Weyl-Schr\"odinger representations of Heisenberg groups in infinite dimensions

TL;DR

This paper explores complexified Heisenberg groups in infinite dimensions, describing their Weyl--Schr"odinger representations on a specialized L^2 space, and applies these findings to heat equations on the group.

Contribution

It introduces a new framework for representing infinite-dimensional Heisenberg groups using invariant measures and Schur polynomials, linking to Fock space and analytic functions.

Findings

Representation of H_C on L^2 space described

Connection established between L^2 space and Fock space

Applications to heat equations on the group

Abstract

A complexified Heisenberg matrix group $\mathrm{H}_\mathbb{C}$ with entries from an infinite-dimensional Hilbert space $H$ is investigated. The Weyl--Schr\"odinger type irreducible representations of $\mathrm{H}_\mathbb{C}$ on the space $L^2_\chi$ of square-integrable scalar functions is described. The integrability is understood under the invariant probability measure $\chi$ which satisfies an abstract Kolmogorov consistency conditions over the infinite-dimensional unitary group $U(\infty)$ irreducible acted on $H$. The space $L^2_\chi$ is generated by Schur polynomials in variables on Paley--Wiener maps over $U(\infty)$. Therewith, the Fourier-image of $L^2_\chi$ coincides with a space of Hilbert--Schmidt entire analytic functions on $H$ generated by suitable Fock space. Applications to linear and nonlinear heat equations over the group $\mathrm{H}_\mathbb{C}$ are considered.

Equations

$$X(a,b,t)=\left[\begin{array}{ccc}1& a& t\\ 0& \text{mathbbm}1& b\\ 0& 0& 1\end{array}\right],X(a,b,t)\cdot X(a^{\prime },b^{\prime },t^{\prime })=\left[\begin{array}{ccc}1& a+a^{\prime }& t+t^{\prime }+⟨a\mid b^{\prime }⟩\\ 0& \text{mathbbm}1& b+b^{\prime }\\ 0& 0& 1\end{array}\right]$$

$${\varphi }_h(\mathfrak{u}):=\sum {\varphi }_i(\mathfrak{u}){\mathfrak{e}}_i^{\ast }(h)\text{with}{\varphi }_i(\mathfrak{u}):=⟨u_i({\mathfrak{e}}_i)\mid {\mathfrak{e}}_i⟩,u_i={\pi }_i(\mathfrak{u}),$$

$$\int\exp\big{\{}\mathop{\mathrm{Re}}\phi_{h}\big{\}}\,d\chi=\exp\Big{\{}\frac{1}{4}\|h\|^{2}\Big{\}},\quad{h\in H}.$$

$$s^{\lambda}_{\imath}(\mathfrak{u}):=\frac{\mathop{\mathrm{det}}\big{[}\phi_{\imath_{i}}^{\lambda_{j}+\eta-j}(\mathfrak{u})\big{]}_{1\leq i,j\leq\eta}}{\prod_{1\leq i<j\leq\eta}[\phi_{\imath_{i}}(\mathfrak{u})-\phi_{\imath_{j}}(\mathfrak{u})]}\quad\text{and}\quad{\phi}^{\lambda}_{\imath}:={\phi}^{\lambda_{1}}_{\imath_{1}}\ldots{\phi}^{\lambda_{\eta}}_{\imath_{\eta}}.$$

$$\Vert {\varphi }_{ı}^{\lambda }{\Vert }_{\chi }^2=\int |{\varphi }_{ı}^{\lambda }{|}^{2}d\chi ={\beta }_{\lambda }\lambda !,{\beta }_{\lambda }:=\frac{(\eta -1)!}{(\eta -1+n)!},\lambda !:={\lambda }_1!\dots {\lambda }_{\eta }!.$$

$$f=\varPsi\circ\psi^{*}_{f},\qquad\psi^{*}_{f}(h)=\sum_{n\geq 0}\frac{1}{n!}\sum_{\imath^{\lambda}\vdash n}\frac{n!}{\lambda!}\mathfrak{e}^{*\lambda}_{\imath}(h)\big{\langle}\mathfrak{e}^{\odot\lambda}_{\imath}\mid\psi_{f}\big{\rangle}_{\Gamma_{\beta}},\quad{h\in{H}},$$

$$\hat{f}(h)=\int \exp ({\bar{\varphi }}_h)fd\chi \text{where}F={\Psi }^{-1}:L_{\chi }^2\ni f⟼\hat{f}:={\psi }_f^{\ast }\in H_{\beta }^2.$$

$$X(a,b,z)\longmapsto\exp(z){W}^{\dagger}(a,b),\quad{W}^{\dagger}(a,b):=\exp\Big{\{}\frac{1}{2}\langle{a}\mid{b}\rangle\Big{\}}T^{\dagger}_{b}M^{\dagger}_{a^{*}}$$

$${W}^{\dagger}(a,b){W}^{\dagger}(a^{\prime},b^{\prime})=\exp\left\{-\big{[}\mathfrak{p}_{a,b}^{\dagger},\mathfrak{p}_{a^{\prime},b^{\prime}}^{\dagger}\big{]}\right\}{W}^{\dagger}(a^{\prime},b^{\prime}){W}^{\dagger}(a,b)$$

$$\frac{dw(r)}{dr}=-\sum\big{(}\partial_{i}^{\dagger}+\bar{\phi}_{i}\big{)}^{2}w(r),\quad w(0)=f,\quad r>0,$$

$${\mathfrak{G}}_r^{†}f$$

$$W_{\tau }^{†}f$$

$$\displaystyle\pi^{m+1}_{m}\colon{u_{m+1}}:=\begin{bmatrix}z_{m}&a\\ b&t\\ \end{bmatrix}\longmapsto u_{m}:=\left\{\begin{array}[]{clc}z_{m}-[a(1+t)^{-1}b]&:&t\operatorname{=/\ }-1\\ z_{m}&:&t=-1\\ \end{array}\right.$$

$${\pi }_m:\mathfrak{U}\ni \mathfrak{u}⟼u_m\in U(m)\text{such that}{\pi }_m={\pi }_m^{m+1}\circ {\pi }_{m+1}.$$

$$\mathfrak{U}\ni \mathfrak{u}⟼\mathfrak{u}.g\in \mathfrak{U}\text{with}g=(v,w)\in U(\infty )\times U(\infty )$$

$$\begin{split}U(m)&\ni u_{m}\longmapsto(u_{k})\in\mathfrak{U},\\ u_{k}&=\left\{\begin{array}[]{cl}\pi^{m}_{k}(u_{m})=(\pi^{k+1}_{k}\circ\ldots\circ\pi^{m}_{m-1})(u_{m})&:k<m,\\ u_{m}&:k\geq m.\end{array}\right.\end{split}$$

$$\chi :=\underleftarrow{\lim }{\chi }_m\text{under}{\chi }_m={\pi }_m^{m+1}({\chi }_{m+1})$$

$$\Vert f{\Vert }_{\chi }=⟨f\mid f{⟩}_{\chi }^{1/2},⟨f_1\mid f_2{⟩}_{\chi }:=\int f_1{\bar{f}}_{2}d\chi .$$

$$\int fd\chi =\lim \int fd({\chi }_m\circ {\pi }_m)=\lim \int (f\circ {\pi }_m^{-1})d{\chi }_m.$$

$$\int f(\mathfrak{u}.g)d\chi (\mathfrak{u})$$

$$\int fd\chi$$

$$\begin{array}{rl}({\chi }_m\circ {\pi }_m)(\mathcal{K})& ={\pi }_m^{m+1}({\chi }_{m+1})\left[{\pi }_m(\mathcal{K})\right]\\ & ={\chi }_{m+1}\left[({\pi }_m^{m+1}{)}^{-1}[{\pi }_m(\mathcal{K})]\right]\ge ({\chi }_{m+1}\circ {\pi }_{m+1})(\mathcal{K}).\end{array}$$

$$({\chi }_m\circ {\pi }_m)(\mathcal{K})\ge 1-\epsilon \text{for all}m\in \mathbb{N}.$$

$$\check{\chi }(\mathcal{K})=\inf ({\chi }_m\circ {\pi }_m)(\mathcal{K})\text{for all}\mathcal{K}\subset \underleftarrow{\lim }\check{U}(m).$$

$${\pi }_m^{m+1}(K_{m+1})\subset K_m\text{and}{\chi }_{m+1}(K_{m+1})>1-\epsilon \text{for all}m\ge 1.$$

$${\chi }_m(K_m)={\chi }_{m+1}[({\pi }_m^{m+1}{)}^{-1}(K_m)]>1-\epsilon .$$

$$K_{m+1}\subset ({\pi }_m^{m+1}{)}^{-1}(K_m)\text{such that}{\chi }_{m+1}(K_{m+1})>1-\epsilon .$$

$$\chi (\mathcal{K})=\inf ({\chi }_m\circ {\pi }_m)(\mathcal{K})=\lim ({\chi }_m\circ {\pi }_m)(\mathcal{K})\text{for all compact}\mathcal{K}\subset \mathfrak{U}.$$

$$\chi (\mathcal{K}.g)=\lim {\chi }_m(K_m.g)=\lim {\chi }_m(K_m)=\chi (\mathcal{K})$$

$$\int d\chi (\mathfrak{u})\int f(\mathfrak{u}.g)d({\chi }_m\otimes {\chi }_m)(g)=\int d({\chi }_m\otimes {\chi }_m)(g)\int f(\mathfrak{u}.g)d\chi (\mathfrak{u})$$

$$d_{HS}(u,v)=\sqrt{m^{-1}\mathrm{tr}|u-v{|}_{HS}}\text{where}|u-v{|}_{HS}=\sqrt{(u-v{)}^{\star }(u-v)}.$$

Full text

Keywords: Infinite-dimensional Heisenberg group; Weyl-Schrödinger representation in infinite dimension; Schur polynomials on Paley-Wiener maps; Fourier analysis on virtual unitary matrices; heat equation over Heisenberg group

††Institute of Mathematics, University of Rzeszów, e-mail: ovlopusz@ur.edu.pl††Mathematics Subject Classification (2010): 81R10;43A65;46E50;35R03

1 Introduction

An aim of this work is to investigate irreducible Weyl–Schrödinger representations of the complexified Heisenberg group $\mathcal{H}_{\mathbbm{C}}$ (see [16, n.9]), consisting of matrix elements $X(a,b,t)$ with any $a,b\in{H}$ and $t\in\mathbb{C}$ such that

[TABLE]

where $H$ is an infinite-dimensional complex Hilbert space and $\mathbbm{1}$ is its identity map.

The group $\mathcal{H}_{\mathbbm{C}}$ has the unit $X(0,0,0)$ and inverse elements of the form $X(a,b,t)^{-1}=X\left(-a,-b,-t+\langle{a}\mid{b}\rangle\right)$.

In what follows, we consider the infinite-dimensional unitary group $U(\infty)=\bigcup{U}(i)$, containing all subgroups $U(i)$ of unitary $i\times i$-matrices, which acts irreducibly on a complex Hilbert space $\left\{H,\langle\cdot\mid\cdot\rangle\right\}$ with an orthonornal basis $(\mathfrak{e}_{i})_{i\in\mathbb{N}}$.

To find the desired representation, we use the space $L^{2}_{\chi}$ of $\mathbb{C}$-valued functions that are quadratically integrable with respect to the probability measure $\chi$. Wherein, according to our assumption $\chi$ has a structure of the projective limit $\chi=\varprojlim\chi_{i}$ of probability Haar’s measures $\chi_{i}$ on $U(i)$, satisfying the Kolmogorov consistency conditions in an abstract Bochner’s formulation (see [22, 26]).

