectral measure of the \nunitary one-parameter group $\\pi_d(t) := \\pi(\\exp(td)$. \nThen the following assertions hold: \n\\begin{description}\n\\item[\\rm(i)] The representation $\\pi\\res_H$ of $H$ is semibounded and, \nfor each bounded measurable subset $B \\subeq \\R$, the \n$H$-representation on $P(B)\\cH$ is bounded. \n\\item[\\rm(ii)] The representation $(\\pi, \\cH)$ is a direct sum of \nrepresentations $(\\pi_j, \\cH_j)$ for which there exist $H$-invariant \nsubspaces $\\cD_j \\subeq (\\cH_j^\\infty)^{\\fp^-}$ for \nwhich the $H$-representation $\\rho_j$ on $V_j := \\oline{\\cD_j}$ is bounded and \n$\\Spann\\big(\\pi_j(G)V_j\\big)$ is dense in $\\cH_j$. \nThen the representations $(\\pi_j, \\cH_j)$ are holomorphically \ninduced from $(\\rho_j, \\beta_j,V)$, where \n$\\beta_j(\\fp^+)= \\{0\\}$. \n\\item[\\rm(iii)] If $(\\pi, \\cH)$ is irreducible and \n$s := \\inf\\Spec(-i\\dd\\pi(d))$, then $P(\\{s\\})\\cH \n= \\oline{(\\cH^\\infty)^{\\fp^-}}$ and \n$(\\pi, \\cH)$ is holomorphically induced \nfrom the bounded $H$-representation $\\rho$ on this space, extended \nby $\\beta(\\fp^+) = \\{0\\}$. \n\\end{description}\n\\end{thm}\n\n\\begin{prf} (i) From  $s_{\\pi\\res_H} = s_\\pi\\res_{\\fh}$ it follows that \n$\\pi\\res_H$ is semibounded with $d \\in W_\\pi \\cap \\fh \\subeq W_{\\pi\\res_H}$. \nFor every bounded measurable subset $B \\subeq \\R$, the relation  \n$d \\in \\z(\\fh)$ entails that $P(B)\\cH$ is $H$-invariant. Let \n$\\rho_B$ denote the corresponding representation of $H$. \nThen the boundedness of $\\dd\\rho_B(d)$ which commutes with \n$\\dd\\rho(\\fh)$ implies that \n$W_{\\rho_B} + \\R d = W_{\\rho_B}$, so that $d \\in W_{\\pi\\res_H} \n\\subeq W_{\\rho_B}$ leads to $0 \\in W_{\\rho_B}$. \nThis means that $\\rho_B$ is bounded. \n\n(ii) We apply Zorn's Lemma to the ordered set of all \npairwise orthogonal systems of closed $G$-invariant subspaces \nsatisfying the required conditions. \nTherefore it suffices to \nshow that if $\\cH \\not=\\{0\\}$, then there exists a non-zero \n$H$-invariant subspaces $\\cD \\subeq (\\cH^\\infty)^{\\fp^-}$ for \nwhich the $H$-representation $\\rho$ on $\\oline{\\cD}$ is bounded. \n\nFor $s := \\inf\\Spec(-i\\dd\\pi(d))$ and $0 < \\eps < \\delta$, we \nmay take the space $\\cD := \\break \\cH^\\infty \\cap P([s,s+\\eps[)\\cH$ \nfrom the proof of Proposition~\\ref{prop:6.2}. \nAs we have seen there, it is annihilated \nby $\\dd\\pi(\\fp^-)$ and the boundedness of the $H$-representation on its \nclosure follows from~(i). \n\nThat the representations $(\\pi_j, \\cH_j)$ are holomorphically induced\n from the bounded $H$-representations on $V_j$ follows from \nTheorem~\\ref{thm:a.3}. \n\n(iii) For $t > s$, let \n\\[ \\cD_U \n:= (\\cH^\\infty)^{\\fp^-} \\cap P([s,t[)(\\cH^\\infty)^{\\fp^-}\n= (\\cH^\\infty)^{\\fp^-} \\cap P(]s-\\delta, t[)(\\cH^\\infty)^{\\fp^-}. \\] \n\n{\\bf Claim 1:} $\\cD_U$ is dense in \n$U := \\oline{P([s,t[)(\\cH^\\infty)^{\\fp^-}}$. \n\nLet $v \\in (\\cH^\\infty)^{\\fp^-}$ and $w := P(]s-\\delta,t[)v$. \nWith Proposition~\\ref{prop:c.3}(iii), we find a sequence $f_n \\in L^1(\\R)$ \nfor which $\\pi_d(f_n)$ converges strongly to $P(]s-\\delta,t[)$ \nand $\\supp(\\hat f_n) \\subeq ]s-\\delta, t[$, so that $\\pi_d(f_n)v  \\to w$ \nand \n\\[ \\pi_d(f_n)v = P(\\hat f_n)v \n= P(]s-\\delta, t[)P(\\hat f_n)v \n\\in P(]s-\\delta, t[)\\cH^\\infty.\\] \nSince $\\fp^-$ is invariant under $e^{\\ad d}$, the closed subspace \n$(\\cH^\\infty)^{\\fp^-}$ is invariant under $\\pi_d(\\R)$, \nso that $\\pi_d(f_n)v\\in (\\cH^\\infty)^{\\fp^-}$. \nThis proves Claim $1$. \n\n{\\bf Claim 2:} $(\\pi, \\cH)$ is holomorphically induced from \nthe bounded $H$-representation $(\\rho, \\beta, U)$, defined by \n$\\beta(\\fp^+) := \\{0\\}$. \n\nFrom (i) we know that the $H$-representation on $U$ is bounded \nand on the dense subspace $\\cD_U$ we have $\\dd\\pi(\\fp^-)\\cD_U = \\{0\\}$. \nTherefore (A1\/2) in Theorem~\\ref{thm:a.3} are satisfied and this proves \nClaim~$2$. \n\n{\\bf Claim 3:} $U = P(\\{s\\})\\cH$ for every $t > s$. \n\nIn view of Claim $2$, Corollary~\\ref{cor:commutant} implies that \nthe $H$-representation on $U$ is irreducible. Since $\\pi_d$ commutes with \n$H$, it follows in particular that $\\rho(\\exp\\R d) \\subeq \\T\\1$. \nThe definition of $s$ now shows that $U \\subeq P(\\{s\\})\\cH$. \n\nFor $0 < \\eps < \\delta$ and $t < \\delta + \\eps$, the proof \nof Proposition~\\ref{prop:6.2} implies that \n$\\big(P([s,t[)\\cH\\big) \\cap \\cH^\\infty$ is dense in \n$P([s,t[)\\cH$ and contained in $(\\cH^\\infty)^{\\fp^-}$, hence \nin $P([s,t[)(\\cH^\\infty)^{\\fp^-}\\subeq U$. \nWe conclude in particular that $P(\\{s\\})\\cH \\subeq U$. \n\n{\\bf Claim 4:} $U = \\oline{(\\cH^\\infty)^{\\fp^-}}$. \n\nFrom the definition of $U$ it is clear that \n$U \\subeq \\oline{(\\cH^\\infty)^{\\fp^-}}$. \nTo see that we actually have equality, we note that Claim $2$ \nshows that $P([s,t[)(\\cH^\\infty)^{\\fp^-} \\subeq P(\\{s\\})\\cH = U$ \nholds for every $t > s$. \nAs $P([s,n]) \\to P([s,\\infty[) = \\id$ holds pointwise, we obtain \n$(\\cH^\\infty)^{\\fp^-} \\subeq U$. \n\nThis completes the proof of (iii). \n\\end{prf}\n\n\\begin{rem} For finite dimensional Lie groups the classification \nof irreducible \nsemibounded unitary representations easily boils down to a situation \nwhere one can apply Theorem~\\ref{thm:6.2b}. Here \n$d \\in \\g$ is a regular element whose centralizer \n$\\fh = \\ft$ is a compactly embedded Cartan subalgebra and \nthe corresponding group $T = H$ is abelian and $V$ is one-dimensional \n(cf.\\ \\cite{Ne00}). In this case $0$ is trivially isolated in \nthe finite set $\\Spec(\\ad d)$. \n\\end{rem}\n\nThe following theorem provides a bridge between the seemingly \nweak positive energy condition and the much stronger semiboundedness \ncondition. \n\n\\begin{thm} \\mlabel{thm:3.15} \nLet $d \\in \\g$ be elliptic with $0$ isolated in $\\Spec(\\ad d)$. \nThen a smooth unitary representation \n$(\\pi, \\cH)$ of $G$ for which the representation $\\rho$ of \n$H$ on $\\oline{(\\cH^\\infty)^{\\fp^-}}$ is bounded \nsatisfies the positive energy condition \n\\begin{equation}\n  \\label{eq:posen}\n\\inf\\Spec(-i\\dd\\pi(d)) > -\\infty\n\\end{equation}\nif and only if $\\pi$ is semibounded with $d \\in W_\\pi$. \n\\end{thm}\n\n\\begin{prf} If $\\pi$ semibounded with $d \\in W_\\pi$, then \nwe have in particular \\eqref{eq:posen}. It remains to show the \nconverse if all the assumptions of the theorem are satisfied. \nRecall that the splitting condition \\eqref{eq:splitcond} \nis satisfied because $0$ is isolated in $\\Spec(\\ad d)$. \nWe note that the representation $\\ad_{\\fp^+}$ of \n$\\fh$ on $\\fp^+$ is bounded with $\\Spec(\\ad_{\\fp^+}(-id)) \\subeq ]0, \\infty[$. \nTherefore the invariant cone \n\\begin{equation}\n  \\label{eq:cone} \nC := \\{ x \\in \\fh \\: \n\\Spec(\\ad_{\\fp^+}(-ix)) \\subeq ]0,\\infty[ \\} \n\\end{equation}\nis non-empty \nand open because it is the inverse image of the \nopen convex cone \n\\[ \\{ X \\in \\Herm(\\fp^+) \\: \\Spec(X) \\subeq ]0, \\infty[ \\}\\] \n(cf.\\ \\cite[Thm.~14.31]{Up85}) under the continuous linear map \n$\\fh \\to \\gl(\\fp^+), x \\mapsto \\ad_{\\fp^+}(-ix)$. \n\nLet $x \\in C$ and \n\\[ s_\\rho(x) := \\sup(\\Spec(i\\dd\\rho(x))\n= -\\inf(\\Spec(-i\\dd\\rho(x)),\\]  so that \n$V := \\oline{(\\cH^\\infty)^{\\fp^-}} \\subeq \\cH^\\infty$ \n(Theorem~\\ref{thm:6.2}) is contained in \nthe spectral subspace $\\cH^\\infty([-s_\\rho(x),\\infty[)$ with respect \nto the one-parameter group $t \\mapsto \\pi(\\exp tx)$. \nSince the map \n\\[ \\g_\\C \\times \\cH^\\infty \\to \\cH^\\infty, \\quad \n(x,v) \\mapsto \\dd\\pi(x)v \\] \nis continuous bilinear and $H$-equivariant, we see with \nProposition~\\ref{prop:spec-add} that \nthe subspace $\\cH^\\infty([-s_\\rho(x),\\infty[)$ of $\\cH^\\infty$ \nis invariant under $\\fp^+$. \n\nFor every $v \\in V \\subeq \\cH^\\omega$, the Poincar\\'e--Birkhoff--Witt \nTheorem shows that it \ncontains the subspace \n\\[ U(\\g_\\C)v = U(\\fp^+)U(\\fh_\\C)U(\\fp^-)v \n= U(\\fp^+)U(\\fh_\\C)v \\subeq U(\\fp^+)V.\\] \nFrom $V \\subeq \\cH^\\omega$ and $\\cH = \\cH_V$ it follows that \n$U(\\g_\\C)V$ is dense in $\\cH$, and hence that \n$\\cH^\\infty([-s_\\rho(x),\\infty[) \\subeq \\cH([-s_\\rho(x),\\infty[)$ \nis dense in $\\cH$. We conclude that \n\\begin{equation}\n  \\label{eq:spirho} \ns_{\\pi}(x) = \\sup(\\Spec(i\\dd\\pi(x))) = s_\\rho(x) \n\\quad \\mbox{ for } \\quad x \\in C.\n\\end{equation}\n\nTo see that $C \\subeq W_\\pi$, it now suffices to show that \n$\\Ad(G)C$ has interior points. \nLet $\\fp := (\\fp^+ + \\fp^-)\\cap \\g$ and note that this is a closed \n$H$-invariant complement of $\\fh$ in $\\g$. \nThe map $F \\: \\fh\\times \\fp \\to \\g, F(x,y) := e^{\\ad y}x$ \nis smooth and \n\\[ \\dd F(x,0)(v,w) = [w,x] +  v.\\] \nSince the operators $\\ad x$, $x \\in \\h$, \npreserve $\\fp$, the operator \n$\\dd F(x,0)$ is invertible if and only if \n$\\ad x \\: \\fp \\to \\fp$ is invertible, and this is the case \nfor any $x \\in C$ because $\\ad x\\res_{\\fp^\\pm}$ are invertible \noperators. This proves that $C$ is contained in the interior \nof $F(C,\\fp)$, and hence that $C \\subeq W_{\\pi}$. \nTherefore $\\pi$ is semibounded with $d \\in W_{\\pi}$. \n\\end{prf}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\subsection{GPU SM Architecture}\n\\label{sec:sm_arch}\n\\noindent  \n\\newedit{Figure~\\ref{fig:gpu1} illustrates a representative SM architecture where shared memory may share a single on-chip memory structure with L1D cache~\\cite{nvidia2009nvidia, nvidia2012nvidia}.\nThe single on-chip memory structure consists of 32 banks with 512 rows, where 128 or 384 contiguous rows can be allocated to shared memory (\\textit{i.e}\\onedot} \\def\\Ie{\\textit{I.e}\\onedot, 16KB or 48KB) based on user configuration and the remaining are allocated as L1D cache~\\cite{gebhart2012unifying}.\nWhile all 32 L1D cache banks operate in tandem for a single contiguous 32$\\times$4-byte (128-byte) L1D cache request, all 32 shared memory banks can be accessed independently and serve upto 32 shared memory requests in parallel.\nL1D cache buffers data from underlying memory and keeps a separate tag array to identify data hit. In such architecture, a L1D cache access is serialized. That is, tag array is accessed before the banks are accessed~\\cite{edmondson1995internal}. \nIn contrast, as shared memory stores intermediate results generated by ALU for each Cooperative Thread Array (CTA) which is explicitly manipulated by programmers, it neither needs tags nor accesses data in underlying memory. \nHence, there is no datapath between shared memory and L2 cache, and no cache write\/eviction policies are applied in shared memory~\\cite{nvidia2009nvidia, nvidia2012nvidia}. In addition, to manage the shared memory space, each SM keeps an independent Shared Memory Management Table (SMMT)~\\cite{yang2012shared}\nwhere each CTA reserves one entry to store the size and base address of allocated shared memory.\n}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=1\\linewidth]{figs\/gpu1.eps}\n\\caption{GPU SM architecture.}\n\\label{fig:gpu1}\n\\end{figure}\n\n\\begin{comment}\n\\end{comment}\n\n\n\\begin{figure}[b]\n\\centering\n\\subfloat[]{\\label{fig:l1_contention}\\rotatebox{0}{\\includegraphics[width=0.4\\linewidth]{figs\/l1_contention}}}\n\\hspace{4pt}\n\\subfloat[]{\\label{fig:VTA_mech}\\rotatebox{0}{\\includegraphics[width=0.57\\linewidth]{figs\/VTA_mech}}}\n\\caption{(a) An example of locality and interference and (b) VTA structure.}\n\\end{figure}\n\n\\subsection{Cache Interference}\n\\label{sec:interfere}\n\\noindent \nAs many warps share small L1D cache, they often contend for the same cache line.\nHence, cached data of an active warp are frequently evicted by cache accesses of other active warps.\nThis phenomenon is referred to as \\textit{cache interference} which often changes supposedly a regular memory access pattern into an irregular one. \nFigure~\\ref{fig:l1_contention} depicts an example of how the cache interference worsens data locality in L1D cache, \nwhere warps \\texttt{W0} and \\texttt{W1} send memory requests to get data \\texttt{D0} and \\texttt{D4}, respectively.\nHowever, since \\texttt{D0} and \\texttt{D4} are mapped to the same cache set \\texttt{S0}, \nrepeated memory requests from \\texttt{W0} and \\texttt{W1} to get \\texttt{D0} and \\texttt{D4} keep evicting \\texttt{D4} and \\texttt{D0} at cycles \\texttt{(a)}, \\texttt{(b)}, \\texttt{(e)}, and \\texttt{(f)}.\nUnless the memory requests from \\texttt{W1} and \\texttt{W0} evicted \\texttt{D0} and \\texttt{D4}, respectively, \nthey should have been L1D cache hits.\nSuch a cache hit opportunity is also called \\textit{potential of data locality}, which can be quantified by the frequency of re-referencing the same data unless cache interference occurs.\n\n\n\\subsection{Potential of Data Locality Detection}\n\\label{sec:vta}\n\\noindent\nTo detect the potential of data locality described in Section~\\ref{sec:interfere}, we may leverage a Victim Tag Array (VTA)~\\cite{rogers2012cache} where\n\nwe store a Warp ID (WID) in each cache tag, as shown in Figure~\\ref{fig:VTA_mech}.\nA WID in a cache tag is to track which warp brought current data in a cache line. \nWhen a memory request of a warp evicts data in a cache line, we first take \n(1) the address in the cache tag associated with the evicted data and \n(2) the WID of the warp evicting the data. \nThen we store (1) and (2) in a VTA entry which is indexed by the WID stored in the cache tag (\\textit{i.e}\\onedot} \\def\\Ie{\\textit{I.e}\\onedot, the WID of the warp which brought the evicted data in the cache line).\nWhen memory requests of an active warp repeatedly incur VTA hits, \nthey exhibit potential of data locality.\n\n\n\\subsection{Methodology}\n\\label{sec:method} \n\\noindent\\textbf{GPU architecture.} \nWe use GPGPU-Sim 3.2.2~\\cite{aaamodt2012gpgpu} and configure it to model a GPU similar to NVIDIA GTX 480;\nsee Table \\ref{tab:config} for the detailed GPGPU-Sim configuration parameters~\\cite{nvidia2009nvidia}.\nBesides, we enhance the baseline L1D and L2 caches with a XOR-based set index hashing technique~\\cite{nugteren2014detailed}, making it close to the real GPU device's configuration. \nSubsequently, we implement seven different warp schedulers: \n(1) \\texttt{GTO} (GTO scheduler with set-index hashing \\cite{nugteren2014detailed});\n(2) \\texttt{CCWS};\n(3) \\texttt{Best-SWL} (best static wavefront limiting);\n(4) \\texttt{statPCAL} (representative implementation of bypass scheme\\cite{li2015priority} that performs similar or better than \\cite{li2015locality,tian2015adaptive});\n(5) \\texttt{CIAO-P} (\\texttt{CIAO} with only redirecting memory requests of interfering warp to shared memory); \n(6) \\texttt{CIAO-T} (\\texttt{CIAO} with only selective warp throttling); and\n(7) \\texttt{CIAO-C} (\\texttt{CIAO} with both \\texttt{CIAO-T} and \\texttt{CIAO-P}).\nNote that \\texttt{CCWS}, \\texttt{Best-SWL}, and \\texttt{CIAO-P\/T\/C} leverage \\texttt{GTO} to decide the order of execution of warps. \n\\texttt{CCWS} and \\texttt{CIAO-T\/C} stall a varying number of warps depending on memory access characteristics monitored at runtime.\nIn contrast, \\texttt{Best-SWL} stalls a fixed number of warps throughout execution of a benchmark; we profile each benchmark to determine the number of stalled warps giving the highest performance for each benchmark; see column $\\rm N_{wrp}$ in Table~\\ref{tab:workload_charac}.