In [20, 23] it was shown that the projective limit $\chi=\varprojlim\chi_{i}$ is well defined over the projective limit $\mathfrak{U}=\varprojlim U(i)$ with respect to the Livšic transforms $\pi_{i}^{i+1}\colon{U(i+1)\to U(i)}$ such that $\chi_{i}=\pi_{i}^{i+1}(\chi_{i+1})$. In this paper, we prove that for such $\chi$ each function from $L^{2}_{\chi}$ admit a superposition (linearization in the sense of [4]) on Paley–Wiener maps associated with $U(\infty)$. As a result, it is shown that Schur polynomials form an orthonormal basis in $L^{2}_{\chi}$ and the Fourier-image of $L^{2}_{\chi}$ consists of Hilbert-Schmidt analytic functions on $H$.

Note also that projective limits of probability measures over various infinite-dimensional manifolds with similar properties were investigated in [29, 30, 31].

If instead of the unitary group $U(\infty)$ we take the infinite-dimensional linear space with a Gaussian measure $\gamma$, a similar construction of the appropriate space $L^{2}_{\gamma}$ can be found in the well-known works [1]. In this case, the Fourier-image of $L^{2}_{\gamma}$ coincides with the Segal–Bargmann space of entire analytic functions over which the Schrödinger type irreducible representations of Heisenberg groups are well defined. In the present paper, we change $\gamma$ by the unitarily-invariant projective limit $\chi=\varprojlim\chi_{i}$ and, as a result, we obtain another irreducible representation, called to be the Weyl-Schrödinger type.

Infinite-dimensional Heisenberg groups over $\mathbb{R}$ was considered in [18] by using the reproducing kernel Hilbert spaces. The Schrödinger representation of such groups using Gaussian measures over a real Hilbert space was described in [2]. Since the group $\mathcal{H}_{\mathbbm{C}}$ in the case of matrix entries $a,b,t\in\mathbb{R}$ coincides with the classical Heisenberg group over $\mathbb{R}$ (see, e.g. [11]), the results of the present paper can be considered as a complexification of previous studies. The Weyl-Schrödinger representation obtained here is not equivalent to that was described earlier.

Further, let us briefly describe the main results. Consider the following mapping $\phi\colon H\ni{h}\longmapsto\phi_{h}\in L^{2}_{\chi}$ defined by Paley–Wiener maps

[TABLE]

where $\mathfrak{e}^{*}_{i}(\cdot):=\left\langle\cdot\mid\mathfrak{e}_{i}\right\rangle$ and the projections $\pi_{i}\colon\mathfrak{U}\ni\mathfrak{u}\to u_{i}\in U(i)$ are uniquely defined by $\pi_{i}^{i+1}$. Every function $\phi_{h}$ of variable ${\mathfrak{u}\in\mathfrak{U}}$ satisfies the equality (Cor.6.6)

[TABLE]

The space $L^{2}_{\chi}$ can be generated by two orthonormal bases, consisting of Schur polynomials and power polynomials of variables ${\phi}_{\imath}=\left({\phi}_{\imath_{1}},\ldots,{\phi}_{\imath_{\eta}}\right)$, respectively,

[TABLE]

These bases are indexed by tabloids $\imath^{\lambda}$ with strictly ordered $\imath=({\imath_{1}},\ldots,{\imath_{\eta}})\in\mathbb{N}^{\eta}$ where $\lambda=(\lambda_{1},\ldots,\lambda_{\eta})\in\mathbb{N}^{\eta}$ is a partition of $n\in\mathbb{N}$ and $\eta=\eta(\lambda)$ stands for the length of $\lambda$. Then we write briefly ${\imath^{\lambda}\vdash n}$. The orthogonal expansion $L_{\chi}^{2}=\bigoplus{L}_{\chi}^{2,n}$ holds (Thm 4.1) where ${L}_{\chi}^{2,n}$ are formed by $n$-homogeneous polynomials ${\phi}^{\lambda}_{\imath}$, normed as follows

[TABLE]

It is also shown that the surjective linear isometry $\varPsi\colon{H}^{2}_{\beta}\ni\psi^{*}_{f}\longmapsto{f}\in{L}^{2}_{\chi}$ holds (Lem. 6.2), where ${H}^{2}_{\beta}=\sum{P}_{\beta}^{n}(H)$ means the Hardy space of entire analytic functions $\psi^{*}_{f}(h)$ of variable $h\in H$ and ${P}_{\beta}^{n}(H)$ is generated by the $n$-homogeneous Hilbert–Schmidt polynomials $\mathfrak{e}^{*\lambda}_{\imath}:=\mathfrak{e}^{*\lambda_{1}}_{\imath_{1}}\ldots\mathfrak{e}^{*\lambda_{\eta}}_{\imath_{\eta}}$, normed as \|\mathfrak{e}^{*\lambda}_{\imath}\|_{{H}^{2}_{\beta}}=\big{(}\beta_{\lambda}{\lambda!}\big{)}^{1/2}.

If the basis of symmetric tensor elements $\mathfrak{e}^{\odot\lambda}_{\imath}:=\mathfrak{e}^{\otimes\lambda_{1}}_{\imath_{1}}\odot\ldots\odot\mathfrak{e}^{\otimes\lambda_{\eta}}_{\imath_{\eta}}$ (associated with $\mathfrak{e}^{*\lambda}_{\imath}$) in the correspondingly weighted Fock space $\Gamma_{\beta}(H)$ is normed as $\|\mathfrak{e}^{\odot\lambda}_{\imath}\|_{\Gamma_{\beta}}=\|\mathfrak{e}^{*\lambda}_{\imath}\|_{{H}^{2}_{\beta}}$ then each function $f\in{L}^{2}_{\chi}$ admits the superposition

[TABLE]

where the Taylor expansion on the right-hand side of any analytic function $\psi^{*}_{f}\in{H}^{2}_{\beta}$ on $H$ is uniquely determined by the corresponding element $\psi_{f}\in\Gamma_{\beta}(H)$.

Our further goal is to analyze the inverse isomorphism $\varPsi^{-1}$ which can be described by the Fourier transform under the measure $\chi$ in following way

[TABLE]

The Fourier transform $F$ acts isometrically on the Hardy space of analytic functions ${H}^{2}_{\beta}$ (Thm 6.8). So, $F$ acts as an analytic extension of the mapping $\phi$.

Applying the superposition with $\varPsi$, we describe two different representations of the additive group $(H,+)$ over $L^{2}_{\chi}$ defined by shift and multiplicative groups (Lem. 7.1). Using this we show (in Thm 8.1) that an irreducible representation of the Heisenberg group $\mathcal{H}_{\mathbb{C}}$ can be realized on ${L}^{2}_{\chi}$ in the Weyl–Schrödinger form

[TABLE]

for all $a,b\in H$ and $z\in\mathbb{C}$, where $T_{b}^{\dagger}$ and $M_{a^{*}}^{\dagger}$ are defined by shift and multiplicative groups, respectively. It is also proved that the Weyl system ${W}^{\dagger}(a,b)$ has the densely-defined generator $\mathfrak{p}^{\dagger}_{a,b}:=\partial_{b}^{\dagger}+\bar{\phi}_{a}$ which satisfies the commutation relation

[TABLE]

where the groups $M_{a^{*}}^{\dagger}$ and $T_{b}^{\dagger}$ are generated by $\bar{\phi}_{a}$ and $\partial_{b}^{\dagger}$, respectively.

Applying the Weyl–Schrödinger representation to the associated with $\mathcal{H}_{\mathbb{C}}$ heat equation, we prove (Thm 9.1) that the following Cauchy problem with $\partial_{i}^{\dagger}:=\partial_{\mathfrak{e}_{i}}^{\dagger}$,

[TABLE]

has the unique solution $w(r)=\mathfrak{G}^{\dagger}_{r}f$ for any function $f$ from a finite sum $\bigoplus{L}^{2,n}_{\chi}$, where the $1$-parameter Gaussian semigroup $\mathfrak{G}_{r}^{\dagger}$ has the form

[TABLE]

Here $\tau=(\tau_{i})$ belongs to the abstract Wiener space $\{w_{0},\|\cdot\|_{w_{0}}\}$ defined by the injections $l_{2}\looparrowright{w}_{0}\looparrowright{c}_{0}$ of real Banach spaces and endowed with the Wiener measure $\mathfrak{w}$ in according to the known Gross’ theorem [10], whereas the sequence of projectors $(p^{\sim}_{n})$ onto $\mathbb{R}^{n}$ is convergent to the identity map on $w_{0}$.

Finally, note that this work is a continuation of previous publications [15, 16]. The novelty results from the observation that the system of Schur polynomials with variables on Paley–Wiener maps form an orthonormal basis in $L^{2}_{\chi}$. This allowed us to investigate irreducible Weyl–Schrödinger representations and Weyl systems of the Heisenberg group $\mathcal{H}_{\mathbb{C}}$ on the whole space $L^{2}_{\chi}$.

2 Invariant probability measure

Consider the unitary group $U(\infty)=\bigcup U(m)$ with $m\in\mathbb{N}_{0}=\mathbb{N}\cup\{0\}$, $\mathbbm{1}=U(0)$, irreducibly acting on a separable Hilbert space ${H}$, where subgroups $U(m)$ are identified with ranges of injections $U(m)\ni u_{m}\longmapsto{\begin{bmatrix}u_{m}&0\\ 0&\mathbbm{1}\\ \end{bmatrix}\in U(\infty)}$. Following to [20], [23], we use the Livšic transforms $\pi^{m+1}_{m}\colon U(m+1)\to U(m)$ of the form

[TABLE]

with $z_{m}\in U(m)$ defined by excluding $x_{1}=y_{1}\in\mathbb{C}$ from $\begin{bmatrix}y_{m}\\ y_{1}\\ \end{bmatrix}=\begin{bmatrix}z_{m}&a\\ -b&-t\\ \end{bmatrix}\begin{bmatrix}x_{m}\\ x_{1}\\ \end{bmatrix}$ for $x_{m},y_{m}\in\mathbb{C}^{m}$ and $a,b\in\mathbb{C}$ [23, Lem. 3.1]. It is surjective (not continuous) Borel mapping [23, Lem. 3.11].

The projective limit $\mathfrak{U}:=\varprojlim U(m)$ under $\pi^{m+1}_{m}$ has surjective Borel (not group homomorphisms) projections

[TABLE]

Their elements ${\mathfrak{u}\in\mathfrak{U}}$ are called the virtual unitary matrices. The right action

[TABLE]

is defined to be $\pi_{m}(\mathfrak{u}.g)=w^{-1}\pi_{m}(\mathfrak{u})v$, where $m$ is large enough that $v,w\in{U}(m)$. On $\mathfrak{U}$ the involution $\mathfrak{u}\mapsto\mathfrak{u}^{\star}=(u_{k}^{\star})$ is well defined, where $u_{k}^{\star}=u_{k}^{-1}$ is adjoint to ${u_{k}\in U(k)}$. Thus, $[\pi_{m}(\mathfrak{u}.g)]^{\star}=\pi_{m}(\mathfrak{u}^{\star}.g^{\star})$ for all $g^{\star}={(w^{\star},v^{\star})}\in{U(\infty)\times U(\infty)}$.

There exists the dense embedding $U(\infty)\looparrowright\mathfrak{U}$ (see [23, n.4]) which assigns the stabilized sequence $\mathfrak{u}=(u_{k})$ to each ${u_{m}\in U(m)}$ such that

[TABLE]

We always assume that the group $U(m)$ is endowed with the probability Haar measure $\chi_{m}$. Using the Kolmogorov consistency theorem (see, e.g. [23, Lem.4.8], [26, Thm 2.2], [28, Cor.4.2]), we determine the probability measure on $\mathfrak{U}$ to be the projective limit

[TABLE]

where $\pi_{m}^{m+1}(\chi_{m+1})$ means an image-measure and ${\chi_{0}=1}$. As is known [28, Thm 2.5], the measure $\chi$ is Radon. We now describe the necessary properties of $\chi$.

Consider the Hilbert space $L^{2}_{\chi}$ of functions $f\colon\mathfrak{U}\to\mathbb{C}$ with the following norm and inner product

[TABLE]

Let $L_{\chi}^{\infty}$ be the space of $\chi$-essentially bounded functions $f\colon\mathfrak{U}\to\mathbb{C}$ with the norm $\|f\|_{\infty}=\mathop{\rm ess\,sup}_{\mathfrak{u}\in\mathfrak{U}}|f(\mathfrak{u})|$. The embedding $L^{\infty}_{\chi}\looparrowright L^{2}_{\chi}$ holds and $\|f\|_{\chi}\leq\|f\|_{\infty}$.

Lemma 2.1**.**

For any $f\in{L}_{\chi}^{\infty}$ there exists the limit

[TABLE]

Moreover, the measure $\chi$ is invariant under the right action, which means that

[TABLE]

Proof.

The sequence $\{(\chi_{m}\circ\pi_{m})(\mathcal{K})\}$ is decreasing for any compact set $\mathcal{K}$ in $\mathfrak{U}$, since $\pi_{m}={\pi^{m+1}_{m}\circ\pi_{m+1}}$ yields $\pi_{m+1}(\mathcal{K})\subseteq(\pi^{m+1}_{m})^{-1}\left[\pi_{m}(\mathcal{K})\right]$. It follows

[TABLE]

This ensures that the necessary and sufficient conditions of the Prokhorov theorem [3, Thm IX.52] and its modification from [28, Thm 4.2] are satisfied.

Indeed, let $\check{U}(m)\subset U(m)$ be the set of matrices with no eigenvalue $\{-1\}$ for ${m\geq 1}$. As is known [23, n.3], $\check{U}(m)$ is open in $U(m)$ and ${\chi_{m}(U(m)\setminus\check{U}(m))=0}$. In virtue of [23, Lem. 3.11] the restrictions $\pi^{m+1}_{m}\colon\check{U}(m+1)\rightarrow\check{U}(m)$ are continuous and surjective. The projective limit $\varprojlim\check{U}(m)$ under these restrictions has continuous surjective projections $\pi_{m}\colon\varprojlim\check{U}(m)\rightarrow\check{U}(m)$. Restrict $\chi_{m}$ to $\check{U}(m)$. By [28, Thm 6], a probability measure $\check{\chi}$ satisfying conditions $\pi_{m}(\check{\chi})=\chi_{m}|_{\check{U}(m)}$ is well defined iff for every $\varepsilon>0$ there exists a compact set $\mathcal{K}\subset\varprojlim\check{U}(m)$ such that

[TABLE]

Then by the Prokhorov theorem $\check{\chi}$ is uniquely determined as

[TABLE]

Let $\varepsilon>0$ and $K_{1}\subset\check{U}(1)$ be a compact set such that $\chi_{1}(K_{1})>1-\varepsilon$. Let a compact sets $K_{m}\subset\check{U}(m)$ be defined inductively such that

[TABLE]

Assume that $K_{1},\ldots,K_{m}$ are constructed. Since $\chi_{m}=\pi^{m+1}_{m}(\chi_{m+1})$, we get

[TABLE]

By regularity of $\chi_{m+1}|_{\check{U}(m)}$, there exists a compact set

[TABLE]

The induction is complete. Then $\mathcal{K}=\varprojlim K_{m}$ with $K_{0}=\mathbbm{1}$ is compact. By virtue of (2.9), we have $\check{\chi}(\mathcal{K})\geq 1-\varepsilon$. Hence, the projective limit $\check{\chi}=\varprojlim\chi_{m}|_{\check{U}(m)}$ is well defined on $\varprojlim\check{U}(j)$ by the Prokhorov criterion.

The measure $\check{\chi}$ can be extended to $\varprojlim{U}(m)\setminus\varprojlim\check{U}(m)$ as zero, since each $\chi_{m}$ is zero on ${U(m)\setminus\check{U}(m)}$. The uniqueness of the projective limits yields $\check{\chi}=\chi$. So, $\chi=\varprojlim\chi_{m}$ is also well defined and by (2.8) and (2.9) we get

[TABLE]

By the known Portmanteau theorem [13, Thm 13.16] it follows that the limit (2.5) exists. Whereas, the property (2.6) is a consequence of the equalities

[TABLE]

for all $g=(v,w)\in{U(\infty)\times U(\infty)}$ where $m$ is large enough that $v,w\in U(m)$.

Finally, the function $(\mathfrak{u},g)\mapsto f(\mathfrak{u}.g)$ with any ${f\in{L}_{\chi}^{\infty}}$ is integrable over ${\mathfrak{U}\times U(m)\times U(m)}$, hence

[TABLE]

by the Fubini theorem. It yields (2.7) since the internal integral on the right-hand side is independent of $g$ by (2.6) and ${\int\,d(\chi_{m}\otimes\chi_{m})(g)=1}.$ The proof is complete. ∎

We now note the concentration property of Haar measures sequence $(\chi_{m})$ satisfying the Kolmogorov conditions $\chi_{m}=\pi_{m}^{m+1}(\chi_{m+1})$ if each group $U(m)$ is endowed with the normalized Hilbert–Schmidt metric

[TABLE]

As is well known (see [9, 33]), $(U(m),d_{HB},\chi_{m})$ is a Lévy family. Namely, the following sequence of isoperimetric constants dependent on $\varepsilon>0$

[TABLE]

with $(\Omega_{m})_{\varepsilon}=\left\{u_{m}\in U(m)\colon d_{HS}\left(u_{m},\Omega_{m}\right)<\varepsilon\right\}$ is such that

[TABLE]

Taking into account the Lemma 2.1, we can formulate the following conclusion.

Corollary 2.2**.**

For any Borel set $\Omega_{\varepsilon}=\varprojlim(\Omega_{m})_{\varepsilon}$ with $\chi_{m}(\Omega_{m})>1/2$ in the projective limit $\mathfrak{U}=\varprojlim U(m)$ the equality

[TABLE]

holds. Consequently, all Borel sets $\mathfrak{U}\setminus\Omega_{\varepsilon}$ with ${\chi_{m}(\Omega_{m})>1/2}$ and any ${\varepsilon>0}$ are $\chi$-measure zero, i.e., the measure $\chi=\varprojlim\chi_{m}$ is concentrated outside these sets.

3 Polynomials on Paley–Wiener maps

Let \mathscr{I}_{\eta}:=\big{\{}\imath=\big{(}{\imath_{1}},\ldots,{\imath_{\eta}}\big{)}\in\mathbb{N}^{\eta}\colon\imath_{1}<\imath_{2}<\ldots<\imath_{\eta}\big{\}} be an integer alphabet of length $\eta$ and $\mathscr{I}=\bigcup\mathscr{I}_{\eta}$. Let $\lambda=(\lambda_{1},\ldots,\lambda_{\eta})\in\mathbb{N}^{\eta}$ with ${\lambda_{1}\geq\lambda_{2}\geq\ldots\geq\lambda_{\eta}}$ be a partition of an $n$-letter word \imath^{\lambda}=\big{\{}\Box_{ij}\colon{1\leq i\leq\eta},j=1,\ldots,\lambda_{i}\big{\}} with ${\imath\in\mathscr{I}_{\eta}}$. A Young $\lambda$-tableau with a partition $\lambda$ is a result of filling the word $\imath^{\lambda}$ onto the matrix [\imath^{\lambda}]=\begin{array}[]{cccc}\Box_{11}&\dots&\dots&\Box_{1\lambda_{1}}\\ \vdots&\vdots&\iddots&\\ \Box_{\eta 1}&\dots&\Box_{\eta\lambda_{\eta}}&\end{array}\!\!\!\!\! with $n$ nonzero entries in some way without repetitions. So, each $\lambda$-tableau $[\imath^{\lambda}]$ can be identified with a bijection $[\imath^{\lambda}]\to\imath^{\lambda}$. The conjugate partition $\lambda^{\intercal}$ corresponds to the transpose matrix $[\imath^{\lambda}]^{\intercal}$.

A Young tableau $[\imath^{\lambda}]$ is called standard (semistandard ) if its entries are strictly (weakly) ordered along each row and strictly ordered down each column. Let $\mathbb{Y}$ denote all Young tabloids $[\imath^{\lambda}]$ and $\mathbb{Y}_{n}$ be its subset such that $\imath^{\lambda}\vdash n$. Assume that $\mathbb{Y}_{0}=\left\{\emptyset\in\mathbb{Y}\colon|\emptyset|=0\right\}$ and $\eta(\emptyset)=0$.

As before, $\left\{H,\langle\cdot\mid\cdot\rangle\right\}$ is a separable complex Hilbert space with an orthonormal basis $\left\{\mathfrak{e}_{i}\colon i\in\mathbb{N}\right\}$ and ${\|\cdot\|={\langle\cdot\mid\cdot\rangle^{1/2}}}$. For its adjoint space $H^{*}$ the conjugate-linear isometry ${*\colon H^{*}\rightarrow H^{**}=H}$ is defined via $a^{*}(h)={\langle h\mid a\rangle}$ for all ${a,h\in H}$. The Fourier expansion $h=\sum\mathfrak{e}^{*}_{i}(h)\mathfrak{e}_{i}$ with $\mathfrak{e}^{*}_{i}(h):={\langle h\mid\mathfrak{e}_{i}\rangle}$ holds. The tensor power ${H}^{\otimes n}\!,$ spanned by elements $\psi_{n}={h_{1}\otimes\ldots\otimes h_{n}}$ with ${h_{i}\in{H}}$ ${(i=1,\ldots,n)}$, is endowed with the norm $\|\psi_{n}\|={\left\langle\psi_{n}\mid\psi_{n}\right\rangle}^{1/2}$ where ${\left\langle\psi_{n}\mid\psi_{n}^{\prime}\right\rangle}:={\langle h_{1}\mid h^{\prime}_{1}\rangle\ldots\langle h_{n}\mid h^{\prime}_{n}\rangle}$.