\n\n\n\n\n\n\n\\noindent \\textbf{Benchmarks.} We evaluate a large collection of benchmarks from \\texttt{PolyBench}~\\cite{grauer2012auto}, \\texttt{Mars}~\\cite{he2008mars} and \\texttt{Rodinia}~\\cite{che2009rodinia}\nwhich are categorized into three classes: \n(1) large-working set (LWS), (2) small-working set (SWS), and (3) compute-intensive (CI). \nTable~\\ref{tab:workload_charac} tabulates chosen benchmarks and their characteristics.\n\n\n\n\n\n\\begin{figure*}\n\\centering\n\\subfloat[IPC]{\\label{fig:ATAX_back_IPC_fig}\\rotatebox{0}{\\includegraphics[width=0.33\\linewidth]{figs\/ATAX_back_IPC_fig}}}\n\\subfloat[Number of active warps]{\\label{fig:ATAX_back_AW_fig}\\rotatebox{0}{\\includegraphics[width=0.33\\linewidth]{figs\/ATAX_back_AW_fig}}}\n\\subfloat[Cache interference]{\\label{fig:ATAX_back_interf_fig}\\rotatebox{0}{\\includegraphics[width=0.33\\linewidth]{figs\/ATAX_back_interf_fig}}}\n\\caption{Comparison between \\texttt{Best-SWL}, \\texttt{CCWS} and \\texttt{CIAO-T} over time: \\texttt{ATAX} and \\texttt{Backprop}}\n\\label{fig:IPCtrace_ATAX_BACK}\n\\end{figure*}\n\n\\begin{figure*}\n\\centering\n\\subfloat[IPC]{\\label{fig:SYRK_KMN_IPC}\\rotatebox{0}{\\includegraphics[width=0.33\\linewidth]{figs\/SYRK_KMN_IPC}}}\n\\subfloat[Number of active warps]{\\label{fig:SYRK_KMN_activewarp}\\rotatebox{0}{\\includegraphics[width=0.33\\linewidth]{figs\/SYRK_KMN_activewarp}}}\n\\subfloat[Cache interference]{\\label{fig:SYRK_KMN_interference}\\rotatebox{0}{\\includegraphics[width=0.33\\linewidth]{figs\/SYRK_KMN_interference}}}\n\\caption{Comparison of \\texttt{CIAO-T}, \\texttt{CIAO-P} and \\texttt{CIAO-C} over time: \\texttt{SYRK} and \\texttt{KMN}.}\n\\label{fig:IPCtrace_SYRK_seperate}\n\\end{figure*}\n\n\n\n\\ignore{\n\\begin{figure*}\n\\centering\n\\subfloat[IPC]{\\label{fig:ATAX_back_IPC_fig}\\rotatebox{0}{\\includegraphics[width=1\\linewidth]{figs\/ATAX_back_SYRK_KMN_IPC_fig}}}\n\\subfloat[Number of active warps]{\\label{fig:ATAX_back_AW_fig}\\rotatebox{0}{\\includegraphics[width=1\\linewidth]{figs\/ATAX_back_SYRK_KMN_AW_fig}}}\n\\subfloat[Cache Interference]{\\label{fig:ATAX_back_interf_fig}\\rotatebox{0}{\\includegraphics[width=1\\linewidth]{figs\/ATAX_back_SYRK_KMN_interf_fig}}}\n\\caption{Performance analysis of \\texttt{ATAX}, \\texttt{Backprop}, \\texttt{SYRK}, and \\texttt{KMN}.\n}\n\\label{fig:IPCtrace}\n\\end{figure*}\n}\n\n\\subsection{Performance Analysis}\n\\label{sec:analy}\n\\noindent\nFigure~\\ref{fig:overall_ipc}\nplots the IPC values with the seven warp schedulers and the \\textbf{geometric-mean} IPC values of three benchmark classes (LWS, SWS, and CI), respectively,\nnormalized to those with \\texttt{GTO}. \nOverall, \\texttt{CCWS}, \\texttt{Best-SWL}, \\texttt{statPCAL}, and \\texttt{CIAO-C} provide 2\\%, 16\\%, 24\\% and 56\\% higher performance than \\texttt{GTO}, respectively.\n\n\\texttt{GTO} performs worst among all evaluated schedulers, because, it shuffles only the order of executed warps and does not notably reduce cache thrashing caused by many active warps accessing small L1D cache. \nIn contrast, \\texttt{Best-SWL} outperforms \\texttt{GTO} as it throttles some warps, reducing the number of memory accesses to small L1D and thus cache thrashing.  \nNonetheless, as \\texttt{Best-SWL} must decide the number of throttled warps before execution of a given application, \nit cannot effectively capture the optimal number of throttled warps varying within an application compared to warp schedulers that dynamically throttle the number of executed warps such as \\texttt{CCWS} and \\texttt{CIAO}.\nFor example, as \\texttt{ATAX} exhibits very dynamic cache access patterns at runtime, \n\\texttt{CCWS} outperforms \\texttt{Best-SWL} by 49\\%. \nNote that \\texttt{CCWS} gives notably lower performance than \\texttt{Best-SWL}\nespecially for CI benchmarks; considerably affecting its performance.\nThat is because running more active warps achieves higher performance for CI benchmarks, whereas \\texttt{CCWS} unnecessarily stalls some active warps to give a higher priority to a few warps exhibiting high data locality.\n\\texttt{statPCAL} gives up to 37\\% higher performance than \\texttt{Best-SWL} by up to 37\\% because \\texttt{statPCAL} offers higher TLP.\nSpecifically, when \\texttt{statPCAL} detects under-utilization of L2 and\/or main memory bandwidth, it activates throttled warps and makes these warp directly access the underlying memory (\\textit{i.e}\\onedot} \\def\\Ie{\\textit{I.e}\\onedot, bypassing L1D cache).\nDue to the long access latency and limited bandwidth of underlying memory, however,\n\\texttt{statPCAL} cannot significantly improve performance of LWS and SWS workloads such as \\texttt{KMN}, \\texttt{SYRK}, etc.\n\n\\texttt{CIAO-T} provides 32\\% and 34\\% higher performance than \\texttt{CCWS} and \\texttt{GTO}, respectively. \nFurthermore, \\texttt{CIAO-T} offers 22\\% higher performance than \\texttt{Best-SWL} for every benchmark except for \\texttt{SYR2K}, \\texttt{II}, and \\texttt{KMN} exhibiting static cache access patterns at runtime.\nBoth \\texttt{CIAO-T} and \\texttt{CCWS} dynamically stall some active warps at runtime, but\nour evaluation shows that it is often more effective to throttle the warps that considerably interfere with other warps than the warps with low potential of data locality\nas \\texttt{CCWS} does.\nFurthermore, for CI benchmarks, \\texttt{CIAO-T} offers as high performance as \\texttt{GTO} in contrast to \\texttt{CCWS}; \nrefer to our earlier comparison between \\texttt{GTO} and \\texttt{CCWS} for CI benchmarks.\n\n\\texttt{CIAO-P}\ngives 34\\% higher performance than \\texttt{GTO}. \n\\newedit{We observe that \\texttt{CIAO-P} offers the highest TLP among all seven warp schedulers, entailing 28\\% higher performance than \\texttt{CIAO-T} for SWS class benchmarks. This is because \\texttt{CIAO-P} fully utilizes the unused space of shared memory (cf. Figure~\\ref{fig:shmutil_fig}). }\nNonetheless, its benefits can be limited for LWS class benchmarks in which the redirected memory requests of interfering warps are often too intensive and thus thrash the shared memory as well.\nIn such a case, \\texttt{CIAO-T} can perform better than \\texttt{CIAO-P}, giving 48\\% and 66\\% higher performance than \\texttt{CIAO-P} and \\texttt{CCWS}, respectively,  as shown in Figure \\ref{fig:IPC_fig}. \nLastly, \\texttt{CIAO-C}, which synergistically integrates \\texttt{CIAO-T} and \\texttt{CIAO-P}, provides 56\\%, 54\\%, 17\\% and 16\\% higher performance than \\texttt{GTO}, \\texttt{CCWS}, \\texttt{CIAO-T}, and \\texttt{CIAO-P}, respectively. \n\n\\begin{figure}[b]\n\\centering\n\\subfloat[Various epoches.]{\\label{fig:epoch}\\rotatebox{0}{\\includegraphics[width=0.48\\linewidth]{figs\/epoch}}}\n\\hspace{2pt}\n\\subfloat[Vairous high cut-off lines.]{\\label{fig:highcutoff}\\rotatebox{0}{\\includegraphics[width=0.48\\linewidth]{figs\/highcutoff}}}\n\\caption{Sensitivity analysis\n}\n\\label{fig:sensi_scheduler}\n\\end{figure}\n\n\\begin{figure*}\n\\centering\n\\subfloat[IPC comparison of varying L1D cache configurations.]{\\label{fig:IPC_fig_sens}\\rotatebox{0}\n{\\includegraphics[width=0.49\\linewidth]{figs\/IPC_fig_sens.eps}}}\n\\subfloat[IPC comparison of vayring DRAM bandwidths.]{\\label{fig:IPC_fig_sens1}\\rotatebox{0}\n{\\includegraphics[width=0.49\\linewidth]{figs\/IPC_fig_sens1.eps}}}\n\\caption{IPC of different L1D cache and DRAM configurations.  \n}\n\\label{fig:sensi1_cache}\n\\end{figure*}\n\n\\subsection{Effectiveness of Interference Awareness}\n\\noindent\nFigure~\\ref{fig:IPCtrace_ATAX_BACK}\nshows the IPC, the number of active warps, and cache interference over time of \\texttt{ATAX} as a representative application that exhibits distinct execution phases in a single kernel execution. \nFor example, \\texttt{ATAX} exhibits two distinct execution phases.\nThe first phase comprised of the first 40-million instructions is very memory-intensive, whereas the second phase is very compute-intensive.\nFigure \\ref{fig:ATAX_back_IPC_fig} shows that \\texttt{CIAO-T} outperforms \\texttt{CCWS} and \\texttt{Best-SWL} for the first 40-million instructions executed. \n\\texttt{CIAO-T} exhibits higher performance during this phase because \\texttt{CIAO-T} more effectively reduces cache interference by throttling severely interfering warps, as shown in Figure \\ref{fig:ATAX_back_interf_fig}.\nAfter the first phase, \\texttt{ATAX} starts a compute-intensive phase, performing the computation by fully exploiting data locality on the GPU caches. \nAs \\texttt{Best-SWL} cannot capture this dynamics at runtime, it executes only 2 warps for the second phase execution of \\emph{ATAX}. \nIn contrast, \\texttt{CCWS} and \\texttt{CIAO-C} dynamically reduce the number of stalled warps as they observe fewer cache misses and less cache interference,\ngiving 4$\\times$ higher geometric-mean performance than \\texttt{Best-SWL}.\n\n\n\n\n\nWe choose \\texttt{Backprop} as a representative application that is very compute-intensive but also experiences many cache misses.\nFigure \\ref{fig:IPCtrace_ATAX_BACK} shows the performance change of \\emph{Backprop} over time.\n\\texttt{Best-SWL} and \\texttt{CIAO-T} provide 500 IPC on average. \nHowever, \\texttt{CCWS} notably degrades the performance, ranging from 320 to 150 IPC because \\texttt{CCWS} ends up giving a higher priority to warps with higher data locality and stalling more than 40 warps (or significantly reducing TLP).\nIn contrast, \\texttt{CIAO-T}, which offers performance similar to \\texttt{Best-SWL}, more selectively throttles warps than \\texttt{CCWS} (i.e., only 10$\\sim$20 most interfering warps), better preserving TLP. \n\n\n\n\\subsection{Sensitivity to Working Set Size}\n\\label{sec:tsa}\n\\noindent \\textbf{Small-working set.}\nFigure~\\ref{fig:IPCtrace_SYRK_seperate} shows the performance of three \\texttt{CIAO} schemes for \\texttt{SYRK} over time.\n\\texttt{SYRK} is a representative application with SWS. \nSpecifically, \nFigure~\\ref{fig:IPCtrace_SYRK_seperate}\nillustrates IPC, the number of active warps, and the number of cache conflicts over time of texttt{SYRK} over time with three \\texttt{CIAO} schemes. \nAs shown in Figure \\ref{fig:SYRK_KMN_IPC}, \\texttt{CIAO-P} offers higher IPC than \\texttt{CIAO-T} overall. \nThis is because, it can secure higher TLP (\\textit{cf}\\onedot} \\def\\Cf{\\textit{Cf}\\onedot Figure \\ref{fig:SYRK_KMN_activewarp}), whereas \\texttt{CIAO-T} alone hurts TLP by throttling many active warps. \nUsing the unused shared memory space, \\texttt{CIAO-P} can effectively reduce cache interference without sacrificing TLP in contrast to \\texttt{CIAO-T}. \nAs expected, \\texttt{CIAO-C} selectively stalls very few warps.\n\n\n\n\\noindent \\textbf{Large-working set.}\nFigure~\\ref{fig:IPCtrace_SYRK_seperate} also depicts the performance of three \\texttt{CIAO} schemes \\texttt{KMN}, a representative application with LWS. \nAs shown in Figure~\\ref{fig:SYRK_KMN_IPC}, \\texttt{CIAO-T} provides 50\\% higher IPC than \\texttt{CIAO-P}, and\n\\texttt{CIAO-C} always achieves the highest performance during the entire execution period amongst all three schemes. \nThis is because, as shown in Figure \\ref{fig:SYRK_KMN_interference}, \\texttt{CIAO-P} still suffers from severe shared memory interference \nas the amount of data requested by the partitioned warps exceeds the amount that shared memory can efficiently accommodate. \nIn contrast, \\texttt{CIAO-C} can better utilize shared memory by selectively throttling only the warps that cause severe interference. \n\n\n\n\n\n\n\n\n\n\n\\ignore{\n\\begin{figure*}\n\\centering\n\\subfloat[IPC comparison]{\\label{fig:IPC_fig_16112}\\rotatebox{0}\n{\\includegraphics[width=0.8\\linewidth]{figs\/IPC_fig_16112.eps}}}\n\\subfloat[Geo-mean IPC]{\\label{fig:IPC_fig2_16112}\\rotatebox{0}\n{\\includegraphics[width=0.18\\linewidth]{figs\/IPC_fig2_16112.eps}}}\n\\caption{IPC of 16KB L1D cache and 112KB shared mem.  \n}\n\\label{fig:sensi_cache}\n\\end{figure*}\n}\n\n\n\n\n\n\\ignore{the highest reduction in L2 miss rate comes from LRR + CIAO -- compared to LRR and LRR + \\texttt{CCWS} by 52.7\\% and 48.9\\%, respectively. This dramatic decrease in miss rate results from a combined effect of the active warp number throttling and the consideration of cache interference upon warp scheduling. On the other hand,  \\texttt{GTO} + CIAO reduces the L2 miss rate by 28.3\\% and 13.8\\% over \\texttt{GTO} and \\texttt{GTO} + \\texttt{CCWS}, respectively. Due to the high number of active warps, even restricting to the oldest warps, \\texttt{GTO} still allows too much data to be contained in L2 cache. Even though, \\texttt{CCWS} can further alleviate the L2-level cache interference by strictly limiting the active warp number, warps with high potential of data locality, which are prioritized in \\texttt{CCWS}, can still contend with each other. The scheduling policy of CIAO successfully exploits this critical observation regarding the potential for further reduction in L2-level interference.\n}\n\n\n\\ignore{\n\\begin{figure}\n\\centering\n\\includegraphics[width=1\\linewidth]{figs\/L1Dynenergy_fig.eps}\n\\caption{L1 total energy analysis.  \n}\n\\label{fig:L1Dynenergy_fig}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=1\\linewidth]{figs\/L2dynenergy_fig.eps}\n\\caption{L2 total energy analysis. \n}\n\\label{fig:L2dynenergy_fig}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=1\\linewidth]{figs\/DRAMdynenergy_fig.eps}\n\\caption{DRAM dynamic energy analysis. \n}\n\\label{fig:DRAMdynenergy_fig}\n\\end{figure}\n}\n\n\n\\subsection{Sensitivity Study}\n\\label{subsubsec:sensi}\n\\noindent \\textbf{Epoch value.} \nFigure~\\ref{fig:epoch} shows the effect of varying \\texttt{high-cutoff} epoch values on the IPC for all the memory-intensive workloads. \nAs we increase the epoch from 1K to 50K instructions, the change in IPC is within 15\\%. \nNote that different workloads can achieve best performance with different epoch values. \nThat is because epoch determines the frequency of checking cache interference for \\texttt{CIAO}.\nA shorter epoch provides fast response to cache interference, while a longer epoch can more accurately detect the warp causing most interference. \nTaking this trade-off into account, we choose 5K instructions as \\texttt{high-cutoff} epoch value. \nAn adaptive scheme can be future work.\n\n\n\n\n\n\\noindent \\textbf{High-cutoff threshold.} \nFigure~\\ref{fig:highcutoff} depicts performance corresponding to different \\texttt{high-cutoff} thresholds, \nwhere the \\texttt{low-cutoff} threshold is fixed to half of it.\nAll benchmarks show steady performance within 5\\% change during the entire execution period.\nThis is because our \\texttt{CIAO} throttles the active warps causing most interference, which can easily exceed the current thresholds we set. 1\\% is chosen in the paper.\n\n\n\\newedit{\n\\noindent \\textbf{L1D cache\/DRAM configurations.} Figure~\\ref{fig:sensi1_cache} illustrates the performance of LWS and SWS workloads by configuring various L1D cache\/DRAM design parameters: \n(1) \\texttt{GTO};\n(2) \\texttt{GTO-cap} (\\texttt{GTO} but increase L1D cache capacity to 48 KB and reduce shared memory size to 16 KB);\n(3) \\texttt{GTO-8way} (\\texttt{GTO} but increase L1D cache associativity to 8 way);\n(4) \\texttt{statPCAL-2X} (\\texttt{statPCAL} but double DRAM bandwidth from 177 GB\/s to 340 GB\/s);\n(5) \\texttt{CIAO-C};\n(6) \\texttt{CIAO-C-2X} (\\texttt{CIAO-C} but double DRAM bandwidth).\nAs shown in Figure \\ref{fig:IPC_fig_sens}, while increasing L1D cache capacity (\\texttt{GTO-cap}) and associativity (\\texttt{GTO-8way}) can effectively improve the overall performance by 108\\% and 51\\% compared to \\texttt{GTO}, \\texttt{CIAO-C} still outperforms \\texttt{GTO-cap} and \\texttt{GTO-8way} by 14\\%, and 57\\%, respectively. This is because, \\texttt{GTO-cap} and \\texttt{GTO-8way} cannot fully eliminate cache interference, as they cannot distinguish the requests between interfering and interfered warps and effectively isolate them. On the other hand, while \\texttt{statPCAL-2X} can benefit from the increased DRAM bandwidth, bypassing requests to underlying DRAM still suffers from long DRAM delay as the latency of DRAM access is much longer than that of L1D cache access. Hence, as shown in Figure \\ref{fig:IPC_fig_sens1}, \\texttt{CIAO-C-2X} outperforms \\texttt{statPCAL-2X} by 16\\%, on average.\n}\n\n\\ignore{\n\\noindent \\textbf{Varying L1D cache sizes.} \nFigure~\\ref{fig:sensi1_cache} \nillustrates the performance of all workloads for the various L1D cache configurations shown in Table~\\ref{tab:config}. \nFor 48KB L1D cache and 16KB shared memory, \\texttt{CIAO-C} gives 29\\%, 27\\%, 13\\%, and 12\\% higher IPC than \\texttt{GTO}, \\texttt{CCWS}, \\texttt{Best-SWL}, and \\texttt{statPCAL}, respectively.\nSince 48KB L1D cache is much larger than the default 16KB size, the performance of \\texttt{GTO} improves greatly and leaves less room for improvement by \\texttt{CIAO}\\xspace. \n}\n\n\\subsection{Overhead Analysis}\n\\noindent \nImplementing the interference detector, \\texttt{CIAO} leverages the VTA structure originally proposed by \\texttt{CCWS}~\\cite{rogers2012cache}, but employs only 8 VTA entries for each warp (\\textit{i.e}\\onedot} \\def\\Ie{\\textit{I.e}\\onedot, half of the VTA entries that \\texttt{CCWS} uses).\nUsing CACTI 6.0~\\cite{muralimanohar2009cacti}, we estimate that the area of one VTA structure is only 0.65 $mm^2$ for 15 SMs, which accounts for only 0.12\\% of the total chip size of NVIDIA GTX480 (529 $mm^2$~\\cite{geforce-gtx-480}).\nIn addition, \\texttt{CIAO} uses 48 registers as VTA-hit counters (one for each warp). \nSince each VTA-hit counter resets at the start of each kernel, a 32-bit counter is sufficient to prevent its overflow. \nThe interference and pair lists are implemented with SRAM arrays indexed by WIDs. \nSince the total number of active warps in a CTA does not exceed 64 (usually, 48 active warps in each SM), we configure the interference and pair lists with 64 entries. \nEach entry of the interference list requires 8 (= 6+2) bits to store one warp index and saturation counter value, while each entry of pair list requires 12 (=6 + 6) bits to store two warp indices.\nUsing CACTI 6.0, we estimate that the combined area of the VTA-hit counters, interference list, and pair lists is 549 $um^{2}$ per SM (8235 $um^{2}$ for 15 SMs).\nOn the other hand, Equation \\ref{eq:irs} is implemented with a few adders, a shifter, and a comparator, which also requires very low cost (2112 gates). \nFor our shared memory modification, the translation unit, multiplexer and MSHR only need 4500 gates and 64B storage per SM.\nWe also track the power consumption of new components employed in \\texttt{CIAO} by leveraging GPUWattch~\\cite{leng2013gpuwattch}, \nwhich reveals the average power is around 79mW. \nOverall, \\texttt{CIAO} improves the performance by more than 50\\% with a negligible area cost (less than 2\\% of the total GTX480 chip area) and power consumption (only 0.3\\% of GTX480 overall power).\n\n\\subsection{Cache Interference Detection}\n\\label{sec:schedule}\n\n\n\\noindent \\textbf{Estimation of cache interference.}\nA level of cache interference experienced by a warp can be quantified by \nan Individual Re-reference Score (IRS)\nwhich can be expressed by: \n\n\n\\begin{equation}\n\\label{eq:irs}\nIRS_i = \\frac{F^i_{VTA-hits}}{N_{executed-inst}\/N_{active-warp}}\n\\end{equation}\n\n\\begin{figure}[b]\n\\centering\n\\includegraphics[width=1\\linewidth]{figs\/vta.eps}\n\\caption{Microarchitecture adaptation for \\texttt{CIAO}\\xspace.}\n\\label{fig:vta}\n\\end{figure}\n\n\n\\noindent where $i$ is active warp number, $F^i_{VTA-hits}$ is the number of VTA hits for warp $i$, $N_{executed-inst}$ is the total number of executed instructions, and $N_{active-warp}$ is the number of active warps running on an SM, respectively. \n$IRS_i$ represents VTA hits per instruction (\\textit{i.e}\\onedot} \\def\\Ie{\\textit{I.e}\\onedot, intensity of VTA hits) for warp $i$.\nHigh $IRS_i$ indicates warp $i$ has experienced severe cache interference in a given epoch.\nBased on $IRS_i$, \\texttt{CIAO}\\xspace (1) decides whether it isolates warps interfering with warp $i$, (2) stalls these interfering warps, or (3) reactivates the stalled warps.\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=1\\linewidth]{figs\/shm.eps}\n\\caption{GPU on-chip memory structure adaptation.}\n\\label{fig:shm}\n\\end{figure*}\n\n\n\\noindent \\textbf{Decision thresholds.} \nFor these aforementioned three decisions we introduce two threshold values: (1) \\texttt{high-cutoff} and (2) \\texttt{low-cutoff}.  \n$IRS_i$ over \\texttt{high-cutoff} indicates that warp $i$ has experienced severe cache interference.\nSubsequently, \\texttt{CIAO}\\xspace decides to isolate or stall the warp that most recently and severely interfered with warp $i$. \n$IRS_i$ below \\texttt{low-cutoff} often indicates that warp $i$ has experienced light cache interference and\/or completed its execution.\nThen, \\texttt{CIAO}\\xspace decides to reactivate previously stalled warps or\nredirect memory requests of these warps back to L1D cache.\nAs these two thresholds influence the efficacy of \\texttt{CIAO}\\xspace, we sweep these two values, evaluate diverse memory-intensive applications, and determine that \\texttt{high-cutoff} and \\texttt{low-cutoff}, which minimize cache interference and maximize performance, are 0.01 and 0.005, respectively.\nSee Section~\\ref{subsubsec:sensi} for our sensitivity analysis.\n\n\n\\noindent \\textbf{Epochs.} \nAs $IRS_i$ changes over time, \\texttt{CIAO}\\xspace should track the latest $IRS_i$ and compare it against \\texttt{high-cutoff} and \\texttt{low-cutoff} to precisely determine whether a warp needs to be isolated, stalled, or reactivated. However, the update of $IRS_i$ calculation consumes more than 6 cycles, which can be on the critical path of performance.\nTo this end, \\texttt{CIAO}\\xspace divides the execution time into \\texttt{high-cutoff} and \\texttt{low-cutoff} epochs, respectively. \nAt the end of each \\texttt{high-cutoff} (or \\texttt{low-cutoff}) epoch, \\texttt{CIAO}\\xspace updates $IRS_i$ and compares it against \\texttt{high-cutoff} (or \\texttt{low-cutoff}).\nThe \\texttt{low-cutoff} epoch should be shorter than the \\texttt{high-cutoff} epoch because of the following reasons. \nAs preserving high TLP is a key to improve GPU performance, \\texttt{CIAO}\\xspace attempts to minimize a negative effect of stalling warps by reactivating stalled warps as soon as these warps start not to notably interfere with other warps at runtime.  \nTo validate this strategy, we sweep \\texttt{high-cutoff} and \\texttt{low-cutoff} epoch values, evaluate diverse memory-intensive applications, and determine that the best \\texttt{high-cutoff} and \\texttt{low-cutoff} epoch values are every 5000 and 100 instructions, respectively.\nSee Section~\\ref{subsubsec:sensi} for our in-depth sensitivity analysis.\n\n\\noindent \\textbf{Microarchitecture support.}\nFigure~\\ref{fig:vta} depicts the necessary hardware, which is built upon the existing VTA organization~\\cite{rogers2012cache}, to implement a cache interference detector.\n\n\nTo capture different levels of cache interference experienced by individual warps, we implement a VTA-hit counter per warp and a total instruction counter per SM (\\texttt{VTACount0-k} and \\texttt{Inst-total} in the figure) atop a VTA.\nEach VTA-hit counter records the number of VTA hits for each warp, and the total instruction counter tracks the total number of instructions executed by a given SM (\\textit{i.e}\\onedot} \\def\\Ie{\\textit{I.e}\\onedot, $N_{executed-inst}$ in Eq.(\\ref{eq:irs})). \nTo compare $IRS_i$ against \\texttt{high-cutoff} and \\texttt{low-cutoff}, we implement the cutoff testing unit which can be implemented by registers, a shifter, and simple comparison logic. \nLastly, we implement the samplers to count the number of executed instructions and determine whether or not the end of a \\texttt{high-cutoff} or \\texttt{low-cutoff} epoch has been reached.\n\n\nTo manage the information related to tracking interfering warps for each warp, we implement the interference list.\nEach entry is indexed by WID of a given warp and stores a 6-bit WID of an interfering warp and a 2-bit saturation counter (\\texttt{C} in the figure). \nWhen a VTA hit occurs, the corresponding entry of interference list is updated, as described in Section~\\ref{sec:interference_detection}.\n\\texttt{CIAO}\\xspace checks the interference list for warp $i$ whenever it needs to isolate or stall an interfering warp based on $IRS_i$.\nTo facilitate this, we also augment a 1-bit active flag (\\texttt{V}) and 1-bit isolation flag (\\texttt{I}) with each ready warp entry in the warp list (\\textit{i.e}\\onedot} \\def\\Ie{\\textit{I.e}\\onedot, a component of warp scheduler). \nUsing \\texttt{V} and \\texttt{I} bits, the warp scheduler can identify whether a given warp is in active (\\texttt{V}=\\texttt{1}, \\texttt{I}=\\texttt{0}), isolated (\\texttt{V}=\\texttt{1}, \\texttt{I}=\\texttt{1}), or stalled state (\\texttt{V}=\\texttt{0}). \n\n\nWe also implement a \\textit{pair list}. \nEach entry is indexed by the WID of a warp at the front of the warp list and composed of two fields \nto record which interfered warp triggered to redirect memory requests of the warp or stall the warp in the past.\nSuppose that warp $i$ is at the front of the warp list.\nBased on WIDs from the first or second field of the entry indexed by warp $i$, \n\\texttt{CIAO}\\xspace checks $IRS_k$ where $k$ is the WID of the interfered warp that previously triggered to either redirect memory requests of warp $i$ or stalling warp $i$. \nThen \\texttt{CIAO}\\xspace decides whether it reactivates warp $i$ or redirects memory requests of warp $i$ back to L1D cache based on $IRS_k$. \nFor example, as \\texttt{W0} is severely interfered by \\texttt{W1}, \\texttt{CIAO}\\xspace decides to redirect memory requests of \\texttt{W1} to unused shared memory space.\nThen \\texttt{W0} is recorded in the first field of the entry indexed by \\texttt{W1} and \\texttt{I} associated with \\texttt{W1} is set, as depicted in Figure~\\ref{fig:vta}. \nSubsequently, \\texttt{W1} begins to send memory requests to the shared memory, but \\texttt{CIAO}\\xspace observes that \\texttt{W1} also severely interferes with \\texttt{W3} that sends its memory requests to the shared memory.\nAs \\texttt{CIAO}\\xspace decides to stall \\texttt{W1}, \\texttt{W3} is recorded in the second field of the entry indexed by \\texttt{W1} and \\texttt{V} associated with \\texttt{W1} is cleared. \nWhen \\texttt{CIAO}\\xspace needs to reactivate \\texttt{W1} later, the second field of the pair list entry and \\texttt{V} corresponding to \\texttt{W1} are cleared to \ninform the warp scheduler of the event that the warp is active. \nWhen \\texttt{CIAO}\\xspace needs to make \\texttt{W1} send its memory request back to L1D cache, the corresponding field in the pair list entry and \\texttt{I} are cleared.\nSee Section~\\ref{sec:pat} for more details on the pair list.\n\n\n\n\\subsection{Shared Memory Architecture}\n\\label{sec:shared_mem_arch}\n\\noindent\nFigure~\\ref{fig:shm}a and b illustrate \\texttt{CIAO}\\xspace on-chip memory architecture and its data placement layout, respectively. \n\n\n\\noindent \\textbf{Determination of unused shared memory space.}\nOne challenge to utilize unused shared memory space is that shared memory is managed  by programmers and the used amount of shared memory space varies across implementations of a kernel. \nTo make \\texttt{CIAO}\\xspace on-chip memory architecture transparent to programmers, we leverage the existing SMMT structure to determine the unused shared memory space. \nWhen a CTA is launched, \\texttt{CIAO}\\xspace checks the corresponding SMMT entry to determine the amount of unused shared memory space (\\textit{cf}\\onedot} \\def\\Cf{\\textit{Cf}\\onedot Section~\\ref{sec:sm_arch}). \nThen, \\texttt{CIAO}\\xspace inserts a new entry in the SMMT with the start address and size of unused shared memory to reserve the space for storing 128-byte data blocks and tags. \n\n\n\\noindent \\textbf{Placement of tags and data.}\nIn contrast to L1D cache, shared memory does not have a separate memory array to accommodate tags~\\cite{gebhart2012unifying}. \nIn this work, instead of employing an additional tag array, we propose to place both 128-byte data blocks and their tags into the shared memory.\nThis is to minimize the modification of the current on-chip memory structure architected to be configured as both L1D cache and shared memory. \nAs shown in Figure~\\ref{fig:shm}b, we partition 32 shared memory banks into two bank groups and stripe a 128-byte data block across 16 banks within one bank group. \nEach 128-byte data block can be accessed in parallel since each shared memory bank allows 64-bit accesses~\\cite{nvidia2012nvidia}.\nSince a tag and a WID require only 31 bits (= 25 + 6 bits), two tags can be placed in a single bank which is different from banks storing the corresponding data blocks.\nThen 32 tags can be grouped together to better utilize a row of one bank group (\\textit{i.e}\\onedot} \\def\\Ie{\\textit{I.e}\\onedot, 16 banks). \nThis design strategy, which puts a tag and the corresponding data block into two different bank groups, shuns bank conflicts and thus allows accesses of a tag and a data block in parallel.\nFurthermore, we only use the unused shared memory space as direct-mapped cache so that a pair of a 128-byte data block and the corresponding tag can be accessed with a single shared memory access.\n\n\n\\noindent \\textbf{Address translation unit.}\nAs shown in Figure~\\ref{fig:shm}b, we introduce a hardware address translation unit in front of shared memory to determine where a target 128-byte data block and its tag exist in the shared memory. \nIn practice, a global memory address can be decomposed by cache-related information such as a tag, block index and byte offset. \nHowever, as the usage of shared memory can be varying based on the needs of each CTA, we put an 8-bit mask into the translation unit to decide how many rows will be used for each CTA at runtime. \nFigure~\\ref{fig:shm}c shows how our translation unit determines locations of a target data block and its tag; \nthe data block address (of shared memory) consists of four fields, the byte offset (``\\texttt{F}''), bank index (``\\texttt{B}''), bank group (``\\texttt{G}''), and row index (``\\texttt{R}''), which are presented from LSB to MSB.  \nSpecifically, we have 8-byte rows per bank, 16 banks per group, two bank groups and 256 rows (at most), \nwhich in turn 3, 4, 1, and 8 bits for \\texttt{F}, \\texttt{B}, \\texttt{G} and \\texttt{R}, respectively. \nThe remaining bits (16 bits in this example) are used as part of the tag. \nNote that our tags also contain 6-bit WID and 9-bit data block index as the number of cache lines required can be greater than the number of rows. \n\nIn \\texttt{CIAO}\\xspace, one row within a bank group can hold 32 tags since a physical row per bank contains two tags. \nThat is, the actual position of a tag can be indicated by 5 bits (\\textit{i.e}\\onedot} \\def\\Ie{\\textit{I.e}\\onedot, 1 \\texttt{F} and  4 \\texttt{B} bits), \nwhich are also used for the row index of the corresponding data block. \nTo access a data block and the corresponding tag in parallel, \\texttt{G} of the data block will be flipped and assigned to such tag's 5 bits as a significant bit. \nThe remaining \\texttt{R} bits are assigned to the row index of the target tag. \nNote that, as shown in the figure, the start of index for both a data block and a tag can be rearranged by considering the data block and tag offset registers, \nwhich are used to adapt the unused shared memory size allocated for cache.\n\n\n\\noindent \\textbf{Datapath connection.}\nWhen we leverage unused shared memory as cache, we need a datapath between shared memory and L2 cache. \nSince the shared memory is disconnected from the global memory in the conventional GPU, \nwe need to adapt the on-chip memory structure, which is partitioned between L1D cache and shared memory, to share some resources of the L1D cache with the shared memory (\\textit{e.g}\\onedot} \\def\\Eg{\\textit{E.g}\\onedot, datapath to L2 cache, MSHR, etc.). \nAs illustrated in Figure~\\ref{fig:shm}a, \na multiplexer is implemented to connect the write queue (WQ) and response queue (RespQ) to either L1D cache or shared memory. \nThe \\texttt{CIAO}\\xspace cache control logic controls the multiplexer based on the isolation flag bit (\\texttt{I}) and the result of checking cache tags associated with accessing L1D cache or shared memory serving as cache.\nWe also augment an extra field with each MSHR entry to store the shared memory address of a memory request from the aforementioned address translation unit.\nOnce the shared memory issues a fill request after a miss, the request reserves one MSHR entry by filling in its global and translated shared memory addresses. \nIf the response from L2 cache matches the global address recorded in the corresponding MSHR entry, the filling data can be directly stored in the shared memory based on the translated shared memory address.\n\n\n\\noindent \\textbf{Performance optimization and coherence.}\nWhen \\texttt{CIAO}\\xspace redirects memory requests of an interfering warp from L1D cache to shared memory, the shared memory does not have any data. \nThis can incur (1) performance degradation because of cold misses and (2) some coherence issues.\nTo address these two issues,\nwhen \\texttt{CIAO}\\xspace needs to access the shared memory, the cache controller first checks the tag array of L1D cache.\nIf a target data resides in L1D cache (not in shared memory), the L1D cache will evict the data directly to the response queue, \nwhich is used to buffer the fetched data from L2 cache and invalidate the corresponding cache line in L1D cache. \nNote that checking the tag array and accessing L1D cache are serialized as described in Section~\\ref{sec:sm_arch}.\nMeanwhile, the shared memory issues a fill request to MSHR, as the shared memory does not have the data yet. \nDuring this process, the target data will be directly fetch from the response queue to the shared memory (\\textit{cf}\\onedot} \\def\\Cf{\\textit{Cf}\\onedot Figure~\\ref{fig:shm}a) \nIn this way, we naturally migrate data from L1D cache to shared memory, hiding the penalty of cold cache misses and coherence issues.\n\n\\begin{algorithm}[t]\n\\scriptsize\n\\DontPrintSemicolon\ni := getWarpToBeScheduled()\\;\nInstNo := getNumInstructions()\\;\nActiveWarpNo := getNumActiveWarp()\\;\n\\uIf{Warp(i).V == 0 \\textbf{and} end of low cut-off epoch}{\n\t\\tcc{Warp(i) is stalled}\n\tk := Pair\\_List[i][1]\\;\n\t$IRS_k$ := $^{VTAHit[k]}\/_{InstNo\/ActiveWarpNo}$\\;\n    \\uIf{$IRS_k$ \\textgreater low-cutoff \\textbf{and} Warp(k) needs executing}{\n        \\textbf{continue}\\;\n    }\n    \\uElse{\n        Warp(i).V := 1\\;\n        Pair\\_List[i][1] := -1 \\tcp{cleared}           }                         }\n\\uElseIf{Warp(i).I == 1 \\textbf{and} end of low cut-off epoch}{\n\t\\tcc{Warp(i) redirects to access shared memory}\n\tk := Pair\\_List[i][0]\\;\n\t$IRS_k$ := $^{VTAHit[k]}\/_{InstNo\/ActiveWarpNo}$\\;\n    \\uIf{$IRS_k$ \\textgreater low-cutoff \\textbf{and} Warp(k) needs executing}{\n        \\textbf{continue}\\;\n    }\n    \\uElse{\n        Warp(i).I := 0\\;\n        Pair\\_List[i][0] := -1 \\tcp{toggling}           }                         }\n\\uIf{Warp(i).V == 1 \\textbf{and} end of high cut-off epoch }{\n\t\\tcc{Warp(i) is active}\n\t$IRS_i$ := $^{VTAHit[i]}\/_{InstNo\/ActiveWarpNo}$\\;\n\tj := Interference\\_List[i]\\;\n\t\\uIf{$IRS_i$ \\textgreater high-cutoff \\textbf{and} $j$ != $i$ }{\n\t\\uIf{ Warp(j).I == 1}{\n\t\tWarp(j).V := 0\\;\n\t\tPair\\_List[j][1] := i\\;\n\t}\n\t\\uElseIf{ Warp(j).I == 0}{\n\t\tWarp(j).I := 1\\;\n\t\tPair\\_List[j][0] := i\\;\n\t}\n     \n     \n         \n    \n\t}\n}\n\\caption{\\texttt{CIAO}\\xspace scheduling algorithm}\n\\label{algo:CIAO}\n\\end{algorithm}\n\n\\subsection{Putting It All Together}\n\\label{sec:pat}\n\\noindent\nAlgorithm~\\ref{algo:CIAO} describes how \\texttt{CIAO}\\xspace schedules warps. \nFor every \\texttt{low-cutoff} epoch, the warp at the front of the warp list (\\textit{e.g}\\onedot} \\def\\Eg{\\textit{E.g}\\onedot, warp $i$), is examined \nto decide whether \\texttt{CIAO}\\xspace redirects memory requests of warp $i$ back to L1D cache or reactivate warp $i$.\nMore specifically, \\texttt{CIAO}\\xspace first checks the first or second field of the \\textit{pair list} entry corresponding to warp $i$. \nOnce \\texttt{CIAO}\\xspace confirms that either \\texttt{CIAO}\\xspace previously redirected memory requests of warp $i$ to shared memory or stalled warp $i$ because warp $i$ severely interfered with another warp (\\textit{e.g}\\onedot} \\def\\Eg{\\textit{E.g}\\onedot, warp $k$),\nit redirects the memory requests of warp $i$ back to L1D cache or reactivate warp $i$,\nunless the following two conditions are satisfied: \n(1) $IRS_k$ is still higher than \\texttt{low-cutoff} and (2) warp $k$ has not completed its execution.\n\nEvery \\texttt{high-cutoff} epoch, \\texttt{CIAO}\\xspace examines $IRS_i$.\nIf warp $i$ is in the active warp list and $IRS_i$ is higher than \\texttt{high-cutoff}, \n\\texttt{CIAO}\\xspace looks up the \\textit{interference} list to determine which warp has most severely interfered with warp $i$. \nOnce \\texttt{CIAO}\\xspace determines the most interfering warp (\\textit{e.g}\\onedot} \\def\\Eg{\\textit{E.g}\\onedot, warp $j$) for warp $i$, \n\\texttt{CIAO}\\xspace checks whether it has redirected memory requests of warp $j$ to shared memory or stalled warp $j$. \nIf \\texttt{CIAO}\\xspace sees that warp $j$ has still sent memory requests to L1D cache, it isolates warp $j$, redirects memory requests of warp $j$ to shared memory,\nand records warp $i$ in the first field of the pair list entry corresponding to warp $j$ to indicate that warp $i$ has triggered to redirect memory requests of warp $j$. \nIf \\texttt{CIAO}\\xspace has already redirected memory requests of warp $j$, then \n\\texttt{CIAO}\\xspace starts to stall warp $j$ and records warp $i$ in second field of the pair list entry corresponding to warp $j$.\nThis record can be referenced when \\texttt{CIAO}\\xspace decides to reactivate warp $i$ in future.\n\n\n\n\n\n\n\n\n\\section{#1}}\n\\newcommand{\\subsect}[1]{\\subsection{#1}}\n\\newcommand{\\subsubsect}[1]{\\subsubsection{#1}}\n\\newcommand{\\mysect}[1]{\\subsect{#1}}\n\n\n\\newtheorem{ourtask}{Task}\n\n\\subsection{Cache Interference Detection}\n\\label{sec:interference_detection}\n\\noindent\nAs introduced in Section~\\ref{sec:interfere}, some warps incur more severe cache interference than other warps (\\textit{i.e}\\onedot} \\def\\Ie{\\textit{I.e}\\onedot, non-uniform cache interference). \nHowever, it is non-trivial to capture such non-uniform interference occurring during the execution of applications \nat compile time \\cite{chenadaptive}.\nThus, we need to determine severely interfering and interfered warps at runtime.\n\n\nAt run time, we may track severely interfered warps, leveraging a VTA structure (\\textit{cf}\\onedot} \\def\\Cf{\\textit{Cf}\\onedot Section~\\ref{sec:vta}).\nA na\\\"ive way to determine severely interfering warps for each warp, however, demands a high storage cost, \nbecause each warp needs to keep track of cache misses incurred by all other $n-1$ warps.\nThis in turn requires a storage structure with $n(n-1)$ entries where $n$ is the number of active warps per SM (\\textit{i.e}\\onedot} \\def\\Ie{\\textit{I.e}\\onedot, 48 warps). \nSearching for a cost-effective way to determine severely interfering warps, we exploit our following observation on an important characteristic of cache interference.\n\n\\begin{figure}\n\\centering\n\\subfloat[]{\\label{fig:unbalance_arrow}\\rotatebox{0}{\\includegraphics[width=0.34\\linewidth]{figs\/unbalance_arrow}}}\n\\subfloat[]{\\label{fig:kmeans_un1}\\rotatebox{0}{\\includegraphics[width=0.61\\linewidth]{figs\/kmeans_un1}}}\n\n\\subfloat[]{\\label{fig:sat_counter}\\rotatebox{0}{\\includegraphics[width=0.92\\linewidth]{figs\/sat_counter}}}\n\\caption{(a) Warps interfering with warp W34 and their interference frequency.\n(b)  Min and max interference frequencies experienced by each warp and each evaluated workload.  (c) Interference detection example.}\n\\end{figure}\n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=1\\linewidth]{figs\/toggle_thrott_exam.eps}\n\\caption{\\texttt{CIAO}\\xspace execution flow.}\n\\label{fig:thrott_exam}\n\\end{figure*}\n\n\n\n\\newedit{ Figure~\\ref{fig:unbalance_arrow} shows that \\texttt{W32} interferes with \\texttt{W34}, more than two thousand times, whereas some warps (\\textit{e.g}\\onedot} \\def\\Eg{\\textit{E.g}\\onedot, \\texttt{W2}) do not interfere with \\texttt{W34} at all in \\texttt{KMEANS}~\\cite{che2009rodinia}; \nwe observe a similar trend on cache interference in all other benchmarks that we tested (cf. Figure~\\ref{fig:kmeans_un1}). }\nObserving such an interference characteristic, we propose to track only the most recently and frequently interfering warp for each warp.\nThis significantly reduces the storage cost required to track every interfering warp for each warp.\nSpecifically, \\texttt{CIAO}\\xspace keeps a small memory structure denoted by \\textit{interference list} \nwhere each entry is indexed by the WID of a currently executed warp.\n\nTo track the most recently and frequently interfering warp for a currently executed warp, \nwe may augment each list entry with a 2-bit saturation counter.\nFigure~\\ref{fig:sat_counter} illustrates how \\texttt{CIAO}\\xspace utilizes the counter to track an interfering warp. \nSuppose that a previously executed warp (\\texttt{W32}) interfered with a currently executed warp (\\texttt{W34}), \nThat is, \\texttt{W32} is an interfering WID and \\texttt{W34} is an interfered WID.\nSubsequently, the interfering WID is stored in the list entry indexed by the interfered WID,\nand the counter in the list entry is set to \\texttt{00};\nthe interfering WID is provided by a VTA entry field that tracks which warp incurred the last eviction (\\textit{cf}\\onedot} \\def\\Cf{\\textit{Cf}\\onedot Section~\\ref{sec:vta}).\n\n\nWhenever \\texttt{W32} interferes with \\texttt{W34} (not shown in the figure), the counter is incremented by 1.\nSuppose that the counter has already reached \\texttt{11} (\\redcircled{\\small{1}}) at a given cycle. \nWhen another warp (\\texttt{W42}) interferes with warp \\texttt{W34} in a subsequent cycle, the counter is decremented by 1 (\\redcircled{\\small{2}}). \nThen, if warp \\texttt{W32} interferes with \\texttt{W34} again, the counter is incremented by 1 (\\redcircled{\\small{3}}). \nThe interfering WID in the list entry is replaced with the most recent interfering WID only when its saturation counter is decreased to \\texttt{00}, \nso that the warp with most frequent cache interference can be kept in the interference list. \n\n\n\n\\subsection{CIAO On-Chip Memory Architecture}\n\\label{sec:warp_partitioning}\n\\noindent\nAn effective way to reduce cache interference is to isolate cache accesses of interfering warps from those of interfered warps \nafter partitioning the cache space and allocating separate cache lines to the interfering warps. \nPrior work proposed various techniques to partition the cache space for CPUs (\\textit{e.g}\\onedot} \\def\\Eg{\\textit{E.g}\\onedot, \\cite{qureshi2006utility, srikantaiah2008adaptive}). \nHowever, the size of L1D cache is insufficient to apply such techniques for GPUs, \nas the number of GPU threads sharing L1D cache lines is very large, compared with that of CPU threads. \nFor example, only two or three cache lines can be allocated to each warp, if we apply a CPU-based cache partitioning technique to the L1D cache of GTX480.\nSuch a small number of cache lines per warp can even worsen cache thrashing. \n\n\n\\newedit{Meanwhile, we observe that programmers prefer L1D cache rather than shared memory for programming simplicity and the limited number of running GPU threads constrains the usage of shared memory, leading to \na large fraction of shared memory unused \n(\\textit{cf}\\onedot} \\def\\Cf{\\textit{Cf}\\onedot $F_{smem}$ of Table~\\ref{tab:workload_charac} in Section~\\ref{sec:method}).\nThis agrees to prior work's analysis~\\cite{hayes2014unified, virtualthread}. }\nExploiting such unused shared memory space, \nwe propose to redirect memory requests of severely interfering warps to the unused shared memory space.\n\n\nAs there is no cache interference at the beginning of kernel execution, \nmemory requests of all the warps are directed to L1D cache, as depicted in Figure~\\ref{fig:thrott_exam}a. \nHowever, as the kernel execution progresses, cache accesses begin to compete one another to acquire specific cache lines in L1D cache. \nAs the intensity of cache interference exceeds a threshold, \\texttt{CIAO}\\xspace determines severely interfering warps (\\textit{cf}\\onedot} \\def\\Cf{\\textit{Cf}\\onedot Section~\\ref{sec:interference_detection}).\nSubsequently, \\texttt{CIAO}\\xspace redirects memory requests of these interfering warps to unused shared memory space, \nisolating the interfering warps from the interfered warps in terms of cache accesses, as depicted in Figure~\\ref{fig:thrott_exam}b.\nThis in turn can significantly reduce cache contentions without throttling warps (\\textit{i.e}\\onedot} \\def\\Ie{\\textit{I.e}\\onedot, hurting TLP). \n\\newedit{After the redirection, the memory requests are forwarded from L1D cache to shared memory but the data may already present in the L1D cache (\\textit{cf}\\onedot} \\def\\Cf{\\textit{Cf}\\onedot W3\/D3 in Figure~\\ref{fig:thrott_exam}b). To guarantee cache coherence between L1D cache and shared memory, single data copy needs to be exclusively stored in either shared memory or L1D cache. \nSuch challenge can be addressed by migrating the data copy from L1D cache to shared memory, which may take the steps as follows: 1) a data miss signal would be raised for shared memory, 2) the data copy in L1D cache would be evicted to response queue, and 3) a new entry of MSHR would be filled with the pointer referring to the location of single data copy in the response queue. Later on, to fill the data miss, shared memory fetches data from response queue based on the location information recorded in MSHR.