Let $S_{n}$ be the group of $n$-elements permutations $\sigma(\psi_{n}):={h_{\sigma(1)}\otimes\ldots\otimes{h}_{\sigma(n)}}$. An orthogonal basis in ${H}^{\otimes n}$ is formed by elements $\sigma(\mathfrak{e}^{\otimes\lambda_{1}}_{\imath_{1}}\otimes\ldots\otimes\mathfrak{e}^{\otimes\lambda_{\eta}}_{\imath_{\eta}})$ with ${\imath^{\lambda}\vdash n}$ and $\eta=\eta(\lambda)$, additionally indexed by all $\sigma\in S_{n}$. The symmetric tensor power $H^{\odot n}\subset H^{\otimes n}$ is defined to be a range of the orthogonal projector ${\mathcal{S}_{n}\colon{H}^{\otimes n}\ni\psi_{n}\longmapsto{h_{1}\odot\ldots\odot h_{n}}:=(n!)^{-1}{\sum}_{\sigma\in S_{n}}\sigma(\psi_{n})}$. We assume that ${H}^{\otimes n}$ is completed and that ${H}^{\otimes 0}=\mathbb{C}$. Let $\psi_{n}:=h^{\otimes n}$ for ${h=h_{i}}$. The embedding $\left\{h^{\otimes n}\colon{h}\in{H}\right\}\subset{H}^{\odot n}$ is total by the polarization formula [6, n.1.5]

[TABLE]

Let $H_{\eta}\subset H$ be spanned by \big{\{}\mathfrak{e}_{\imath_{1}},\ldots,\mathfrak{e}_{\imath_{\eta}}\big{\}}. We can uniquely assign to any semistandard tableau $[\imath^{\lambda}]$ with $\imath^{\lambda}\vdash n$ the element in $H_{\eta}^{\otimes n}$ for which there exists the permutation ${\sigma^{\prime}\in{S}_{n}}$ such that {\sigma^{\prime}\big{(}\mathfrak{e}^{\otimes\lambda_{1}}_{\imath_{1}}\otimes\ldots\otimes\mathfrak{e}^{\otimes\lambda_{\eta}}_{\imath_{\eta}}\big{)}}={\mathfrak{e}^{\otimes\lambda_{1}}_{\imath_{1}}\odot\ldots\odot\mathfrak{e}^{\otimes\lambda_{\eta}}_{\imath_{\eta}}\in H_{\eta}^{\odot n}}. Taking all $\imath\in\mathscr{I}$, we conclude that the system indexed by semistandard $\lambda$-tabloids

[TABLE]

forms an orthogonal basis in the symmetric tensor power $H_{\eta}^{\odot n}$.

The system \big{\{}\mathfrak{e}_{\imath}^{\otimes\lambda}:=\mathcal{S}_{n}\big{(}\mathfrak{e}^{\otimes\lambda_{1}}_{\imath_{1}}\otimes\ldots\otimes\mathfrak{e}^{\otimes\lambda_{\eta}}_{\imath_{\eta}}\big{)}\colon\imath^{\lambda}\vdash n,\ \lambda\in\mathbb{Y}_{n},\ \imath\in\mathscr{I}\big{\}}, additionally indexed by all $\sigma\in S_{n}$, forms an orthonormal basis in the whole tensor power $H^{\otimes n}$.

As usually, the symmetric Fock space is defined to be the Hilbertian orthogonal sum $\Gamma(H)={\bigoplus}_{n\geq 0}H^{\odot n}$ with the orthogonal basis \mathfrak{e}^{\mathbb{Y}}:={\bigcup\big{\{}\mathfrak{e}^{\mathbb{Y}_{n}}\colon{n}\in\mathbb{N}_{0}\big{\}}} of elements $\psi=\bigoplus\psi_{n}$ with $\psi_{n}\in H^{\odot n}$ endowed with the inner product and norm

[TABLE]

Note that by tensor multinomial theorem the Fourier expansion under $\mathfrak{e}^{\mathbb{Y}_{n}}$

[TABLE]

holds in $H^{\odot n}$ for all $h\in H$. Consequently, the linearly independent, so-called, coherent states {\big{\{}\exp(h)\colon{h}\in{H}\big{\}}} in $\Gamma(H)$ have the expansion under the basis $\mathfrak{e}^{\mathbb{Y}}$

[TABLE]

with $h^{\otimes 0}=1$, that is convergent, since $\|\mathfrak{e}^{\odot\lambda}_{\imath}\|^{2}_{\Gamma}=n!\|\mathfrak{e}^{\odot\lambda}_{\imath}\|^{2}$ and

[TABLE]

Definition 3.1**.**

For any $h\in H$ and $\mathfrak{u}\in\mathfrak{U}$ the Paley–Wiener maps are defined to be

[TABLE]

where projections $\pi_{i}\colon\mathfrak{U}\ni\mathfrak{u}\to u_{i}\in U(i)$ are uniquely defined by $\pi_{i}^{i+1}$.

These maps satisfy the orthogonal conditions $\phi_{\mathfrak{e}_{i}}=\phi_{i}$ and have the natural extension ${\phi}_{h^{*}}=\bar{{\phi}_{h}}$ onto the adjoint space $H^{*}$.

Note that, as in the case of linear spaces (see e.g. [8, n.4.4], [27]), the Paley–Wiener maps uniquely determine the embedding $\phi\colon H\ni{h}\longmapsto\phi_{h}\in L^{2}_{\chi}$.

For every ${h\in H}$ the ${l}_{2}$-valued function ${\phi}_{h}(\mathfrak{u})$ of variable ${\mathfrak{u}\in\mathfrak{U}}$ is well-defined, since $(\mathfrak{e}_{i}^{*}(h))\in{l}_{2}$ and $\left|\left\langle u_{i}(\mathfrak{e}_{i})\mid\mathfrak{e}_{i}\right\rangle\right|\leq 1$. We show that ${\phi}_{h}\in L^{2}_{\chi}$. Assign for any partition $\lambda=(\lambda_{1},\ldots,\lambda_{\eta})\in\mathbb{N}^{\eta}$ of the weight $|\lambda|={\lambda_{1}+\ldots+\lambda_{\eta}}$ the constant

[TABLE]

Lemma 3.2**.**

To every semistandard tableau $[\imath^{\lambda}]$ one can uniquely assign the function

[TABLE]

of variable $u\in\mathfrak{U}$ belonging to $L^{\infty}_{\chi}$. The system of $\chi$-essentially bounded functions

[TABLE]

is orthogonal in the space $L^{2}_{\chi}$ and is normed as follows

[TABLE]

Proof.

According to (2.3), we have $(\pi_{m}\circ\pi_{m+l}^{-1})u_{m+l}(\mathfrak{e}_{m})=u_{m}(\mathfrak{e}_{m})$ for $t=-1$ and $(\pi_{m}\circ\pi_{m+l}^{-1})u_{m+l}(\mathfrak{e}_{m})=u_{m}(\mathfrak{e}_{m})-[a(1+t)^{-1}b]\mathfrak{e}_{m}$ for $t\operatorname{=/\ }-1$ for any integer $l\geq 1$. This means that ${({\phi}_{k}\circ\pi_{m}^{-1})(u_{m})}=\left\langle u_{m}(\mathfrak{e}_{m})\mid\mathfrak{e}_{k}\right\rangle\operatorname{\equiv/\ }0$ for all $k\leq m$ and that

[TABLE]

Let $U(\eta)$ with $\eta=\eta(\lambda)$ be the unitary group acting over the linear complex $\mathop{\rm span}\left\{\mathfrak{e}_{\imath_{1}},\ldots,\mathfrak{e}_{\imath_{\eta}}\right\}$ in $H$. Let $\chi_{\eta}$ be the probability Haar measure on $U(\eta)$ and $\pi_{\eta}\colon\mathfrak{U}\to U(\eta)$ be the corresponding projector. Using (2.5) and (3.7), we obtain

[TABLE]

By (3.8) and the known integral formula for unitary groups $U(\eta)$ [32, 1.4.9], we get

[TABLE]

On the other hand, the invariant property (2.7) provides the formula

[TABLE]

From (3.9) it follows the orthogonality relations ${\phi^{\lambda^{\prime}}_{\jmath}\perp\phi^{\lambda}_{\imath}}$ with ${|\lambda^{\prime}|\operatorname{=/\ }|\lambda|}$, since

[TABLE]

for any $\lambda^{\prime},\lambda\in\mathbb{Y}\setminus\{\emptyset\}$. Let $|\lambda^{\prime}|=|\lambda|$ and $\eta(\lambda^{\prime})>\eta(\lambda)$ for definiteness. Then there exists an index $k$ with a nonzero integer $\lambda^{\prime}_{k}$ in \lambda^{\prime}=\big{(}\lambda^{\prime}_{1},\ldots,\lambda^{\prime}_{k},\ldots,\lambda^{\prime}_{\eta(\lambda^{\prime})}\big{)}\in\mathbb{Y}\setminus\{\emptyset\} such that $\eta(\lambda)<k\leq\eta(\lambda^{\prime})$. In this case ${{\phi}^{\lambda^{\prime}}_{\jmath}\perp{\phi}^{\lambda}_{\imath}}$ because (3.9) yields

[TABLE]

Consider the case $|\lambda^{\prime}|=|\lambda|$ and $\eta(\lambda^{\prime})=\eta(\lambda)$. If ${{\phi}^{\lambda^{\prime}}_{\jmath}\operatorname{=/\ }{\phi}^{\lambda}_{\imath}}$ then $\lambda^{\prime}\operatorname{=/\ }\lambda$. There exists an index $0<k\leq\eta(\lambda)$ such that $\lambda^{\prime}_{k}\operatorname{=/\ }\lambda_{k}$. As above, ${{\phi}^{\lambda^{\prime}}_{\jmath}\perp{\phi}^{\lambda}_{\imath}}$, because

[TABLE]

This proves that the system ${\phi}^{\mathbb{Y}}$ is orthogonal. ∎

4 Orthonormal basis of Schur polynomials

Let $\imath^{\lambda}\vdash n$, $\eta=\eta(\lambda)$ and $t_{\imath}=(t_{\imath_{1}},\ldots,t_{\imath_{\eta}})$ be a complex variable. Let $t^{\lambda}_{\imath}:=\prod t_{\imath_{j}}^{\lambda_{j}}$. The $n$-homogenous Schur polynomial is defined (see, e.g. [17]) to be

[TABLE]

$\Delta(t_{\imath})={\prod_{1\leq i<j\leq\eta}(t_{\imath_{i}}-t_{\imath_{j}})}$ is Vandermonde’s determinant. It can be written as $s^{\lambda}_{\imath}(t_{\imath})={\sum}_{[\imath^{\lambda}]}t^{\lambda}_{\imath}$ with summation over all semistandard Young tabloids [7, I.2.2].

We construct an orthonormal basis in $L^{2}_{\chi}$ consisting of Schur polynomials on Paley–Wiener maps. Assign (uniquely) to $\imath\in\mathscr{I}_{\eta}$ the vector {\phi}_{\imath}:=\big{(}{\phi}_{\imath_{1}},\ldots,{\phi}_{\imath_{\eta}}\big{)}. Let $s^{\lambda}_{\imath}(\mathfrak{u})=(s^{\lambda}_{\imath}\circ{\phi}_{\imath})(\mathfrak{u})$ be $n$-homogeneous functions of variable ${\mathfrak{u}\in\mathfrak{U}}$ with $\lambda\in\mathbb{N}^{\eta}$, defined by the formulas (1.3). Denote

[TABLE]

Theorem 4.1**.**

The system of Schur polynomials $s^{\mathbb{Y}}$ forms an orthonormal basis in $L^{2}_{\chi}$ and $s^{\mathbb{Y}}_{n}$ is the same basis in $L_{\chi}^{2,n}$. The following orthogonal decomposition holds,

[TABLE]

For any ${h\in{H}}$ the equality (1.2) uniquely defines the conjugate-linear embedding

[TABLE]

Proof.

Let $U(\eta)$ be the unitary group over the linear complex $\mathop{\rm span}\left\{\mathfrak{e}_{\imath_{1}},\ldots,\mathfrak{e}_{\imath_{\eta}}\right\}$ with $\eta=\eta(\lambda)$. Taking into account (3.7) similarly as (3.8), we obtain

[TABLE]

for all $[\imath^{\lambda}]$, $[\imath^{\mu}]$ with $\imath=(\imath_{1},\ldots,\imath_{\eta})$ and $\lambda,\mu\in\mathbb{N}^{\eta}$. In fact, the corresponding Schur polynomials \big{\{}s^{\lambda}_{\imath}\colon\lambda\in\mathbb{N}^{\eta}\big{\}} are characters of the group $U(\eta)$. Hence, by the Weyl integration formula, the right-hand side integral is equal to Kronecker’s delta $\delta_{\lambda\mu}$ [25, Thm 8.3.2 & Thm 11.9.1].

The family of finite alphabets ${\imath\in\mathscr{I}}$ is directed and for any $\imath,\imath^{\prime}$ there exists $\imath^{\prime\prime}$ such that $\imath\cup\imath^{\prime}\subset\imath^{\prime\prime}$. This means that the whole system $s^{\mathbb{Y}}_{n}$ is orthonormal in $L^{2}_{\chi}$.

The property ${s_{\jmath}^{\mu}\perp s_{\imath}^{\lambda}}$ with ${|\mu|\operatorname{=/\ }|\lambda|}$ for any ${\imath,\jmath\in\mathscr{I}}$ follows from (3.9), since

[TABLE]

for all $\lambda\in\mathbb{Y}$ and $\mu\in\mathbb{Y}\setminus\{\emptyset\}$. This yields $L_{\chi}^{2,|\mu|}\perp L_{\chi}^{2,|\lambda|}$ in the space $L^{2}_{\chi}$. Taking $\lambda=\emptyset$ with $|\emptyset|=0$, we get $1\perp L_{\chi}^{2,|\mu|}$ for all $\mu\in\mathbb{Y}\setminus\{\emptyset\}$. Hence, (4.1) is proved.

By Lemma 3.2 the subsystem $\phi_{k}=s^{1}_{k}$ is orthonormal in $L^{2}_{\chi}$, hence by Definition 3.1 it instantly follows that $\|{\phi}_{h}\|_{\chi}^{2}=\sum|\mathfrak{e}_{k}^{*}(h)|^{2}\int|\phi_{k}|^{2}d\chi=\|h\|^{2}.$ It follows the isometric embedding (4.2).

The set $\check{U}(m)$ of matrices with no eigenvalue $\{-1\}$ has Stone–Ĉech compactification $\tilde{U}(m)$ such that the mapping $\check{\pi}^{m+1}_{m}$ has a continuous ${U}(m)$-valued extension

[TABLE]

This fact follows from [35, Thm 19.5] by virtue of that ${U}(m)$ is compact. Hence, the projective limit $\tilde{\mathfrak{U}}:=\varprojlim\tilde{U}(m),$ determined by $\tilde{\pi}^{m+1}_{m}$, is a compact set in $\mathfrak{U}$ with continuous ${U}(m)$-valued projections $\tilde{\pi}_{m}\colon\tilde{\mathfrak{U}}\rightarrow{U}(m)$.

Since $U(\infty)$ on $H$ acts irreducibly, for any $\mathfrak{u}^{\prime}\operatorname{=/\ }\mathfrak{u}^{\prime\prime}$ there is $m$ such that

[TABLE]

i.e., $\phi^{\mathbb{Y}}$ separates $\mathfrak{U}$ and so $\tilde{\mathfrak{U}}$. Hence, the system of Schur polynomials $s^{\mathbb{Y}}$ also separates $\tilde{\mathfrak{U}}$. Moreover, each complex-conjugate function $\bar{\phi}_{m}(\mathfrak{u})=\left\langle\mathfrak{e}_{m}\mid\pi_{m}(\mathfrak{u})(\mathfrak{e}_{m})\right\rangle=\left\langle\pi_{m}(\mathfrak{u}^{\star})(\mathfrak{e}_{m})\mid\mathfrak{e}_{m}\right\rangle$ belongs to $\phi^{\mathbb{Y}}$. Thus, by the Stone–Weierstrass approximation theorem the complex linear span of polynomials $\phi^{\mathbb{Y}}$, as well as, of $s^{\mathbb{Y}}$, forms a dense subspace in the Banach space of all continuous functions $C(\tilde{\mathfrak{U}})$.

Let $\tilde{\chi}_{m}$ means the image of $\chi_{m}$ under $\check{U}(m)\looparrowright{U}(m)$. In Lemma 2.1 it inductively was shown that for every $\varepsilon>0$ there exists a compact set $\varprojlim K_{m}\subset\check{\mathfrak{U}}$ such that

[TABLE]

where $\tilde{\chi}_{m}(K_{m})=\check{\chi}_{m}(K_{m})=\chi_{m}(K_{m})$, by definition of the measure $\tilde{\chi}_{m}$ as an image. Hence, by the Prokhorov theorem the projective limit $\tilde{\chi}=\varprojlim\tilde{\chi}_{m}$, defined by mappings $\tilde{\pi}^{m+1}_{m}$, possesses the properties

[TABLE]

for all Borel $\Omega$ in $\check{\mathfrak{U}}$ or otherwise $\tilde{\chi}|_{\check{\mathfrak{U}}}=\chi|_{\check{\mathfrak{U}}}$. Consequently,

[TABLE]

In particular, $\tilde{\chi}=\varprojlim\tilde{\chi}_{m}$ is regular on $\tilde{\mathfrak{U}}$ by the Riesz–Markov theorem [19, 1.1].

As a consequence, the space $L^{2}_{\chi}$ coincides with the completion of $C(\tilde{\mathfrak{U}})$ and for any ${f\in L_{\chi}^{2}}$ there exists a sequence ${(f_{n})\subset\mathop{\rm span}(s^{\mathbb{Y}})}$ such that ${\int|f-f_{n}|^{2}d\chi\to 0}$. Hence, the system $s^{\mathbb{Y}}$ forms an orthogonal basis in $L_{\chi}^{2}$.

Finally, $s^{\mathbb{Y}}_{n}\cap L^{2}_{\chi}$ is total in $L_{\chi}^{2,n}$ and $s^{\mathbb{Y}}_{n}\perp s^{\mathbb{Y}}_{m}$ if $n\operatorname{=/\ }m$. This yields (4.1). ∎

5 Unitarily-weighted symmetric Fock space

Define on the tensor power ${H}^{\otimes n}$ the unitarily-weighted norm $\|\cdot\|_{H^{\otimes n}_{\beta}}={\langle\cdot\mid\cdot\rangle^{1/2}_{H^{\otimes n}_{\beta}}}$ where the inner product ${\langle\cdot\mid\cdot\rangle^{1/2}_{H^{\otimes n}_{\beta}}}$ is determined by the relations

[TABLE]

Here $\mathfrak{e}^{\otimes\lambda}_{\imath}:=\sigma^{\prime}(\mathfrak{e}^{\otimes\lambda_{1}}_{\imath_{1}}\otimes\ldots\otimes\mathfrak{e}^{\otimes\lambda_{\eta}}_{\imath_{\eta}})$ with $\eta=\eta(\lambda)$ and $\sigma^{\prime}\in S_{n}$ is fixed. Let ${H}^{\otimes n}_{\beta}$ be the completion of \big{\{}{H}^{\otimes n},\|\cdot\|_{H^{\otimes n}_{\beta}}\big{\}}. Its closed subspace, defined by the projection

[TABLE]

forms an unitarily-weighted symmetric tensor power ${H}^{\odot n}_{\beta}\subset{H}^{\otimes n}_{\beta}$ with the inner product determined by relations $\langle\mathfrak{e}^{\odot\lambda}_{\imath}\mid\mathfrak{e}^{\odot\lambda^{\prime}}_{\imath^{\prime}}\rangle_{H^{\otimes n}_{\beta}}=\beta_{\lambda}\langle\mathfrak{e}^{\odot\lambda}_{\imath}\mid\mathfrak{e}^{\odot\lambda^{\prime}}_{\imath^{\prime}}\rangle$ or more specific

[TABLE]

Definition 5.1**.**

The unitarily-weighted symmetric Fock space is defined to be the Hilbertian orthogonal sum ${\Gamma_{\!\beta}(H)=\bigoplus_{n\geq 0}{H}_{\beta}^{\odot n}}$ of elements $\psi=\bigoplus\psi_{n}$, $\psi_{n}\in{H}_{\beta}^{\odot n}$ with the orthogonal basis \mathfrak{e}^{\mathbb{Y}}=\bigcup\big{\{}\mathfrak{e}^{\mathbb{Y}_{n}}\colon{n}\in\mathbb{N}_{0}\big{\}} and the following inner product and norm

[TABLE]

We immediately notice that $\|h\|_{\beta}^{2}=\sum|\mathfrak{e}_{i}^{*}(h)|^{2}=\|h\|^{2}$ for all $h=\sum\mathfrak{e}_{i}\mathfrak{e}_{i}^{*}(h)\in{H}$.

Lemma 5.2**.**

The set of coherent states $\left\{\exp(h)\colon{h}\in{H}\right\}$ is total in $\Gamma_{\beta}(H)$ and the expansion (3.3) is convergent in $\Gamma_{\beta}(H)$. The injections

[TABLE]

are contractive and dense. The $\Gamma_{\beta}(H)$-valued function ${H\ni{h}\longmapsto\exp(h)}$ is entire analytic. The shift group, defined to be

[TABLE]

for $a,h\in H$, has a unique linear extension ${\mathcal{T}_{a}\colon\Gamma_{\beta}(H)\ni\psi\longmapsto\mathcal{T}_{a}\psi\in\Gamma_{\beta}(H)}$ such that

[TABLE]

Proof.

Taking into account that $\beta_{\lambda}\leq 1$, we get the following inequalities

[TABLE]

Hence, (3.2), (3.3) are convergent in $\Gamma_{\beta}(H)$. This implies that ${h\mapsto\exp(h)}$ is analytic and inclusions $\Gamma(H)\looparrowright\Gamma_{\beta}(H)$ and ${{H}^{\odot n}\looparrowright{H}_{\beta}^{\odot n}}$ are contractive. By the polarization formula (3.1) their ranges are dense.

Using the binomial formula ${(h+za)^{\otimes n}}=\bigoplus_{m=0}^{n}\binom{n}{m}(za)^{\otimes m}\odot h^{\otimes(n-m)}$, we find

[TABLE]

with the orthogonal projector $\mathcal{S}_{n/m}$ defined as $\psi_{m}\odot\psi_{n-m}=\mathcal{S}_{n/m}\left(\psi_{m}\otimes\psi_{n-m}\right)\in{H}^{\odot n}_{\beta}$ for all $\psi_{m}\in{H}^{\odot m}_{\beta}$ and $\psi_{n-m}\in{H}^{\odot(n-m)}_{\beta}$. By orthogonality $\|\mathcal{S}_{n/m}\|\leq 1$.