\n}\nWhen \\texttt{CIAO}\\xspace detects significant decrease in cache contentions due to a change in cache access patterns or completion of execution of some warps, \nit redirects the memory requests of these interfering warps from shared memory back to L1D cache (\\textit{cf}\\onedot} \\def\\Cf{\\textit{Cf}\\onedot Figure~\\ref{fig:thrott_exam}c).\n\n\nTo exploit the unused shared memory space for the aforementioned purpose, however, there are two challenges. \nFirst, the shared memory has its own address space separated from the global memory, \nand there is no hardware support that translates a global memory address to a shared memory address. \nSecond, the shared memory does not have a direct datapath to L2 cache and main memory~\\cite{jamshidi2014d}. \nThat is, it always receives and sends data only through the register file.\nTo overcome these limitations, we propose to adapt shared memory architecture as follows. \nFirst, we implement a address translation unit in front of shared memory to translate a given global memory address to a local shared memory address.\nSecond, we slightly adapt the datapath between L1D and L2 caches such that the shared memory can also access L2 cache when the unused shared memory space serves as cache.\n\n\n\\subsection{CIAO Warp Scheduling}\n\\label{sec:warp_throttling}\n\\noindent\nAlthough \\texttt{CIAO}\\xspace on-chip memory architecture\ncan effectively isolate cache accesses of interfering warps from those of interfered warps, its efficacy depends on various run-time factors, such as the number of interfering warps and the amount of unused shared memory space. \nFor example, \nthe interfering warps end up thrashing the shared memory as well when the amount of unused shared memory space\nis insufficient to handle a large number of memory requests from the interfering warps in a short time period (cf. Figure~\\ref{fig:thrott_exam}d).\n\n\nTo efficiently handle such a case, we propose to throttle interfering warps \\textit{only} when it is not effective to redirect memory requests of interfering warps to the shared memory.\nSpecifically, sharing the same cache interference detector used for \\texttt{CIAO}\\xspace on-chip memory architecture,\n\\texttt{CIAO}\\xspace monitors the intensity of interference at the shared memory at runtime.\nOnce the intensity of interference at the shared memory exceeds a threshold, \\texttt{CIAO}\\xspace stalls\nthe most severely interfering warp at the shared memory (\\textit{e.g}\\onedot} \\def\\Eg{\\textit{E.g}\\onedot, \\texttt{W2} in Figure~\\ref{fig:thrott_exam}e). \n\\texttt{CIAO}\\xspace repeats this step until the intensity of interference at the shared memory falls below the threshold. \nAs some warps complete their execution and subsequently the intensity of interference at the shared memory falls below the threshold,\n\\texttt{CIAO}\\xspace starts to reactivate the stalled warp(s) in the reverse order to keep high TLP and maximize the utilization of shared memory (\\textit{cf}\\onedot} \\def\\Cf{\\textit{Cf}\\onedot Figure~\\ref{fig:thrott_exam}f). \n\n\nNote that \\texttt{CIAO}\\xspace warp scheduling shares the same interference detector with \\texttt{CIAO}\\xspace on-chip memory architecture, instead of \nkeeping two separate interference detectors for L1D and shared memory, respectively. \nThis is because isolated interfering warps do not compete L1D cache with warps that exclusively access L1D cache, and memory accesses of isolated interfering warps often interfere with one another. \nIn other words, L1D cache and shared memory interferences do not affect each other. \nHence, L1D cache and shared memory can share the same VTA array to detect interferences. \n\n\\section{Introduction}\n\\label{sec:introduction}\n\\input{introduction}\n\n\\section{Background}\n\\label{sec:background}\n\\input{background}\n\n\\section{Architecture and Scheduling}\n\\label{sec:overview}\n\\input{overview}\n\n\\section{Implementation}\n\\label{sec:implementation}\n\\input{implementation}\n\n\\section{Evaluation}\n\\label{sec:result}\n\\input{evaluation}\n\n\\section{Discussion and Related Work}\n\\label{sec:relatedwork}\n\\input{related}\n\n\n\\section{Conclusion}\n\\label{sec:conclusion}\n\\input{conclusion}\n\n\\section{Acknowledgement}\n\\label{sec:ack}\n\\input{acknowledge}\n\n\n\\bibliographystyle{IEEEtran}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\subsection{Variance patterns in space-time\\label{mot}}\n\n\nIn many fields of science interest lies on extreme events such as large temperatures or ozone crossing a threshold. Often these processes are observed over space and time and common characteristics are non-normality of observations, presence of outliers or non-constant variance. These characteristics are even more noticeable if data are obtained through long temporal windows, in which case it is often unrealistic to assume that variances are constant for the whole period. In the context of environmental applications, even if seasonality is accounted for, it is rather common to observe changes in variance depending on the influence of air flows or ocean currents. This heterogeneity when not considered in the modelling might lead to poor predictions in out-of-sample locations or future time points.\n\n\nTo illustrate this characteristic of environmental processes, consider the daily maximum ozone data in the United Kingdom observed across 61 locations (Panel (a) of Figure \\ref{figUK1}) from March to November of 2017. This period was chosen because it comprises  the highest levels of ozone (Panel (b) of Figure \\ref{figUK1}). Ground level ozone is created by chemical reactions when pollutants emitted by cars, industry, to mention a couple of examples, react with sunlight. Moreover, high levels of ozone can also be found in rural areas due to wind transportation. It is well known that high levels of ozone can be harmful to human health and this problem has motivated several new modelling developments over the last years.\n \nWe  start by fitting a multivariate dynamic linear model (MDLM) to this data \\citep{West97}. The mean structure of the MDLM  includes time varying effects of latitude, longitude, daily mean temperature and wind speed. In space, we assume a Cauchy correlation function \\citep{Gneit00}, that is, $c(s,s')=   \\left[1+ \\left( ||s-s'||\/\\phi \\right)^{\\alpha} \\right]^{-1}$ with \n$s,s'$ any two locations in $D$, $\\phi >0 $ the spatial range parameter and $\\alpha$ the shape parameter.\nPanels (c) to (f) of Figure \\ref{figUK1} show temporal and spatial residuals based on the MDLM fitting. Panel (c) presents the residual temporal precision, whereas panel (d) shows the scatter plot of wind speed versus the residual precision. It is clear that there is some temporal structure left in the residual of this fitted MDLM. Panels (e) and (f), on the other hand, show the spatial residual precision after fitting the MDLM. It is clear that there are smaller residual precisions in the south-eastern portion of the region, and a non-linear relationship of the spatial precision with latitude.\n\nIn this data the heterogeneity is mostly due to volatility in time with peaks of small precision (and large variance) in the months of June and July. This suggests that the proposed model should account for these patterns\nto explain the volatility of ozone observed across the different locations.\nIn what follows we review some attempts to treat the volatility in spatiotemporal applications and present our proposed approach based on modelling the variance laws through a dynamic linear model.\n\n\n\\begin{figure}[H]\n\\centering\n\\begin{tabular}{cc}\n\\includegraphics[width=5cm]{MapUKfull.pdf} &\n\\includegraphics[width=5cm]{MeanTimeUKmax.pdf} \\\\\n(a) UK map and spatial locations. & (b) Empirical temporal mean. \\\\\n \\includegraphics[width=5cm]{PrecisionTimeUKmaxResNew.pdf}&   \\includegraphics[width=5cm]{VarianceTemporalWindUKmaxRes.pdf}\\\\\n\\\\\n{(c) Residual temporal precision.} & {(d) Residual temporal precision versus wind.} \\\\\n \\includegraphics[width=5cm]{MapVarSpaceUK.pdf}&   \\includegraphics[width=5cm]{VarianceLatitudeUKmaxRes.pdf}\\\\\n\\\\\n{(e) Residual spatial precision.} & {(f) Residual spatial precision versus latitude.} \\\\\n\n\\end{tabular}\n\\caption{Data summaries for the ozone data observed over the UK. Panel (a) displays the UK map with the training locations (solid circles) and the testing locations ($\\bf{\\times}$). Panel (b) presents the empirical mean over the year. Panels (c)-(f) present the precision over space and time of the residuals based on the fitting of a multivariate dynamic linear model.\\label{figUK1}\n}\n\\end{figure}\n\n\\clearpage\n\n\\subsection{Related literature}\n\nSeveral papers have investigated the presence of patterns in the variance of spatiotemporal processes and its effects on the predictive performance of the process of interest.\n\\cite{Stein09} discusses the presence of peaks in the temporal variance in the modelling of atmospheric pressure even after including altitude in the mean function. In particular, the author suggests that the observed patterns is possibly due to the passage of weather fronts over the region. Often transformations such as the log or squared root are applied to the data aiming to stabilise the variance \\citep{DeOliveira97,Johns03} or to account for truncated domains \\citep{Allcroft03}. Recently, \\cite{Gent17} proposed to add flexibility to the usual transformed Gaussian fields by considering a large family of possible transformations. However, the transformation approaches will not result in reasonable predictions if changes in variance have a pattern over time. That is, in many applications, even after fitting a Gaussian process to the data, the residuals still present varying variances which might depend on covariates which were already included in the mean function \\citep[see e.g. ][]{Bueno2017}.\nMoreover,\nthe transformation approach may have difficult interpretations and may obscure the relationship between the response and the covariates \\citep[see][for an example]{Bolin15}. In these situations, keeping the observations in their original scale and modelling the variance laws is an appealing alternative. \n\n\n\\cite{Gelf2005} constructed a spatial model based on mixtures via a Dirichlet process which is non-stationary and non-Gaussian. \\cite{Ge07} extend this idea to allow a random surface to be selected  in each site based on latent covariates. The approach is non parametric and replications are required for full inference, in which case dynamical models are considered to model temporal dependence.\n\nTo account for outliers \\cite{Bai15} \npropose an estimator to robustify the kriged Kalman filter, extending the spatio-temporal approach of \\cite{Mardia98} which is highly affected by outlying observations. \\cite{bevil20} propose a skew-t model for geostatistical data aiming to accommodate fat tails and asymmetric marginal distributions.\n\nIn the context of variance modelling, \\cite{PSteel06} propose a non-Gaussian process for geostatistical data which accommodates fat tails by scale mixing a Gaussian process; the Gaussian model is a limiting case. This approach was extended by \\cite{Fons11} to account for non-gaussianity in spatio-temporal processes. The proposed model for the variance is the product of two separable mixing processes, one in space by another in time and both are assumed continuous.\n\\cite{Bueno2017} extended \\cite{Fons11}  by allowing the use of covariate information\nto explain the spatial patterns observed in the variances, and time is also assumed to vary continuously in $\\mathbb{R}_+$.\n More recently, \\cite{Tadayon2018} propose a modelling approach that considers the use of covariates in the measurement error and can capture the effects of the skewness and heavy tails for datasets with non-Gaussian characteristics. \\cite{Chu18} consider hierarchical modelling of Student-t processes with heterogeneous variance. The dynamic mean and variances depend on the lagged observations in time instead of past states.  \n\nNote that, if time is assumed to be continuous in the variance model, correlation matrices will have large dimension and inference becomes too costly for reasonably long temporal windows. Thus, to allow for computational feasibility of real data applications, {different from \\cite{Fons11}}, this paper considers discrete time and dynamic linear models for the spatio-temporal variance process.\nThis proposal modifies the well known multivariate dynamic linear model (MDLM) \\citep{West97} which assumes Gaussianity to account for heterogeneity in spatio-temporal data analysis by modelling the variance laws over space and time. In the context of temporal evolution of variances, \\cite{Uhlig94} extends the usual Gaussian dynamic model by including a sequential evolution for the precision matrix by assuming a Matrix-Beta evolution. An alternative specification considers the Wishart sequential filtering for the variance matrix. \\cite{Liu00b} presents further discussion and model implementations for these proposals. In the context of more flexible state space models, \\cite{Liu00} propose a conditional dynamical model specification which allows for non-Gaussian errors\naccounting for outliers.\nHowever, the model does not consider possible patterns in the variance model and the distributions are the same over time. \n\n\\cite{West97} proposed to model variance laws  \nby letting the observational variance to be a function of a known weight. In the context of spatial data, it is reasonable to assume that the weighting depends on  Euclidean distances and smoothness in space should be ensured.\nIn the usual multivariate dynamical modelling approach, the variance may vary stochastically as an inverse Gamma (or inverse Wishart) distribution, in which case the resulting sampling distribution for the response is Student-t. However, this extension is not flexible enough to capture spatial heterogeneity as discussed in \\cite{PSteel06} and \\cite{Fons11}. In our proposed solution to this issue, the variance is assumed to vary according to a log-Gaussian process \\citep{PSteel06} and the mixing distribution varies in discrete-time assuming smooth transitions. Besides, the variance laws are allowed to depend on covariates. In this case, recurrence equations for filtering and smoothing are presented for the variance process allowing for feasible computations even for large temporal windows.\n\n\n\n\nThe remaining of the paper is organised as follows. Section \\ref{sec:propmodel} describes the proposed model and its properties. In particular, Sections \\ref{sec:2.2} and \\ref{sec:2.3} describe the inference and prediction procedures for dynamical spatial modelling over time with stochastic variance.\nSections \\ref{sec:real} and  \\ref{sec:real2} present the analysis of the maximum temperature in the Spanish Basque Country and the  maximum ozone levels in the United Kingdom, respectively. Different models are fitted and these analyses illustrate the effectiveness of our proposal in modelling varying variances over both time and space and the improvement it provides in the precision of predictions.  \nSection \\ref{sec:conclusion} concludes with some discussion. Some simulated examples are presented in  Appendix \\ref{ApSimD} to verify that our proposed predictive comparison measures indicate the correct data generating models and do not result in overfitting.\n\n\n\n\\section\nNon-Gaussian state-space modelling }\\label{sec:propmodel}\n\nThis section extends the multivariate Gaussian dynamic model by allowing for stochastic variance over space and time.\n\n\\subsection{Spatial Dynamic Linear Models with stochastic variance \n}\\label{sec2.2}\n\n\n\n\n\n\n\n\n\n\nConsider $\\{Z_t(\\mathbf{s}): \\mathbf{s} \\in D\\subseteq \\mathbb{R}^d, \\, \nt\\in T\\subseteq \\mathbb{Z}\\}$ a spatio-temporal random field. We assume $Z_t(\\mathbf{s})$ follows a spatial mixture model, that is,\n\n\n\n\\begin{equation}\\label{model:eq2}\nZ_t(\\bm{s}) = \\mathbf{x}_t(\\mathbf{s})' \\boldsymbol{\\theta}_t + \\sigma \\frac{{\\epsilon}_t(\\mathbf{s})}{\\sqrt{\\lambda_t(\\mathbf{s})}}+\\tau  \\rho_t(\\mathbf{s}), \n\\end{equation}\nwhere $\\mathbf{x}_{t}(\\mathbf{s})$ is a vector of observed covariates, $\\bm{\\theta}_t$ is a vector of time varying regression coefficients, $\\sigma$ is a scale parameter and $\\tau$ is a nugget effect. The process  ${\\epsilon}_t(\\cdot)$ is a zero mean Gaussian process with correlation function $c(\\cdot,\\cdot)$,\n$\\rho_t(\\cdot)$ is an uncorrelated Gaussian noise with zero mean and unit variance responsible for small scale variation and $\\lambda_t(\\mathbf{s})$ is a mixing process. Conditionally on the mixing process $\\lambda_t(\\cdot)$ and on the coefficients $\\boldsymbol{\\theta}_t$, the process $Z_t(\\cdot)$ has mean function $m_t(\\mathbf{s})=\\mathbf{x}_t(\\mathbf{s})'\\boldsymbol{\\theta}_t$ and covariance function $K(\\mathbf{s},\\mathbf{s}')=\\sigma^2c(\\mathbf{s},\\mathbf{s}')\/\\sqrt{\\lambda_t(\\mathbf{s})\\lambda_t(\\mathbf{s}')}$, $\\mathbf{s},\\mathbf{s}'\\in D$, $t\\in T$. For $\\lambda_t(\\mathbf{s})\\not = 1$ the process $Z(\\mathbf{s})$ has heterogeneous spatiotemporal variance and if $\\lambda_t(s)$ is integrated out the resulting process is non-Gaussian. If $\\lambda_t(\\mathbf{s})=1$ and an evolution state equation is assumed for ${\\boldsymbol{\\theta}_t}$ then the resulting model is the usual Gaussian Dynamic Linear Model \\citep{West97}.\n\n\nIn the sequel, we discuss the mixing process specification which is the crucial part of this spatiotemporal mixture model. Assuming $\\lambda_t(\\mathbf{s}) =\\lambda \\sim {\\rm Gamma}(v\/2, v\/2)$, $\\thinspace \\forall \\thinspace \\mathbf{s} \\in D$ implies that the distribution of $Z_t(\\cdot)$ is a Student-t process \\citep{Omre06} with $v$ degrees of freedom. \\cite{PSteel06} discuss the limitations of assuming a Student-t process for spatial observations. In short, the Student-t process is not able to account for spatial heterogeneity as it inflates the variance of the whole process whenever outliers or spatial heterogeneity is observed. \nOn the other hand, if we assume the Gaussian-log-Gaussian (GLG) model proposed by \\cite{PSteel06}, then ${\\rm ln} \\thinspace[ \\lambda_t(\\cdot)]$ is a Gaussian Process with mean $-\\nu\/2$ and covariance function $\\nu c(\\cdot,\\cdot)$ such that ${\\rm E}[\\lambda_t(\\mathbf{s}) ] = 1$, ${\\rm Var}[\\lambda_t(\\mathbf{s})] = e^{\\nu} -1$ and the kurtosis of the process $Z_t(\\cdot)$ is $3e^{\\nu}$ which is controlled by $\\nu$. This implies that the marginal distribution of $\\lambda_t(\\mathbf{s})$ is concentrated around one for very small value of $\\nu$ (of the order $\\nu= 0.01$) and as $\\nu$ increases, the distribution becomes more spread out and more right-skewed, while the mode becomes zero.\nOur proposed model extends the GLG specification by defining a model for $\\lambda_t(\\mathbf{s})$ through state space equations which assume that, conditionally on state parameters, the variances are independent in time, resulting in computationally efficient estimation algorithms. This approach takes advantage of the recurrence equations of DLM while accounting for more flexible variance laws for spatiotemporal data. \n\n\nNext we specify the dynamical evolution for both the mean and variance states and we discuss the connection between the usual Gaussian dynamic spatial model and the proposed non-Gaussian extension in equation (\\ref{model:eq2}). Let $\\mathbf{Z}_t= (Z_t(\\bm{s}_1), \\ldots, Z_t(\\bm{s}_n))'$ be the data collected at $n$ spatial locations in $D$. Conditional on the latent variables $\\Lambda_t=diag({\\lambda}_t(\\bm{s}_1),\\ldots,{\\lambda}_t(\\bm{s}_n))$, the observation and system equations obtained by integrating $\\epsilon_t(s)$ and $\\rho_t(s)$ out are given by\n\\begin{subequations}\\label{eq5}\n\\begin{equation}\\label{eq5a}\n \\mathbf{Z}_t\\mid \\boldsymbol{\\theta}_t,\\Lambda_t \\sim N\\left ( \\bm{F}_t' \\boldsymbol{\\theta}_t,\\;\\sigma^2  \\Lambda_t^{-1\/2}C_{{\\bm \\psi}}\\Lambda_t^{-1\/2}+\\tau^2I_n\\right ),\n \\end{equation}\n \\begin{equation}\\label{eq5b}\n \\boldsymbol{\\theta}_t\\mid \\boldsymbol{\\theta}_{t-1} \\sim N\\left ( { G}_t \\boldsymbol{\\theta}_{t-1} ,W_t\\right ),\n \\end{equation}\n\\end{subequations}\nwhere $\\bm{F}_t=(\\mathbf{x}_t(\\bm{s}_1),\\ldots,\\mathbf{x}_t(\\bm{s}_n))$ is the $p \\times n$ design matrix with observed $p$ covariates,  $\\bm{\\theta}_t$ is the $p-$dimensional state vector,  $C_{{\\bm \\psi}}$ represents the correlation matrix with elements computed by $C_{\\bm {\\psi},ij}=c(\\bm{s}_i,\\bm{s}_j)$ that depends on parameters ${\\bm \\psi}$ and the Euclidean distance among locations, $G_t$ represents the evolution matrix and $W_t$ is a $p$-dimensional covariance matrix of the states.\nEquation \\eqref{eq5b} defines the temporal evolution of state variables in the mean function and the smoothness of this evolution is controlled by $W_t$.\n\n\n\n\nWe now focus on the specification of the spatio-temporal mixing process $\\lambda_t(\\bm{s})$, $\\bm{s}\\in D$, $t \\in T$. To keep the model parsimonious,\n we define $\\lambda_t({\\bm s}) = \\lambda_1(\\bm{s})\\lambda_{2t}$ as a separable process.\nThe mixing distributions and the evolution equation for the state space parameters in the variance model are defined as \n\\begin{eqnarray}\\label{eq:lambda1}\n{\\rm ln}(\\mathbf{\\boldsymbol{\\lambda}}_1) & \\sim &   N \\left ( -\\frac{\\nu_1}{2}\\bm{1}_n+\\bm{F}_{1}'\\boldsymbol{\\beta},\\nu_1 C_{{\\bm{\\xi}}}\\right ), \n\\end{eqnarray}\n\\begin{subequations}\\label{eqL}\n\\begin{equation}\\label{eq:lambda21}\n{\\rm ln}(\\lambda_{2t}) =    \\bm{F}_{2t}'\\boldsymbol{\\eta}_t+v_{2t},  \\; v_{2t}\\sim N \\left ( -\\frac{\\nu_2}{2},\\nu_2 \\right ), \n\\end{equation}\n\\begin{equation}\\label{eq:lambda22}\n\\boldsymbol{\\eta}_t  = G_{2t}\\boldsymbol{\\eta}_{t-1}+\\mathbf{\\omega}_{2t}, \\; \\mathbf{\\omega}_{2t}\\sim N \\left ( 0,W_{2t} \\right ),\n\\end{equation}\n\\end{subequations}\nwhere, in equation (\\ref{eq:lambda1}), ${\\rm ln}(\\mathbf{\\boldsymbol{\\lambda}}_1)= ({\\rm ln}(\\lambda(\\bm{s}_1), \\ldots, {\\rm ln}(\\lambda(\\bm{s}_n))'$ and $C_{{\\bm \\xi}}$ the spatial correlation matrix that depends on parameter $\\bm{\\xi}$ and the Euclidean distance between locations. Note that $C_{\\bm{\\xi},ij}=c^*(\\bm{s}_i,\\bm{s}_j)$ which could differ from $c(\\bm{s}_i,\\bm{s}_j)$, that is, in the spatio-temporal context it is possible to estimate a different correlation structure for the process $\\epsilon_t(\\cdot)$ and the process ${\\rm ln}[\\lambda_1(\\cdot)]$. In equation \\eqref{eq:lambda1}, $\\bm{F}_{1}=(\\bm{\\tilde x}(\\bm{s}_1),\\ldots, \\bm{\\tilde x}(\\bm{s}_n))$  is a $p_1\\times n$ design matrix that will allow for the effect of covariates in the spatial variance, and $\\boldsymbol{\\beta}$ is a $p_1$-dimensional vector of coefficients to be estimated. In equation \\eqref{eq:lambda21}, $\\bm{F}_{2t}=\\bm{x}_{t}^*$ is a $p_2$-dimensional vector that will allow for the effect of covariates in the temporal variance. Equation \\eqref{eq:lambda22} defines the temporal evolution of state parameters $\\boldsymbol{\\eta}_t$ in the variance model, with ${W}_{2t}$ controlling the temporal smoothness, and $G_{2t}$ representing the evolution matrix. \n\n\n\n\n\n \n\n\nThe resulting covariance function of  $\\left\\{  Z_t(\\bm{s}): \\bm{s} \\in D; t \\in T \\right\\}$\n, defined in \\eqref{model:eq2}, is obtained by integrating out the mixing processes $\\lambda_1(\\bm{s})$ and $\\lambda_{2t}$.\nIf $t_1=t_2=t$ and $\\bm{s}_1=\\bm{s}_2=\\bm{s}$ we obtain the spatio-temporal variance as \\begin{equation}\n    Var\\left(Z_t({\\bm s})\\mid \\boldsymbol{\\eta}_{1:T},\\boldsymbol{\\theta}_{1:T}\\right) = \\sigma^2 \\thinspace \\exp\\left\\{\\nu_1  +\\nu_2  -\\bm{F}_1'(\\bm{s})\\bm{\\beta}-\\bm{F}_{2t}'\\boldsymbol{\\eta}_{{t}}\\right\\},\n\\end{equation} \nwith $F_1(\\bm{s})=\\tilde{\\bm{x}}(\\bm{s})$ the vector of spatial covariates at site $\\bm{s}\\in D$. The temporal dependence is carried out by the states $(\\boldsymbol{\\theta}_t,\\boldsymbol{\\eta}_t)$, $t=1,\\ldots,T$ and the conditional spatial correlation is given by\n\\begin{equation}\\label{eq:corr}\nCorr\\left[Z_{t}(\\bm{s}_1), Z_{t}(\\bm{s}_2)\\mid \\boldsymbol{\\eta}_{1:T},\\boldsymbol{\\theta}_{1:T}\\right] = C_{{\\bm \\psi}}(\\mathbf{s}_1,\\mathbf{s}_2)\\exp\\left\\{ \\frac{\\nu_1}{4} \\left( C_{{\\bm \\xi}}(\\mathbf{s}_1,\\mathbf{s}_2) -1 \\right) \\right\\}. \n\\end{equation}\n\nThe kurtosis in each location unconditional on $\\lambda_t({\\bm s})$ is given by\n\\begin{equation}\\label{eq:kurt}\nKurt\\left[Z_t({\\bm s})\\right] = 3 \\thinspace \\exp\\left\\{\\nu_1 + \\nu_2  \\right\\}.\n\\end{equation}\nSee Appendix \\ref{ApB} for the proofs of these results. A particular case of the model proposed in equation \\eqref{model:eq2} is obtained for $\\lambda_t(\\mathbf{s})= 1$ and, consequently, the non-Gaussian distribution converges to the Gaussian distribution for small values of $\\nu_1$ and  $\\nu_2$.\n\n\n\\subsection{Resultant posterior distribution and inference procedure}\\label{sec:2.2}\n\nWe follow the Bayesian paradigm to make inference, predictions and model comparisons that are obtained from the joint posterior distribution of the parameters. In particular, we take advantage of the hierarchical structure of our proposal in our iterative estimation algorithm to sample from the joint posterior and to make predictions. In what follows we present the prior, the joint posterior distributions and briefly describe the steps to obtain samples from the posterior distribution.\n\nIn our motivating example, we assume a Cauchy correlation function with range parameter $\\phi>0$ and shape parameter $\\alpha>0$. This function is flexible allowing for long-memory dependence and also correlations at short and intermediate lags.\nWe assume an exponential correlation function for the spatial mixing process $ln[\\lambda_1(\\bm{s})]$ given by $c^*(\\bm{s},\\bm{s}') = \\exp\\left\\{ -||\\bm{s}-\\bm{s}'||\/\\gamma\\right\\}$, where $\\gamma> 0$.  \nModel specification is complete after assigning a prior distribution for the static parameters $\\Phi=(\\sigma^2,\\tau^2,\\nu_1,\\nu_2, {\\boldsymbol{\\beta}}, {\\bm \\psi}=(\\alpha,\\phi), {\\bm \\xi}=(\\gamma))$. We assign vague independent priors to the static parameters in $\\Phi$.\n In particular, we assume $\\sigma^{-2} \\sim Gamma(a_{\\sigma^2},b_{\\sigma^2})$ with small values for $a_{\\sigma^2}$ and $b_{\\sigma^2}$.\nFor the range parameter $\\phi$, we take into account that the prior is critically dependent on the scale of the observed distances among locations. For the Cauchy correlation function, we assign a gamma prior $\\phi$, i.e. $\\phi \\sim Gamma\\left(1, c\/med(d) \\right)$, with $med(d)$ representing the median of observed distances and the shape parameter follows a uniform prior, that is, $\\alpha \\sim U(a_{\\alpha};b_{\\alpha})$. For the exponential correlation function parameter, we assume $\\gamma \\sim Gamma(a_{\\gamma}, b_{\\gamma})$. For the mixing parameters $\\nu_i$, $i=1,2$, we assign a $Gamma(a_\\nu,b_{\\nu})$ prior. Notice that very small values of $\\nu_i$ (around 0.01) lead to approximate normality while large values of $\\nu_i$ (of the order of say 3) suggest very thick tails. \n\nFollowing Bayes' theorem, the posterior distribution of model parameters and latent variables given the observed data, $\\mathbf{z}_t = (z_t(\\bm{s}_1), \\ldots, z_t(\\bm{s}_n))'$, $t=1,\\ldots,J$,  is proportional to \n\\begin{eqnarray}\\label{eq:post} \\nonumber\n   \n    p\\left( \\boldsymbol{\\theta}_{1:J}, \\boldsymbol{\\eta}_{1:J} \\boldsymbol{\\lambda}_{1},\\boldsymbol{\\lambda}_{2}, \\Phi  \\mid \\mathbf{z}\\right) &\\propto& \\prod_{t=1}^{J}  f_{N_{n}}(\\mathbf{z}_t|\\bm{F}'_t\\boldsymbol{\\theta}_t,\\Sigma_t)  \\nonumber \\\\ \n   \n& \\times &  f_{N_{n}}(\\boldsymbol{\\Delta}|{\\bf 0},\\nu_1 C_{\\bm{\\xi}}) \\prod_{t=1}^{J} f_{N_{1}}(L_t|\\bm{F}'_{2t}\\boldsymbol{\\eta}_t,\\nu_2) \\nonumber \\\\ \n   \n    & \\times &  f_{N_{p}}(\\boldsymbol{\\theta}_0\\mid \\bm{m}_0,C_0) \\prod_{t=1}^{J} f_{N_{p}}(\\boldsymbol{\\theta}_t \\mid \\boldsymbol{\\theta}_{t-1},W_t)  \\\\\n & \\times & f_{N_{p_{2}}}(\\boldsymbol{\\eta}_0\\mid \\bm{m}_0^*,C_0^*) \\prod_{t=1}^{J} f_{N_{p_{2}}}(\\boldsymbol{\\eta}_t \\mid \\boldsymbol{\\eta}_{t-1},W_{2t}) \\;  \\pi(\\Phi), \\nonumber\n\\end{eqnarray}\n\\noindent where $f_{N_{p}}(\\cdot\\mid A,B)$ denotes the density function of a $p$-variate multivariate normal distribution with mean A and covariance matrix B, $\\boldsymbol{\\Delta}= ln (\\boldsymbol{\\lambda}_1) +  \\nu_1\/2 \\,  {\\bf 1}_n  {- \\bm{F}_1'\\boldsymbol{\\beta}}$, ${L}_t= ln \\lambda_{2t} + \\nu_2\/2$, $\\Sigma_t = \\sigma^2 \\Lambda_t^{-1\/2} C_{\\bm{\\psi}} \\Lambda_t^{-1\/2} + \\tau^2 I_n$, $\\Lambda_t = diag(\\lambda_1(\\bm{s}_1), \\ldots, \\lambda_J(\\bm{s}_n))$ and  $\\pi(\\cdot)$ the prior distribution of static parameters. Finally, $f_{N_{p}}(\\boldsymbol{\\theta}_0\\mid \\bm{m}_0,C_0)$ and $f_{N_{p_{2}}}(\\boldsymbol{\\eta}_0\\mid \\bm{m}_0^*,C_0^*)$ are the densities for the initial prior information at time $t=0$ for $\\boldsymbol{\\theta}_0$ and $\\boldsymbol{\\eta}_0$, respectively.\n\nThe resultant posterior distribution does not have closed form and we resort to Markov chain Monte Carlo methods \\citep{gamerman} to obtain samples from the posterior. In particular, posterior samples are obtained through a Gibbs sampler algorithm with  steps of the Metropolis-Hastings algorithm for $\\phi$, $\\alpha$, $\\gamma$ and $\\nu_i$, $i=1,2$  which are based on random walk proposals.\n\n\n\n\n\n\n\\paragraph{Brief description of the MCMC algorithm} Conditional on the latent variables $\\boldsymbol{\\lambda}_1$ and $\\boldsymbol{\\lambda}_2$, Gaussianity is preserved and samples from the posterior full conditional distributions for the state vectors $\\boldsymbol{\\theta}_t$ in the mean are obtained through the usual forward filtering and backward smoothing recursions (FFBS) proposed by \\cite{Sylvia1994} and \\cite{Carter1994}.\nAnalogously, conditionally on $\\boldsymbol{\\lambda}_2$, the posterior distribution of states $\\boldsymbol{\\eta}_t$ are also obtained through the FFBS algorithm. \nAppendices \\ref{ApA1} and \\ref{ApA2} provide the equations to run the FFBS for  $\\boldsymbol{\\theta}_t$ and $\\boldsymbol{\\eta}_t$, respectively. Note that, \ndifferent from \\cite{Bueno2017} and \\cite{Fons11}, as we assume time to be discrete, we do not need to rely on computing the inverse of high dimensional  covariance matrices at each iteration of the MCMC. \nConditional on the regression coefficients $\\bm{\\beta}$, the spatial latent mixing variable  $\\boldsymbol{\\lambda}_{1}=(\\lambda_{1}(\\bm{s}_1),\\cdots,\\lambda_{1}(\\bm{s}_n))'$ is sampled as part of a Gibbs algorithm using blocks of random walks or the independent sampler proposed in \\cite{PSteel06}. To sample $\\bm{\\beta}$ the Gibbs step is given by\n\\begin{equation}\n    \\bm{\\beta}\\mid \\bm{\\lambda}_1,\\nu_1,\\bm{\\xi}\\sim N_{p_1}((\\bm{F}_1'C_{\\bm{\\xi}}^{-1}\\bm{F}_1)^{-1}\\bm{F}_1'C_{\\bm{\\xi}}^{-1}(ln \\bm{\\lambda}_1+\\nu_1\/2\\; \\bm{1}_n),\\nu_1(\\bm{F}_1'C_{\\bm{\\xi}}^{-1}\\bm{F}_1)^{-1}).