Applying the expansions (3.2) to $a^{\otimes m}$ and $h^{\otimes(n-m)}$, by (5.1), we get

[TABLE]

with summations over semistandard tableaux $[\imath^{\lambda}],[\jmath^{\mu}]$ and $\imath,\jmath\in\mathscr{I}$. Let $(\lambda,\mu)\in\mathbb{N}^{\eta(\lambda,\mu)}$ be the smallest partition of number $n$ with the length $\eta(\lambda,\mu)$ containing the partitions $\lambda$ for $m$ and $\mu$ for $n-m$. Then $\eta(\lambda,\mu)\geq\max\{\eta(\lambda),\eta(\mu)\}$ and so

[TABLE]

since $\frac{(\eta-1)!}{(\eta-1+n)!}$ is decreasing in variable $\eta$. Thus, the following inequality

[TABLE]

holds. Using this inequality and that $\|\mathcal{S}_{n/m}\|\leq 1$, we find

[TABLE]

Summing with coefficients $1/m!$, we get \|\mathcal{T}_{a}\exp(h)\|^{2}_{\beta}\leq\exp\big{(}\|a\|^{2}\big{)}\|\exp(h)\|^{2}_{\beta}. This inequality and totality of $\left\{\exp(x)\colon h\in{H}\right\}$ in $\Gamma_{\beta}(H)$ yield the required inequality (5.3). It also follows that $\Gamma_{\beta}(H)$ is invariant under $\mathcal{T}_{a}$ and that the group property (5.3) holds, since $\partial_{a+b}=\partial_{a}+\partial_{b}$ for all $a,b\in H$ by linearity. ∎

Lemma 5.3**.**

The mapping $\phi\colon H\ni{h}\longmapsto\phi_{h}\in L^{2}_{\chi}$, extended onto $\mathcal{T}_{a}\exp(h)$ as

[TABLE]

has the unique isometric conjugate-linear extension

[TABLE]

defined to be $\langle\varPhi\mathfrak{e}^{\odot\lambda}_{\imath}\mid f\rangle_{\chi}=\langle\mathfrak{e}^{\odot\lambda}_{\imath}\mid\varPhi^{*}f\rangle_{\beta}$ for all ${f\in L_{\chi}^{2}}$ in such way that

[TABLE]

As a result, the conjugate-linear isometries $\Gamma_{\beta}(H)\stackrel{{\scriptstyle\varPhi}}{{\simeq}}L^{2}_{\chi}$ and ${H}^{\odot n}_{\beta}\stackrel{{\scriptstyle\varPhi}}{{\simeq}}L^{2,n}_{\chi}$ hold.

Proof.

By Lemma 5.2 the $\Gamma_{\beta}(H)$-valued function $H\ni h\mapsto\mathcal{T}_{a}\exp(h)$ is well defined for all ${a\in H}$. Let us use the expansion $\phi_{h+a}={\sum\mathfrak{e}^{*}_{i}(h+a)\phi_{i}}$. By Lemma 3.2 and Theorem 4.1, $\phi\colon H\ni{h}\longmapsto\phi_{h}\in L_{\chi}^{2}$ may be extended to $\varPhi$ in following way

[TABLE]

is an orthogonal component of $\varPhi\mathcal{T}_{a}\exp(h)$ in $L^{2}_{\chi}$. It follows that

[TABLE]

Hence, the composition $\mathfrak{U}\ni\mathfrak{u}\longmapsto[\varPhi\exp(h+a)](\mathfrak{u})$ is well defined in $L^{2}_{\chi}$.

Now, we consider the ordinary irreducible representation of permutation group $S_{n}$ on the Specht $\lambda$-module $S^{\lambda}_{\imath}$ that is corresponded to the standard Young tableau $[\imath^{\lambda}]$. The following known hook formula (see [7, I.4.3]) holds,

[TABLE]

with h(i,j)\!=\!\#\big{\{}\Box_{i^{\prime}j^{\prime}}\in[\imath^{\lambda}]\colon i^{\prime}\geq i,j^{\prime}=j\big{\}}\!=\!\#\big{\{}\Box_{i^{\prime}j^{\prime}}\in[\imath^{\lambda}]\colon{i^{\prime}=i,j^{\prime}\geq j}\big{\}} independed of $\imath\in\mathscr{I}$. Assign to $\imath\in\mathscr{I}_{\eta}$ the vectors

[TABLE]

Let $s^{\lambda}_{\imath}(\mathfrak{u},h):=s^{\lambda}_{\imath}(t_{\imath})$ with $t_{\imath}=t_{\imath}(\mathfrak{u},h)$ for all ${\mathfrak{u}\in\mathfrak{U}}$, where polynomial terms are ${\phi}^{\lambda}_{\imath}(\mathfrak{u}){\mathfrak{e}}_{\imath}^{*\lambda}(h)={\phi}^{\lambda_{1}}_{\imath_{1}}(\mathfrak{u}){\mathfrak{e}}^{*\lambda_{1}}_{\imath_{1}}(h)\ldots{\phi}^{\lambda_{\eta}}_{\imath_{\eta}}(\mathfrak{u}){\mathfrak{e}}^{*\lambda_{\eta}}_{\imath_{\eta}}(h)$. Applying the Frobenius formula [17, I.7] and taking into account (1.2), (1.3), (5.4), we obtain

[TABLE]

where $s^{\lambda}_{\imath}=0$ if $\lambda^{\intercal}_{1}>l_{\lambda}$ and the summation is over all standard tabloids. Hence, \big{\{}\phi_{h}^{n}\colon h\in H\big{\}} is total in $L_{\chi}^{2,n}$ by Theorem 4.1. In consequence, $\left\{\exp(\phi_{h})\colon h\in H\right\}$ is total in $L_{\chi}^{2}$. This yields surjectivity of $\varPhi$ and of all its restrictions to ${H}^{\odot n}_{\beta}$. ∎

Corollary 5.4**.**

The sets \big{\{}\phi_{h}^{n}\colon{h}\in{H}\big{\}} in $L^{2,n}_{\chi}$ and $\left\{\exp\phi_{h}\colon{h}\in{H}\right\}$ in $L^{2}_{\chi}$ are total.

6 Fourier analysis on virtual unitary matrices

Consider the isometry ${H}^{*\odot n}_{\beta}\stackrel{{\scriptstyle\mathcal{P}}}{{\simeq}}{P}_{\beta}^{n}(H)$ (see e.g., [6, 1.6]), where the space ${P}_{\beta}^{n}(H)$ of unitarily-weighted $n$-homogeneous Hilbert–Schmidt polynomials of variable $h\in H$ is defined to be a restriction to the diagonal in ${H\times\ldots\times H}$ of the $n$-linear forms ${\mathcal{P}\circ\psi_{n}}$ endowed with the norm $\|\psi_{n}^{*}\|_{P_{\beta}^{n}}=\|\psi_{n}\|_{H^{\otimes n}_{\beta}}$ where

[TABLE]

Let ${H}^{2}_{\beta}=\sum_{n\geq 0}{P}_{\beta}^{n}(H)$ be the direct sum of functions $\psi^{*}(h)=\sum\psi_{n}^{*}(h)$ of variable ${h\in H}$ with summands $\psi_{n}^{*}=\mathcal{P}\circ\psi_{n}\in{P}_{\beta}^{n}(H)$ where $\psi=\sum\psi_{n}\in\Gamma_{\beta}(H)$. Since the set $\{\exp(h)\colon h\in H\}$ is total in $\Gamma_{\beta}(H)$, elements of ${H}^{2}_{\beta}$ can be written as

[TABLE]

The analyticity of $H\ni{h}\mapsto\psi^{*}(h)$ is a result of the composition $\exp(\cdot)$ and $\psi^{*}(\cdot)$.

Definition 6.1**.**

Let ${H}^{2}_{\beta}$ be defined as a Hardy space of unitarily-weighted Hilbert–Schmidt analytic functions $\psi^{*}(h)$ of variable ${h\in H}$ endowed with the inner product

[TABLE]

The conjugate-linear surjective isometry from ${H}^{2}_{\beta}$ onto $\Gamma_{\beta}(H)$ is realized by the conjugate-linear mapping

[TABLE]

On the other hand, the correspondence $\varPhi\colon\mathfrak{e}^{\odot\lambda}_{\imath}\rightleftarrows\phi_{\imath}^{\lambda}$ with $\lambda\in\mathbb{Y}$ and $\imath\in\mathscr{I}_{\eta(\lambda)}$ allows us to determine the conjugate-linear isometry from $\Gamma_{\beta}(H)$ onto $L^{2}_{\chi}$. As a result, the mapping

[TABLE]

defines the surjective isometry

[TABLE]

Lemma 6.2**.**

The systems of Hilbert–Schmidt polynomials of variable ${h\in{H}}$,

[TABLE]

where ${\mathfrak{e}^{*\emptyset}_{\imath}=1}$, form orthogonal bases in ${P}_{\beta}^{n}(H)$ and ${H}^{2}_{\beta}$, respectively, such that

[TABLE]

Every function ${\psi^{*}\in{H}^{2}_{\beta}}$ with $\psi\in\Gamma_{\beta}(H)$ has the expansion with respect to $\mathfrak{e}^{*\mathbb{Y}}$

[TABLE]

with summation in the inner sum over all semistandard tabloids ${[\imath^{\lambda}]}$ such that ${\imath^{\lambda}\vdash n}$. Each function ${\psi^{*}\in{H}^{2}_{\beta}}$ is entire Hilbert–Schmidt analytic and can be also written as

[TABLE]

The following linear isometries, defined by linearization via coherent states, hold

[TABLE]

Proof.

Taking into account (3.3) and (5.2), we conclude that every $\psi^{*}\in{H}^{2}_{\beta}$ such that $\psi=\bigoplus\psi_{n}\in\Gamma_{\beta}(H)$ with ${\psi_{n}\in{H}^{\odot n}_{\beta}}$ has the following expansion

[TABLE]

On the other hand, in relative to the inner product $\langle\cdot\mid\cdot\rangle_{\Gamma}$, we have

[TABLE]

Verify the first equality in (LABEL:cformula) by substituting (6.1) into the formula (LABEL:cformula). We get

[TABLE]

If ${\omega^{*}(h^{\prime}):=\psi^{*}(h)\exp\langle h\mid h^{\prime}\rangle[\exp\langle h^{\prime}\mid h^{\prime}\rangle]^{-1}}$ then $\omega^{*}(h)=\psi^{*}(h)$ for ${h=h^{\prime}\in{H}}$. Now, putting \omega^{*}(h^{\prime}):=\big{\langle}\psi^{*}(\cdot)\mid\exp\langle h^{\prime}\mid\cdot\rangle[\exp\langle h^{\prime}\mid h^{\prime}\rangle]^{-1}\exp\langle\cdot\mid h^{\prime}\rangle\big{\rangle}_{H^{2}_{\beta}}, we obtain

[TABLE]

Hence, the second equality in (LABEL:cformula) holds. Lemma 5.3 yields (6.3). ∎

Remark 6.3**.**

Since $\phi_{h}=\sum\mathfrak{e}^{*}_{i}(h)\phi_{i}$ for all $h=\sum\mathfrak{e}^{*}_{i}(h)\mathfrak{e}_{i}$, a range of the embedding (4.2) coincides with $L_{\chi}^{2,1}$.

Lemma 6.4**.**

Denote $\exp\langle h^{\prime}\mid h\rangle:=K(h^{\prime},h)$. The functions

[TABLE]

with $\mathfrak{u}\in\mathfrak{U}$ take values in $L^{2}_{\chi}$ and can be represented as follows

[TABLE]

where the last exponential function has the power series expansion

[TABLE]

with coefficients in the form of complex Hermite polynomials $\mathfrak{h}_{n,m}(z,\bar{z})$, $z\in\mathbb{C}$.

Proof.

Applying the transform $\varPsi$ to ${K}(h^{\prime},h)$ in variable ${h^{\prime}\in{H}}$, we obtain

[TABLE]

Similarly, applying $\varPsi$ to ${E}(h^{\prime},h)$ in variable ${h^{\prime}\in{H}}$, we obtain

[TABLE]

By Lemma 5.3, $(\varPsi\circ{K})(\cdot,h)$ and $(\varPsi\circ{E})(\cdot,h)$ with ${h\in{H}}$ take values in $L_{\chi}^{2}$. The expansion (6.4) follows from [12, n.12] where polynomials $\mathfrak{h}_{n,m}(z,\bar{z})$ were introduced. ∎

Theorem 6.5**.**

For any $f={\sum f_{n}\in L^{2}_{\chi}}$ with $f_{n}\in L^{2,n}_{\chi}$ the entire function

[TABLE]

and its Taylor coefficients at zero $d^{n}_{0}\hat{f}$ have the integral representations

[TABLE]

respectively. The Fourier transform ${F\colon L^{2}_{\chi}\ni{f}\longmapsto\hat{f}\in{H}^{2}_{\beta}}$ provides the isometries

[TABLE]

Proof.

Since $\varPsi=\varPhi\circ*^{-1}$, we obtain $\varPsi^{*}=*\circ\varPhi^{*}$. From (LABEL:cformula) it follows that \hat{f}(h)=\left\langle\exp(h)\mid\varPhi^{*}f\right\rangle_{\beta}={\big{\langle}(\varPsi^{*}\circ f)(\cdot)\mid{K}(\cdot,h)\big{\rangle}_{{H}^{2}_{\beta}}}=\big{\langle}(\varPsi^{*}\circ f)(\cdot)\mid{E}(\cdot,h)\big{\rangle}_{{H}^{2}_{\beta}}. Thus,

[TABLE]

by Lemma 6.4. On the other hand, according to the same claim

[TABLE]

It particularly follows that for all $h=\alpha{x}$ with $x\in H$,

[TABLE]

Using the $n$-homogeneity of derivatives, we find

[TABLE]

Finally, we notice that the isometry ${L^{2}_{\chi}\stackrel{{\scriptstyle F}}{{\simeq}}{H}^{2}_{\beta}}$ holds, since the isometry $\varPhi^{*}$ is surjective by Lemma 6.2. Similarly, we get $L^{2,n}_{\chi}\stackrel{{\scriptstyle F}}{{\simeq}}{P}_{\beta}^{n}(H)$. ∎

Corollary 6.6**.**

For any $h\in{H}$ the Paley–Wiener map $\phi_{h}$ satisfies the equality

[TABLE]

Proof.

It is enough to put $f\equiv 1$ and to replace $h$ by $h/2$ in the formula (6.5). ∎

Corollary 6.7**.**

The isometry ${*\colon\Gamma_{\beta}(H)\longrightarrow{H}^{2}_{\beta}}$ has the factorization $*={F}\circ\varPhi$.

Proof.

In fact, ${\varPhi\colon\Gamma_{\beta}(H)\ni\psi\longmapsto\varPhi\psi=f\in L^{2}_{\chi}}$ and ${F}\colon L^{2}_{\chi}\ni f\longmapsto\hat{f}\in{H}^{2}_{\beta}$. ∎

Corollary 6.8**.**

For every $f\in L^{2}_{\chi}$ the Taylor expansion at zero of the function

[TABLE]

has the coefficients

[TABLE]

with summation over all standard Young tabloids ${[\imath^{\lambda}]}$ such that ${\imath^{\lambda}\vdash n}$ where $s^{\lambda}_{\imath}=0$ if the conjugate partition $\lambda^{\intercal}$ has $\lambda^{\intercal}_{1}>\eta(\lambda)$ and ${s}^{\lambda}_{\imath}[{f}_{\imath}\,{\mathfrak{e}}_{\imath}^{*}(h)]:=s^{\lambda}_{\imath}(t_{\imath})$ with $t_{\imath}={f}_{\imath}\,{\mathfrak{e}}_{\imath}^{*}(h)$.

Proof.

By the Frobenius formula [17, I.7] we find that $\phi_{h}^{n}(\mathfrak{u})=\sum_{\imath^{\lambda}\vdash n}\hbar_{\lambda}s^{\lambda}_{\imath}(\mathfrak{u},h)$, where $s^{\lambda}_{\imath}=0$ if $\lambda^{\intercal}_{1}>\eta(\lambda)$, and $s^{\lambda}_{\imath}(\mathfrak{u},h)$ is defined by (1.3), whereas $\hbar_{\lambda}$ by (5.4). Thus,

[TABLE]

Using (6.7) in combination with Theorem 4.1, we find

[TABLE]

where the derivative at zero may be defined as

[TABLE]

In fact, for $zh$ with ${z\in\mathbb{C}}$ and $\imath^{\lambda}\vdash n$ with $\lambda^{\intercal}_{1}>\eta(\lambda)$ we find

[TABLE]

Hence, the derivative $d^{n}_{0}\hat{f}(h)=(d^{n}/dz^{n})\hat{f}(zh)|_{z=0}$ is a Taylor coefficient of $\hat{f}$.

Now, the Frobenius formula and Theorem 4.1 yield the first equality in (6.6). By Lemmas 6.2 and 6.4 the second formula in (6.6) also holds. ∎

Remark 6.9**.**

In the finite-dimensional case $\mathfrak{U}=U(m)$, the Hardy space ${H}^{2}_{\beta}$ of entire analytic functions of variable $h\in\mathbb{C}^{m}$ has the following orthogonal basis \big{\{}\mathfrak{e}^{*\lambda}=\mathfrak{e}^{*\lambda_{1}}_{1}\ldots\mathfrak{e}^{*\lambda_{m}}_{m}\colon{\lambda=(\lambda_{1},\ldots,\lambda_{m})\in\mathbb{Y}}\big{\}}. The Fourier transform

[TABLE]

provides the surjective isometry ${F\colon L_{\chi_{m}}^{2}\ni{f}\longmapsto\hat{f}\in{H}^{2}_{\beta}}$, defined by mappings

[TABLE]

where the space $L_{\chi_{m}}^{2}$ with the Haar measure $\chi_{m}$ on $U(m)$ has the orthogonal basis \big{\{}\phi^{\lambda}={\phi}^{\lambda_{1}}_{1}\circ\pi_{m}^{-1}\ldots{\phi}^{\lambda_{m}}_{m}\circ\pi_{m}^{-1}\colon\lambda\in\mathbb{Y}\big{\}}.

7 Intertwining properties of Fourier transform

The shift group on ${H}^{2}_{\beta}$ is defined as $T_{a}\psi^{*}(h):=\left\langle\mathcal{T}_{a}\exp(h)\mid\psi\right\rangle_{\beta}$ for all $\psi\in\Gamma_{\beta}(H)$, ${a,h\in H}$. By (LABEL:cformula), \left\langle\mathcal{T}_{a}\exp(h)\mid\psi\right\rangle_{\beta}=T_{a}\psi^{*}(h)=\big{\langle}T_{a}\psi^{*}(\cdot)\mid\exp\langle\cdot\mid h\rangle\big{\rangle}_{H^{2}_{\beta}}. Hence,

[TABLE]

where ${M}_{a^{*}}\exp\langle\cdot\mid h\rangle:=\exp{a}^{*}(\cdot)\exp\langle\cdot\mid h\rangle=\exp\langle\cdot\mid h+a\rangle$ is defined to be the multiplicative group onto the total set $\{{\exp\langle\cdot\mid h\rangle}\colon h\in H\}$ in ${H}^{2}_{\beta}$.

Comparing the above formulas, we obtain that ${M}_{a^{*}}$ is adjoint to ${T}_{a}$ on ${H}^{2}_{\beta}$. By virtue of adjoint relations, $\|{T}_{a}\psi^{*}\|_{H^{2}_{\beta}}=\|{M}_{a^{*}}\psi^{*}\|_{H^{2}_{\beta}}$. The isometry $H^{2}_{\beta}\simeq\Gamma_{\beta}(H)$ yields $\|{T}_{a}\psi^{*}\|_{H^{2}_{\beta}}=\|\mathcal{T}_{a}\psi\|_{\beta}$. According to (5.3), we have

[TABLE]

for ${a,b\in H}$. Thus, these groups are strongly continuous with densely defined closed generators $\partial^{*}_{a}\psi^{*}:={\lim_{z\to 0}(T_{za}\psi^{*}-\psi^{*})/z}$ and ${a}^{*}\psi^{*}:={\lim_{z\to 0}(M_{za^{*}}\psi^{*}-\psi^{*})/z}$.

Hence, the additive group $(H,+)$ on ${H}^{2}_{\beta}$ is represented by ${M}_{a^{*}}\colon{H}^{2}_{\beta}\rightarrow{H}^{2}_{\beta}$ and the generator $dM_{za^{*}}/dz\mid_{z=0}=a^{*}$ of its $1$-parameter subgroup $M_{za^{*}}$ is strongly continuous with the dense domain \mathfrak{D}(a^{\!*})={\big{\{}\psi^{*}\in{H}^{2}_{\beta}\colon a^{*}\psi^{*}\in{H}^{2}_{\beta}\big{\}}}. On the other hand, the group $(H,+)$ can be represented as $M_{a^{*}}^{\dagger}=\varPsi{M}_{a^{*}}\varPsi^{*}\colon L^{2}_{\chi}\rightarrow L^{2}_{\chi}$. The generator of its strongly continuous subgroup

[TABLE]

has the dense domain \mathfrak{D}(\bar{\phi}_{a})={\big{\{}f\in L^{2}_{\chi}\colon\bar{\phi}_{a}f\in L^{2}_{\chi}\big{\}}} and is closed, since $a^{*}$ is closed.

The group $(H,+)$ on $L^{2}_{\chi}$ can be also represented by $T_{a}^{\dagger}:=\varPsi{T}_{a}\varPsi^{*}\colon L^{2}_{\chi}\rightarrow L^{2}_{\chi}$. From Lemmas 5.2, 6.2 it follows that the generator of strongly continuous subgroup

[TABLE]

has the dense domain \mathfrak{D}(\partial_{a}^{\dagger})={\big{\{}f\in L^{2}_{\chi}\colon\partial_{a}^{\dagger}f\in L^{2}_{\chi}\big{\}}} and is closed, since $\partial_{a}^{*}$ is closed. By (LABEL:cformula) \hat{f}(h)=\left\langle\exp(h)\mid\varPhi^{*}f\right\rangle_{\beta}=\big{\langle}(\varPsi^{*}\circ f)(\cdot)\mid\exp\langle\cdot\mid h\rangle\big{\rangle}_{H^{2}_{\beta}}. Hence, by Lemma 6.4,

[TABLE]

Lemma 7.1**.**

The additive group $(H,+)$ on $L^{2}_{\chi}$ has two representations ${a\mapsto M_{a^{*}}^{\dagger}}$ and ${a\mapsto T_{a}^{\dagger}}$ which are adjoint, strongly continuous with closed densely defined generators $\bar{\phi}_{a}$ and $\partial_{a}^{\dagger}$, respectively. For every f\in\mathfrak{D}(\bar{\phi}_{a}^{m})={\big{\{}f\in L^{2}_{\chi}\colon\bar{\phi}_{a}^{m}{f}\in L^{2}_{\chi}\big{\}}} with $m\in\mathbb{N}_{0}$,

[TABLE]

For every f\in\mathfrak{D}(\partial_{a}^{\dagger m})={\big{\{}f\in L^{2}_{\chi}\colon\partial_{a}^{\dagger m}{f}\in L^{2}_{\chi}\big{\}}} with $m\in\mathbb{N}_{0}$,

[TABLE]

As a conclusion, $\partial_{\mathbbm{i}a}^{\dagger}=-\mathbbm{i}\partial_{a}^{\dagger}$. Moreover, the following commutation relations hold,

[TABLE]

for all $f$ from the dense subspace $\mathfrak{D}(\bar{\phi}_{a}^{2})\cap\mathfrak{D}(\partial_{b}^{\dagger 2})\subset{L}^{2}_{\chi}$ and nonzero $a,b\in{H}$.