\n\\end{equation}\nA summary of our proposed sampling algorithm is described in Appendix \\ref{ApPostComp}. The algorithm was coded in {\\tt R} using RStudio \nVersion 1.1.442 \\citep{RCoreTeam} and the source code can be obtained at: https:\/\/github.com\/thaiscofonseca\/DynGLG.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Predictions in space-time}\\label{sec:2.3}\n\n\nFor spatial interpolation {for given observed times}, consider the vector $(\\mathbf{Z}_t^{obs}, \\mathbf{Z}_t^{pred})$, with $\\mathbf{Z}_t^{obs}$ and $\\mathbf{Z}_t^{pred}$ representing, respectively, observed and out-of-sample values of $Z_t(\\bm{s})$, at each time $t= 1, \\ldots, J$. Let $\\Phi= ( \\sigma^2,\\tau^2, \\nu_1,\\nu_2, \\bm{\\beta}, \\bm{\\psi},\\bm{\\xi})$ be the static parameters in the proposed model in equation \\eqref{eq5}.\nIn order to obtain samples from the posterior predictive distribution $p(\\mathbf{Z}_t^{pred} \\mid \\mathbf{Z}_t^{obs})$ we resort to composition sampling;  assume that $\\Phi$,\n$\\bm{\\lambda}_{1}^{obs}=(\\lambda_{1}(\\bm{s_1}),\\ldots,\\lambda_{1}(\\bm{s}_n))'$, $\\bm{\\lambda}_{2}^{obs}=(\\lambda_{2,1},\\ldots,\\lambda_{2J})'$, $\\boldsymbol{\\theta}^{obs}=(\\bm{\\theta}_{1},\\ldots,\\bm{\\theta}_J)'$, $\\boldsymbol{\\eta}^{obs}=(\\bm{\\eta}_{1},\\ldots,\\bm{\\eta}_J)'$ were sampled from the joint posterior distribution $p(\\Phi, \\bm{\\lambda}_{1}^{obs},\\bm{\\lambda}_{2}^{obs},\\boldsymbol{\\theta}^{obs},\\boldsymbol{\\eta}^{obs} \\mid \\mathbf{Z}_t^{obs})$.\nThus, samples from $p(\\mathbf{Z}_t^{pred} \\mid \\mathbf{Z}_t^{obs})$ may be obtained by sampling\n\\begin{enumerate}\n    \\item[(i)] ${\\rm ln}(\\bm{\\lambda}_{1}^{pred})\\mid \\bm{\\lambda}_{1}^{obs},\\nu_1, \\bm{\\xi} $ and \\item[(ii)] $\\bm{Z}_t^{pred}\\mid \\bm{Z}_t^{obs}, \\bm{\\lambda}_{1}^{pred},\\bm{\\lambda}_2^{obs},\\bm{\\theta}^{obs},\\Phi$. \n\\end{enumerate}\nBoth distributions in (i) and (ii) are Gaussian, the second is the observational model and the first is given by\n\\begin{equation}\\label{eqpredL1}\n{\\rm ln}(\\bm{\\lambda}_1^{pred}) \\mid \\bm{\\lambda}_1^{obs} , \\nu_1, \\bm{\\xi}  \\sim  N_n \\left[ -\\nu_1\/2 \\; {\\bf 1}_n+{C}_{o,p} {C}_{o,o}^{-1} \\mathbf{a} ;   \\nu_1 \\left({C}_{p,p} - {C}_{p,o}{C}_{o,o}^{-1}{C}_{o,p} \\right)\\right] \n\\end{equation}\n\\noindent with $\\mathbf{a} = \\left(ln(\\bm{\\lambda}_1^{obs}) +\\nu_1\/2 \\; {\\bf 1}_n - {\\bm{F}'_1 \\boldsymbol{\\beta}}\\right)$ and\n${C}_{\\bm{\\xi}} = \\begin{pmatrix} \nC_{p,p} & C_{p,o} \\\\\nC_{o,p} & C_{o,o} \n\\end{pmatrix}$. {This result follows from the properties of the partition of the multivariate normal distribution.}\n\n\nSuppose now that interest lies in forecasting future observations at a set of locations given historical data $\\bm{Z}^{obs}=(Z_{1}^{obs},\\ldots,Z_J^{obs})'$. Consider that at time $J$ we want to predict $h$ instants ahead and $h>0$. We define  $\\bm{\\lambda}_{2}^{pred}=(\\lambda_{2,J+1},\\ldots,\\lambda_{2,J+h})'$. Thus, samples of $\\bm{Z}_t^{pred}$ may be obtained by sampling from\n\\begin{enumerate}\n    \\item[(i)] $\\bm{\\eta}^{pred},\\mid \\bm{\\eta}^{obs},\\bm{\\lambda}_{2}^{obs} $ and $\\bm{\\theta}^{pred}\\mid \\bm{\\theta}^{obs},\\bm{Z}^{obs} $,\n    \\item[(ii)] $ln(\\bm{\\lambda}_{2}^{pred})\\mid \\bm{\\eta}^{pred},\\nu_2 $ and \n    \\item[(iii)] $\\bm{Z}_t^{pred}\\mid \\bm{Z}_t^{obs}, \\bm{\\lambda}_{1}^{obs},\\bm{\\lambda}_2^{pred},\\bm{\\theta}^{pred},\\Phi$. \n\\end{enumerate}\nIf we are predicting in the future for ungauged locations we replace $\\bm{\\lambda}_1^{obs}$  with $\\bm{\\lambda}_1^{pred}$ obtained using equation (\\ref{eqpredL1}). Steps (ii) and (iii) are performed simulating from the variance and observational models which are all conditionally Gaussian distributions. Step (i) depends on the usual forecast distributions available for the Gaussian Multivariate Dynamical Model \\citep{West97}. \n\n\n\n\\paragraph{Model Comparison}\nTo check the predictive accuracy of competing models, measures based on scoring rules are considered. Scoring rules provide summaries for the evaluation of probabilistic forecasts by comparing the predictive distribution with the actual value which is observed for the process \\citep{GneitRaf07}. In particular, we consider the Interval Score, the Logarithmic Predictive Score and the Variogram Score. Note that the Logarithmic Predictive Score and the Variogram Score are multivariate measures for a $d$-dimensional vector. { We briefly describe how to compute each of these criteria. }\\\\\n\n\\noindent \\textit{Interval Score:}  Interval forecast is a crucial special case of quantile prediction \\citep{GneitRaf07}. It compares the predictive credibility interval with the true observed value (validation observation), and it considers the uncertainty in the predictions such that the model is penalised if an interval is too narrow and misses the true value. The Interval Score is given by \n\\begin{equation}\\label{eq:IS}\nIS(u,l;z) = (u-l) + \\frac{2}{\\gamma}(l - z)I_{\\left[z< l\\right]}  + \\frac{2}{\\gamma}( z - u)I_{\\left[z> u\\right]},\n\\end{equation}\n\\noindent where $l$ and $u$ represent for the forecaster quoted $\\frac{\\gamma}{2}$ and $1-\\frac{\\gamma}{2}$ quantiles based on the predictive distribution and $z$ is the validation observation. If $\\gamma=0.05$ the resulting interval has 95\\% credibility.\\\\\n\n\\noindent \\textit{Log Predictive Score:} The log predictive score evaluates the predictive density at the observed validation value $\\bm{z}$. It is given by  \n\\begin{equation}\\label{eq:LPS}\n    LPS(\\bm{z})= -log\\left\\{p(\\bm{z}\\mid \\bm{z}^{obs})\\right\\}.\n\\end{equation}\nThe smaller the log predictive score, the better the model does at forecasting $\\bm{z}^{obs}$.\\\\\n\n\\noindent \\textit{Variogram Score:}\nThe variogram score of order $p$ \\citep{Sche15} was proposed to evaluate forecasts of multivariate quantities. It depends on a matrix $w$ of non-negative weights specified subjectively that allow to emphasize or downweight pairs of observations, for instance, based on Euclidean distances. It is defined as \n\\begin{equation}\n   \\mbox{VS-p}(\\bm{z},\\bm{z}^{obs}) = \\sum_{i,j=1}^{d} w_{ij}\\left (|\\bm{z}^{obs}_{i}-\\bm{z}^{obs}_j|^p-\\frac{1}{m}\\sum_{k=1}^M|\\bm{z}_i^{(k)}-\\bm{z}^{(k)}_j|^p\\right )^2,\n\\end{equation}\nwhere $\\{\\bm{z}^{(k)};\\, k=1,\\ldots,M\\}$ are simulated values from the predictive distribution. The smaller the variogram score, the better the model does at forecasting $\\bm{z}^{obs}$. Empirical studies presented in \\cite{Sche15} suggest that $p=0.5$ leads to good model discrimination, however, if the predictive distribution is skewed, then values of $p<0.5$ may lead to better results.\\\\\n\n\n\\section{Data analysis}\\label{sec4}\n\nThis section presents two data analyses relevant in the discussion about extremes in environmental applications: the first application considers the maximum temperature data in the Spanish Basque Country. These data have been previously analysed by \\cite{PSteel06}, \\cite{Fons11} and \\cite{Bueno2017}. As our proposal is able to account for longer temporal windows than \\cite{Fons11}, the analysis shown in Section \\ref{sec:real} considers one year of daily observations instead of one month as in \\cite{Fons11} and \\cite{Bueno2017}. The second application focuses on the maximum ozone data described in Section \\ref{mot}, which illustrates the use of spatial and temporal covariates in the variance model.\nWe define $\\lambda_t(\\bm{s})=\\lambda_1(\\bm{s})\\lambda_{2t}$ and based on equations (\\ref{eq5}) and (\\ref{eqL}) we fit the models described in Table \\ref{tabmodels} which are particular cases of the general model proposed in the previous section.\n\n \n  \n   \n       \n   \n   \n    \n\n\n\n\\begin{table}\n  \\caption{Competing models fitted to data applications: Gaussian (G), Student-t (ST), Spatial GLG (GLG), Dynamical (Dyn), Dynamical with covariates (CovDyn), Dynamical GLG (DynGLG), Dynamical GLG with covariates (CovDynGLG) and the complete model (Full).  \\label{tabmodels}}\n\\centering\n  \\fbox{  \\begin{tabular}{lll}\n    \\hline\n  \n  Model  & $\\lambda_1(\\bm{s})$ & $\\lambda_{2t}$\\\\\n    \\hline\nG & $1$ & $1$ \\\\\nST & $\\lambda \\sim Gamma(\\nu_1\/2,\\nu_1\/2)$ & $1$\\\\\nGLG & \n    $ln (\\bm{\\lambda}_{1})\\sim N\\left (-\\frac{\\nu_1}{2} \\; \\mathbf{1}_n , \\nu_1 C_{\\bm{\\xi}}\\right )$ & $1$\\\\ \nDyn & $1$ & $ln(\\lambda_{2t})\\sim N\\left (-\\frac{\\nu_2}{2} +\\eta_{0t}, \\nu_2\\right )$\\\\\nCovDyn & $1$ &  $ln(\\lambda_{2t}) \\sim N\\left (-\\frac{\\nu_2}{2}+\\bm{F}_{2t}'\\bm{\\eta}_t, \\nu_2\\right )$\\\\\nDynGLG &  \n     $ln (\\bm{\\lambda}_{1})\\sim N\\left (-\\frac{\\nu_1}{2} \\; \\mathbf{1}_n, \\nu_1 C_{\\bm{\\xi}}\\right )$ & $ln(\\lambda_{2t})\\sim N\\left (-\\frac{\\nu_2}{2}+\\eta_{0t}, \\nu_2\\right )$\\\\\nCovDynGLG & $ln (\\bm{\\lambda}_{1})\\sim N\\left (-\\frac{\\nu_1}{2} \\; \\mathbf{1}_n, \\nu_1 C_{\\bm{\\xi}}\\right )$ & $ln(\\lambda_{2t}) \\sim N\\left (-\\frac{\\nu_2}{2}+\\bm{F}_{2t}'\\bm{\\eta}_t, \\nu_2\\right )$\\\\\n Full & $ln (\\bm{\\lambda}_{1})\\sim N\\left (-\\frac{\\nu_1}{2}\\; \\mathbf{1}_n+\\bm{F}_1'\\bm{\\beta} , \\nu_1 C_{\\bm{\\xi}}\\right )$ & $ln(\\lambda_{2t}) \\sim N\\left (-\\frac{\\nu_2}{2}+\\bm{F}_{2t}'\\bm{\\eta}_t, \\nu_2\\right )$\\\\\n\\hline\n    \\end{tabular}}\n\\end{table}\n\n\n\n\\subsection{Application to temperature data in the Spanish Basque Country}\\label{sec:real}\n\nThis dataset refers to the maximum temperature recorded in 2006 in the Spanish Basque Country (Figure \\ref{sec6fig1}(a)).\n{Part of these data was analysed by} \\cite{PSteel06}, \\cite{Fons11} and \\cite{Bueno2017} where they only used the maximum temperature recorded in July 2006 at 70 locations.\n\n\\cite{PSteel06} considered only spatial data while \\cite{Fons11} and \\cite{Bueno2017} considered spatio-temporal data. As this region is quite mountainous, with altitude of the monitoring stations lying between 0 and 1188 meters, altitude is included as an explanatory variable in the dynamic mean of the process that also depends on the spatial coordinates, that is, $m_{t}(\\mathbf{s}) = \\theta_{0t} + \\theta_{1t}\\thinspace lat(\\mathbf{s}) + \\theta_{2t}\\thinspace long (\\mathbf{s}) + \\theta_{3t} \\thinspace alt(\\mathbf{s})$, $\\forall \\thinspace t=1, \\ldots J$.\nFor the stations with missing observations, data were inputted using a random forest algorithm  \\citep{Stek12}. We considered stations with no more than 5\\% missing data resulting in 68 locations.\n\n\nPanels of Figure \\ref{sec6fig1}(c)--(e) show that the empirical mean, empirical precision over time and space for the residuals of a Gaussian dynamical model is far from constant, suggesting that a spatial model with constant variance might be unsuitable. Panel {(e)} of Figure \\ref{sec6fig1} shows the behaviour of the variability across the region. The diameter of the solid circles is proportional to the value of the residual precision at the respective location.  The map suggests that there is a spatial trend left in the variance of the residuals.\n\n\n\n\n\n\\begin{figure\n\\centering\n\\begin{tabular}{cc}\n\\includegraphics[width=5.5cm]{MapSpainShape.pdf} \n&\n\\includegraphics[width=4.6cm]{MeanDaysTemp.pdf} \\\\\n(a) Spain Map and spatial locations & (b) Empirical temporal mean  \\\\ \n\n  \\includegraphics[width=4.6cm]{PrecisionTimeTempmaxResNew.pdf} &\\includegraphics[width=4.6cm]{PrecisionAltitudeTempmaxRes.pdf} \n    \\\\\n{(c) Residual temporal precision }& (d) Residual spatial precision versus altitude \\\\\n    \\includegraphics[width=5.5cm]{PrecisionSpaceSpainTemp.pdf} &\n\\includegraphics[width=4.6cm]{PrecisionLatitudeTempmaxRes.pdf} \\\\% &\\includegraphics[width=4.6cm]{VarianceLongitudeTemmaxRes.pdf}  \\\\\n(e) Residual spatial precision  & (f) Residual spatial precision versus latitude \\\\\n\\end{tabular}\n\\caption{Data summaries { for the maximum temperature data observed in the Spanish Basque Country}. Panel (a) displays the map with spatial locations (solid circles) and the crosses are the ones left out from the inference procedure to check the predictive ability of the different models. Panel (b) presents the empirical mean over the year. Panels (c)-(f) represent the empirical precision of the maximum temperature observed data for the residuals of a Gaussian (G) model.\\label{sec6fig1}}\n\\end{figure}\n\nGiven the results from Figure \\ref{sec6fig1}, we move forward and fit models: G, ST, GLG, Dyn, DynGLG and Full as described in Table \\ref{tabmodels} with covariates in the spatial variance. We leave out three locations for predictive comparison (represented by `$\\times$' in Panel (a) of Figure \\ref{sec6fig1}). The parameters to be estimated are the dynamic coefficients ($\\theta_{0t}, \\theta_{1t}, \\theta_{2t},\\theta_{3t}$), $\\boldsymbol{\\eta}_t$, the covariance parameters ($\\sigma^2, \\tau^2, \\phi, \\alpha, \\gamma$), the mixing parameters ($\\nu_1, \\nu_2$),  the latent mixing variables ($\\lambda_{1}(\\bm{s})$, $\\lambda_{2t}$) and the variance regression coefficients $\\bm{\\beta}$. As already mentioned in section \\ref{sec2.2} the variances $W_t$ and $W_{2t}$ are estimated through discount factors. We must specify two discount factors referring to the structure of the mean process $\\mathbf{Z}_t$ and the mean of the variance process ${\\lambda}_{2t}$, respectively. In a general context, the value of the discount factor is usually fixed between 0.90 and 0.99, or it is chosen by model selection diagnostics, e.g, looking at the predictive performance of the model for different values of $\\delta = (\\delta_1, \\delta_2)$ using some comparison criteria \\citep{petris2009dynamic}. To illustrate the performance of the competing models, we fixed $\\delta_{1}=0.99$ (for all competing models) and $\\delta_{2}=0.99$ (for the Dyn, the DynGLG and the Full models) for evaluating the behaviour and goodness of fit. \n\n { Following the values of the different model comparison criteria shown in Table \\ref{sec4tab1}, the G model is the one with the worst predictive performance.} As mentioned previously, the G model is not able to accommodate the uncertainty for some observations which presented larger maximum temperature values. { Under LPS the Full and the DynGLG models provide quite similar values, whereas DynGLG results in the smallest value of VS-0.25. Note that the LPS under DynGLG is similar to the one under the Full model. Therefore, in what follows we discuss the main results obtained for the DynGLG model where we do not consider spatial covariates.}\n\n\n\n\\begin{table\n \n   \\caption{Model comparison based on the Interval Score (IS), the Log Predictive Score (LPS) and the Variogram Score of order 0.25 (VS-0.25) criteria for the predicted observations at the out-of-sample locations under all fitted models for the maximum temperature dataset. The smallest values are highlighted in boldface.  \n\\label{sec4tab1}}\n     \\centering\n  \\fbox{  \\begin{tabular}{lcccccc}\n    \\hline\n  \n    & G  & ST & GLG & Dyn & DynGLG &{Full} \\\\\n    \\hline\nIS &6.85 & 4.62  & 4.54 &  {\\bf 4.21}& 4.34 & { 4.25}\\\\\nLPS & 1565 & 1355  & 1206 & 1286& { 1097}& {\\bf 1095}\\\\\nVS-0.25 & 1658 & 1304 & 1222 & 1283 & {\\bf 1194} & 1240\\\\\n\\hline\n    \\end{tabular}}\n  \n\\end{table}\n\nPosterior summaries (limits of the 95\\% posterior credible intervals) of the time varying coefficients (see Figure \\ref{sec6fig2a}) in the mean of the process do not include zero suggesting that latitude, longitude and altitude are important covariates to explain levels of temperature. In particular, and as expected, the coefficient associated with altitude (see panel (d) of Figure \\ref{sec6fig2a}) is negative over time, suggesting that the maximum temperature decreases as the altitude increases.\n\n\\begin{figure\n\\centering\n\\begin{tabular}{cc}\n\\includegraphics[width=5cm]{d0glgTemp.pdf}\n&\n \\includegraphics[width=5cm]{d1glgTemp.pdf} \\\\ {(a) Intercept }  &{(b) Latitude effect} \\\\ \n \\includegraphics[width=5cm]{d2glgTemp.pdf} &\n \\includegraphics[width=5cm]{d3glgTemp.pdf}\\\\\n{(c) Longitude effect} & {(d) Altitude effect} \\\\\n\\end{tabular}\n\\caption{Temperature data: posterior summaries for the dynamic mean effects $\\theta_t$ under the DynGLG model.\\label{sec6fig2a}}\n\\end{figure}\n\nPanels (a)-(c) of Figure \\ref{sec6fig3} present the dynamic mixing effect indicating that the model captures variability over time and it is able to identify some stations that are potential outliers over space and time. Clearly, the variance is not constant over space-time as previously suggested by Figure \\ref{sec6fig1}.\n\n Figure \\ref{figTempvar1} presents the posterior summaries for the predictive standard deviation of $z_t(s)$, ${ s}=(43.16, -3.28)$ conditional on the latent mixing variables for the DynGLG model compared to the Gaussian model. The posterior predictive standard deviation is obtained numerically by composition sampling that simulates replicated observations from the observational model and computes the empirical standard deviation of these artificial data. Note that the variance is non-constant with some peaks over time. This behaviour cannot be captured by the G model which estimates the standard deviations as almost constant over time. The advantage of our proposed model is clear from  panels (a)-(b) of Figure \\ref{figTempvar2}. For this application, the DynGLG model tends to have shorter ranges of the 95\\% posterior predictive intervals whereas uncertainty seems small and it presents larger intervals in periods of more volatile behaviours.\n\n\nAs the Full model provided a similar value of LPS and a smaller value of IS in comparison to the DynGLG we briefly discuss the posterior summaries of the parameters in the model for $\\lambda_1(\\bm{s})$.\nThe Full model includes covariates in spatial variance $\\lambda_1(\\bm{s})$ and the regression coefficients indicate that latitude and longitude do not impact on the variability over space with the 95\\% posterior credible interval for $\\beta_1$ being $(-0.0093 , 0.0132 )$ and, for $\\beta_2$ $(-0.1156,  0.0834 )$, respectively. On the other hand, the effects of altitude $IC(95\\%, \\beta_3)= (-0.0001, 0.0000)$ show that it influences spatial heterogeneity not only in the dynamical mean but also in the variability of the process. Note that the range of $\\beta_3$ is very small and it does not improve the predictive performance  substantially.\n\n\n\\begin{figure\n\\begin{center}\n\\begin{tabular}{ccc}\n\\includegraphics[width=4.6cm]{MutTemp.pdf} &\n \\includegraphics[width=4.6cm]{LambdasTemp.pdf} &\n \\includegraphics[width=4.6cm]{LambdatTemp.pdf}\\\\\n (a) $\\eta_{0t}$    &{(b) $\\lambda_{1}(\\bm{s})$}  & (c) $\\lambda_{2t}$  \\\\\n\\end{tabular}\n\\caption{Temperature data: posterior summaries for the DynGLG model: (a) dynamic variance mean (solid line), (b) mixing space and (c) mixing temporal.}\n\\label{sec6fig3}\n\\end{center}\n\\end{figure}\n\n\n\\begin{figure}[!ht]\n\\begin{center}\n\\begin{tabular}{c}\n \\includegraphics[width=11cm]{SeriesEstimatedVarFullTemp2.pdf} \\\\\n \\end{tabular}\n\\caption{Temperature data: Approximated posterior predictive standard deviation over time for the DynGLG model and the Gaussian model for ${\\bf s}=(43.16, -3.28)$. }\n\\label{figTempvar1}\n\\end{center}\n\\end{figure}\n \\begin{figure}[!ht]\n\\begin{center}\n\\begin{tabular}{c}\n\n\\includegraphics[width=12cm]{Site3predTemp.pdf} \\\\\n \\\\ \n (a) ${\\bf s}= (43.18, -2.77)$\\\\\n \\includegraphics[width=12cm]{Site2predTemp.pdf} \\\\\n(b) ${\\bf s}=(43.16, -3.28)$\\\\\n\\end{tabular}\n\\caption{Temperature data: Predictive posterior distribution ($95\\%$ interval) over time for the DynGLG model and the Gaussian model for {${\\bf s}= (43.18,-2.77 )$ and ${\\bf s}=(43.16, -3.28)$.}}\n\\label{figTempvar2}\n\\end{center}\n\\end{figure}\n\n\n\n\n\\clearpage\n\n\n\\subsection{Application to ozone data in the UK}\\label{sec:real2}\n\n\nThis section analyses the ozone data presented in Section \\ref{mot}. The proposed mean function is $m_t(\\mathbf{s}) = \\theta_{0t} + \\theta_{1t}\\thinspace lat(\\mathbf{s}) + \\theta_{2t}\\thinspace long (\\mathbf{s}) + \\theta_{3t} \\thinspace temp_t(\\mathbf{s})+ \\theta_{4t} \\thinspace wind_t(\\mathbf{s})$, $\\forall \\thinspace t=1, \\ldots J$. For the stations with missing observations, data were inputted using a random forest algorithm \\citep{Stek12}. We considered stations with less than $5\\%$ of missing data in all variables resulting in 61 stations, with 56 stations used for model fitting and 5 stations used for prediction comparison. \nThe parameters to be estimated for the complete model are the dynamic coefficients $\\bm{\\theta}_{t}$ and $\\bm{\\eta}_t$, the covariance parameters ($\\sigma^2, \\tau^2, \\phi, \\alpha, \\gamma$), the mixing parameters $\\nu_1$, $\\nu_2$, the latent mixing processes  $\\lambda_{1}(\\bm{s})$ and $\\lambda_{2t}$, and the variance regression coefficients $\\bm{\\beta}$. Analogous to the temperature application, smooth evolutions are assumed for the temporal evolution of trend and variance coefficients with discount factors $\\delta_1=0.99$ and $\\delta_2=0.99$, respectively. \nIn what follows we discuss the main results obtained for the best model (CovDynGLG) according to our predictive comparison measures (see Table \\ref{tabozone1}). The different criteria indicate that the Gaussianity is unlikely to hold for this dataset. The most complete models with dynamical effects in the variance have superior predictive performances under all criteria. \n\n\n\n\\begin{table}\n \n     \\caption{Model comparison based on the Interval Score (IS), the Log Predictive Score (LPS) and the Variogram Score of order 0.25 (VS-0.25) criteria for the predicted observations at the out-of-sample locations under all fitted models for the maximum ozone dataset.\\label{tabozone1}}\n     \\centering\n \\fbox{   \\begin{tabular}{lccccccc}\n    \\hline\n   \n&  G  & ST & GLG & CovDyn & DynGLG & CovDynGLG & Full\\\\\n    \\hline\n IS &  78 & 75 & 76 &  77 & 70 & {\\bf 68} & 71\\\\\n LPS & 5960  & 5883 & 5696 & 5940  & 5631 & 5563 & {\\bf 5343}\\\\\nVS-25 & 10116  & 9760 & 10081 & 9842 &  9698 & {\\bf 9518} & 9535\\\\\n         \\hline\n    \\end{tabular}}\n\n\\end{table}\n\nPanels of Figure \\ref{figUK2} present the posterior summaries of the time varying coefficients in the mean \nfor the CovDynGLG model indicating that the maximum ozone mean changes substantially from March to November (Panel (a)). Latitude, longitude and wind are associated with ozone levels resulting in a non-constant behaviour across time, while temperature is mostly not associated with ozone levels as $0$ is within the limits of the 95\\% posterior credible interval.\n\n Panels of Figure \\ref{figUK3} present the time varying coefficients for temperature and wind in the precision model.\nNote that the coefficient for temperature is mostly negative in the precision model while in the mean model the 95\\% posterior credible interval contains zero for most of the instants in time. \nSpecifically, the temperature effect in the precision is negative in July, indicating smaller precision in the exponential scale, when indeed we observe the largest empirical temporal volatility of maximum ozone.  For the time-varying coefficients of wind we observe a positive association both in the mean and variance models, however, in the mean model the coefficient has a decreasing pattern (Figure 7 (e)), whereas in the variance model it has an increasing pattern with time (Figure 8 (b)).\n\nFigure \\ref{figUKvar1} presents the posterior summaries for the standard deviation of $z_t(\\mathbf{s})$, $\\mathbf{s}=(50.74,-1.83)$ for the CovDynGLG model compared to the Gaussian model. Note that the variance is non-constant with large peaks in June and July. Differently, the Gaussian dynamic model suggests a nearly constant standard deviation across time.\nThis pattern has a direct effect on the predictive uncertainty of the Gaussian Model which does not capture many extreme observations and tend to have greater variability across the observed period, which is clear from panels of Figure \\ref{figUKvar2}.  Note that model CovDynGLG captures the periods of extreme values of ozone while it has shorter ranges of the 95\\% credible intervals for those periods that observations do not change much across time. Regarding the complete model that includes the regression components in the equation for $\\lambda_1(\\bm{s})$, the 95\\% posterior credible interval for latitude is $(-0.0107,-0.0043)$ and for longitude is $(-0.1778,-0.0442)$, suggesting that both variables have a negative association with the precision over space. This results in smaller predictive precision for south-eastern locations. \n\n\n\n\n\n\n\n\n\\begin{figure\n\\begin{center}\n\\begin{tabular}{ccc}\n\\includegraphics[width=4.6cm]{d0UK2a.pdf}\n&\n \\includegraphics[width=4.6cm]{d1UK2a.pdf} &\n \\includegraphics[width=4.6cm]{d2UK2a.pdf}\\\\ {(a) Intercept. } &{(b) Latitude effect} & {(c) Longitude effect.} \\\\  \n \\includegraphics[width=4.6cm]{d3UK2a.pdf} &\n \\includegraphics[width=4.6cm]{d4UK2a.pdf} & \\\\\n{(d) Temperature effect.} & {(e) Wind effect.} & \\\\\n\\end{tabular}\n\\caption{Ozone data: Posterior summaries for the dynamic mean effects, $\\mathbf{\\theta}_t$ in equation (\\ref{eq5b}), under the CovDynGLG model.}\n\\label{figUK2}\n\\end{center}\n\\end{figure}\n\n\\begin{figure\n\\begin{center}\n\\begin{tabular}{cc}\n\\includegraphics[width=5cm]{dd1UK2a.pdf}\n&\n \\includegraphics[width=5cm]{dd2UK2a.pdf} \\\\ \n {(a) Temperature effect. } &{(b) Wind effect}  \\\\  \n\n\\end{tabular}\n\\caption{Ozone data: Posterior summaries of the coefficients included the equation for the time-varying variance (CovDynGLG model).\n\\label{figUK3}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[!ht]\n\\begin{center}\n\\begin{tabular}{c}\n \\includegraphics[width=12cm]{SeriesEstimatedVarFullnew.pdf} \\\\\n \\end{tabular}\n\\caption{Ozone data: Approximated predictive standard deviation over time for the CovDynGLG model and the Gaussian model for ${\\bf s}=(50.74,-1.83)$.}\n\\label{figUKvar1}\n\\end{center}\n\\end{figure}\n\n \\begin{figure}[!ht]\n\\begin{center}\n\\begin{tabular}{c}\n\\includegraphics[width=12cm]{predozonefinals2.pdf} \\\\\n(a) ${\\bf s}=(50.74,-1.83)$\\\\\n\\includegraphics[width=12cm]{predozonefinals5.pdf} \\\\\n(b) ${\\bf s}=(52.95,-1.15)$\\\\\n\\end{tabular}\n\\caption{Ozone data: Predictive distribution ($95\\%$ interval) over time for the CovDynGLG model and the Gaussian model for ${\\bf s}=(50.74,-1.83)$ and ${\\bf s}=(52.95,-1.15)$. }\n\\label{figUKvar2}\n\\end{center}\n\\end{figure}\n\n\n\n\n\\newpage\n\\clearpage\n\n\\section{Conclusions}\\label{sec:conclusion}\nWe have proposed a flexible dynamical non-Gaussian spatio-temporal model that extends the well known multivariate dynamic linear model and accommodates both outliers and regions in space or time with larger observational variance. The dynamic evolution in the variance model proposed in equation (\\ref{eq:lambda21}) is able to account for different regimes of variability over time, which is a desirable feature when modelling environmental data in large temporal windows. For instance, the most complete models with covariates aiding in the representation of uncertainty over space and time presented the best performances in predicting the maximum ozone in the UK. This result indicates that patterns in periods of large variability could be explained by changes in wind and temperature that not only influence the mean but also have an impact on the description of the variance of the process. This results in a better description of the uncertainty associated with temporal predictions and spatial interpolations of interest. As inference is performed under the Bayesian paradigm using MCMC methods, we proposed an efficient sampling algorithm for inference and prediction. It takes advantage of the conditionally Gaussian distributions obtained when we condition the distribution of $Z_t({\\bf s})$ on the mixing latent variables. \n\nAs shown in Section \\ref{sec2.2}, the proposed model allows the resultant variance structure to change across space and time depending on the effect of covariates. Moreover, the kurtosis depends on the mixing scales $\\nu_1$ and $\\nu_2$, which reflect the inflation in the tails when necessary. The correlation structure, on the other hand will not change with the covariates but will have the effect of the correlation structure assumed for the variance model. The model generalizes the well known Gaussian model for spatio-temporal data and adds flexibility to the alternative Student-t model. Although the Student-t model allows for variance inflation, it increases the variance of the process in every location and does not allow for local changes in variability as our proposal does.\n\n\nWe performed extensive simulation studies to investigate the ability of the proposed model to capture different structures of the spatio-temporal process of interest. Our simulated examples in Section D of the Supplementary Material indicate that the correct model is selected with the complete model having worse performance when the data does not have the effect of covariates in the spatial mixing process. The non-Gaussian proposals have equivalent performance when fitted to the Gaussian simulated data. Thus, complexity is not always preferred suggesting that our model does not lead to overfitting. Moreover it seems  that the predictive scoring rules used to compare the models are adequate measures of good predictive performance. \n\nWe conclude that allowing for a flexible model for the variance of the process provides coherent posterior predictive credible intervals that accommodates well the structure of the spatio-temporal process under study. A possible drawback of the proposed approach is that prediction of the process to future instants in time depend on covariates that are themselves spatio-temporal processes that need to be predicted. One possible solution is to consider a multivariate spatio-temporal process which is subject for future research.\n\n\n\n\n\n\n\\section*{Acknowledgments}\nSchmidt is grateful for financial support from the Natural Sciences and Engineering Research Council (NSERC) of Canada (Discovery Grant RGPIN-2017-04999).\n\n\n\n\\newpage\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}