Proof.

Using that ${T}_{a}$ and ${M}_{a^{\!*}}$ are adjoint, we find that

[TABLE]

for all ${f\in L^{2}_{\chi}}$. This gives (7.2). Since M_{a^{*}}\psi^{*}(h)=\big{\langle}\psi^{*}(\cdot)\mid{M}_{a^{*}}\exp\langle\cdot\mid h\rangle\big{\rangle}_{H^{2}_{\beta}}=\exp{a^{*}}(h)\,\psi^{*}(h), we obtain

[TABLE]

This together with the group property by applying ${F}$ and ${F}^{-1}$ yields (7.3).

Now, we prove the commutation relations. For any ${f\in L^{2}_{\chi}}$ and $h\in{H}$, we have

[TABLE]

For each $\hat{f}\in\mathfrak{D}(b^{*2})\cap\mathfrak{D}(\partial_{a}^{2})$ and ${t\in\mathbb{C}}$ by differentiation, we obtain

[TABLE]

Subsequently, taking into account (7.6) together with ${(d/dt)[\exp\langle ta\mid\bar{t}b\rangle M_{tb^{*}}T_{ta}]}$ $=[(d/dt)\exp\langle{ta}\mid\bar{t}b\rangle]M_{tb^{*}}T_{ta}+\exp\langle{ta}\mid{\bar{t}b}\rangle[(d/dt)M_{tb^{*}}T_{ta}],$ we find

[TABLE]

Hence, for each $\hat{f}$ from the dense subspace $\mathfrak{D}(b^{*2})\cap\mathfrak{D}(\partial_{a}^{2})\subset{H}^{2}_{\beta}$, which includes all polynomials generated by finite sums $\varPsi^{*}(f)=\bigoplus\psi_{n}\in\Gamma_{\beta}(H)$ with $\psi_{n}\in{H}^{\odot{n}}_{\beta}$,

[TABLE]

Corollary 6.7 yields ${F}=*\circ\varPhi^{*}$ and ${F}^{-1}=\varPhi\circ*^{-1}$. The equality (7.5) for $m=0$ can be rewritten as $M_{b^{*}}\hat{f}(a)=\left\langle\exp(a)\mid{T}_{b}\varPhi^{*}f\right\rangle_{\beta}$ with ${f\in L^{2}_{\chi}}$ or in another way $*\circ{T}_{b}=M_{b^{*}}\circ*$. Hence, $T_{b}^{\dagger}=\varPhi\,{T}_{b}\varPhi^{*}=\varPhi\circ{*}^{-1}\circ{M}_{b^{*}}\circ*\circ\varPhi^{*}={F}^{-1}M_{b^{*}}\,{F}$ and $\partial_{b}^{\dagger}={F}^{-1}b^{*}\,{F}$. Similarly, $M_{a^{*}}^{\dagger}={F}^{-1}T_{a}\,{F}$ and $\bar{\phi}_{a}={F}^{-1}\partial_{a}^{*}\,{F}$. Finally,

[TABLE]

for all $f$ from the dense subspace $\mathfrak{D}(\bar{\phi}_{a}^{2})\cap\mathfrak{D}(\partial_{b}^{\dagger 2})\subset{L}^{2}_{\chi}$, which includes all functions generated by finite sums $\varPhi\left(\bigoplus\psi_{n}\right)$ with $\psi_{n}\in{H}^{\odot{n}}_{\beta}$. ∎

8 Infinite-dimensional Heisenberg group

Our goal is to describe an irreducible representation on the space $L^{2}_{\chi}$ of the group $\mathcal{H}_{\mathbb{C}}$, defined by (1.1). We will use the appropriate generalization of Weyl’s system which in our case is written in the form of $L^{2}_{\chi}$-valued function of variable ${h\in H}$

[TABLE]

For convenience, we will use the quaternion algebra $\mathbbm{H}=\mathbb{C}\oplus\mathbb{C}\mathbbm{j}$ of numbers $\zeta={{(\alpha_{1}+\alpha_{2}\mathbbm{i})}+{(\alpha^{\prime}_{1}+\alpha^{\prime}_{2}\mathbbm{i})}\mathbbm{j}}={\alpha+\alpha^{\prime}\mathbbm{j}}$ such that $\mathbbm{i}^{2}=\mathbbm{j}^{2}=\mathbbm{k}^{2}=\mathbbm{i}\mathbbm{j}\mathbbm{k}=-1$, $\mathbbm{k}=\mathbbm{i}\mathbbm{j}=-\mathbbm{j}\mathbbm{i}$, $\mathbbm{k}\mathbbm{i}=-\mathbbm{i}\mathbbm{k}=\mathbbm{j}$, where $(\alpha,\alpha^{\prime})\in\mathbb{C}^{2}$ with $\alpha={\alpha_{1}+\alpha_{2}\mathbbm{i}},\alpha^{\prime}={\alpha^{\prime}_{1}+\alpha^{\prime}_{2}\mathbbm{i}\in\mathbb{C}}$ and $\alpha_{\imath},\alpha^{\prime}_{\imath}\in\mathbb{R}$ $(\imath=1,2)$ [25, 5.5.2]. Let us denote $\alpha^{\prime}:=\Im{\zeta}$ for all $\zeta=\alpha+\alpha^{\prime}\mathbbm{j}\in\mathbbm{H}$.

Consider the Hilbert space ${H}\oplus{H}\mathbbm{j}$ with $\mathbbm{H}$-valued inner product

[TABLE]

where ${h}={a}+{b}\mathbbm{j}$ with ${a}$, ${b}\in{H}$. Hence,

[TABLE]

Theorem 8.1**.**

The representation of  $\mathcal{H}_{\mathbbm{C}}$ over $L^{2}_{\chi}$ in the Weyl-Schrödinger form

[TABLE]

is well defined and irreducible. The Weyl system satisfies the relation

[TABLE]

which on any real subspace $\{\tau{h}\colon\tau\in\mathbb{R}\}$ transforms to the $1$-parameter group

[TABLE]

with the densely defined generator on $L^{2}_{\chi}$ of the form $\mathfrak{p}^{\dagger}_{h}:=\partial_{b}^{\dagger}+\bar{\phi}_{a}$. Moreover, the following commutation relations hold,

[TABLE]

on the dense subspace $\mathfrak{D}(\bar{\phi}_{a}^{2})\cap\mathfrak{D}(\partial_{b}^{\dagger 2})\subset{L}^{2}_{\chi}$.

Proof. Let us consider the auxiliary group $\mathbb{C}\times(H\oplus{H}\mathbbm{j})$ with multiplication $(t,{h})(t^{\prime},{h}^{\prime})=\left(t+t^{\prime}-\frac{1}{2}\Im\langle{h}\mid{h}^{\prime}\rangle,\,{h}+{h}^{\prime}\right)$ for all ${h}={a}+{b}\mathbbm{j}$, ${h}^{\prime}={a}^{\prime}+{b}^{\prime}\mathbbm{j}\in{H}\oplus{H}\mathbbm{j}$. The mapping ${G}\colon X({a},{b},t)\longmapsto\left(t-\frac{1}{2}\langle{a}\mid{b}\rangle,\,{a}+{b}\mathbbm{j}\right)$ is a group isomorphism, since

[TABLE]

On the other hand, let us define the auxiliary Weyl system

[TABLE]

Using group properties and the commutation relation (7.7), we obtain

[TABLE]

Hence, the mapping ${\mathbb{C}\times({H}\oplus{H}\mathbbm{j})\ni(t,{h})\longmapsto\exp(t){W}({h})}$ acts as a group isomorphism into the operator algebra over ${H}^{2}_{\beta}$. So, the representation

[TABLE]

is also well defined over ${H}^{2}_{\beta}$, as a composition of group isomorphisms.

Let us check the irreducibility. Suppose the contrary. Assume there exist an element ${h}_{0}\operatorname{=/\ }0$ in ${H}$ and an integer ${n>0}$ such that

[TABLE]

But, this is only possible for ${h}_{0}=0$. It gives a contradiction. Finally, using that

[TABLE]

we obtain that $S^{\dagger}={F}^{-1}S\,{F}$ is irreducible. Applying ${F}$, ${F}^{-1}$ to (8.5) we get (8.1).

Consider the Weyl system ${W}^{\dagger}$ on the space $L^{2}_{\chi}$. By (8.1) we obtain the equality

[TABLE]

Using this equality, we get (8.2) for any fixed ${h}={a}+{b}\mathbbm{j}\in{H}\oplus{H}\mathbbm{j}$. The $1$-parameter group ${W}^{\dagger}(\tau{a},\tau{b})={W}^{\dagger}(\tau{h})$ with real $\tau$ has the generator $\mathfrak{p}_{h}^{\dagger}=\mathfrak{p}_{a,b}^{\dagger}$, since

[TABLE]

Taking into account the inequalities (7.1) and that $F$ is isometric, we get

[TABLE]

Hence, the group ${W}^{\dagger}(\tau{a},\tau{b})$ in variable $\tau\in\mathbb{R}$ is strongly continuous on $L^{2}_{\chi}$ and therefore has the dense domain \mathfrak{D}(\mathfrak{p}_{h}^{\dagger})={\big{\{}f\in L^{2}_{\chi}\colon\mathfrak{p}^{\dagger}_{h}f\in L^{2}_{\chi}\big{\}}}. Moreover, its generator $\mathfrak{p}^{\dagger}_{h}$ is closed (see, e.g., [34]). Note also that $\mathfrak{p}^{\dagger}_{\tau{h}}=\tau\mathfrak{p}^{\dagger}_{h}$ for $\tau\in\mathbb{R}$.

Finally, applying the commutation relation (7.4) and commutability of group generators in different directions over the dense set $\mathfrak{D}(\bar{\phi}_{a}^{2})\cap\mathfrak{D}(\partial_{b}^{\dagger 2})\subset{L}^{2}_{\chi}$, we have

[TABLE]

9 Heat equation associated with Weyl system

In what follows, we will consider the real Banach space $c_{0}$ and let $\xi_{n}^{*}$ be the coordinate functional, i.e., $\xi_{n}^{*}(\xi)=\xi_{n}$ for ${\xi\in c_{0}}$. Since, the embedding $\mathcal{I}\colon{l}_{2}\looparrowright c_{0}$ is continuous, the Gelfand triple $l_{1}\stackrel{{\scriptstyle\mathcal{I}^{*}}}{{\longrightarrow}}l_{2}\looparrowright c_{0}$ with adjoint $\mathcal{I}^{*}$ holds. The mapping $Q\colon{l}_{1}\to c_{0}$ with $Q:=\mathcal{I}\circ\mathcal{I}^{*}$ is positive and $\langle Q\xi^{*}\mid Q\xi^{*}\rangle_{l_{2}}:=\xi^{*}(Q\xi^{*})=\sum\xi_{n}^{2}=\|\xi\|^{2}_{l_{2}}$ where $\xi=Q\xi^{*}\in\mathscr{R}(Q)$ and $\xi^{*}\in{l}_{1}=c^{*}_{0}$. By the Aronszajn–Kolmogorov decomposition theorem (see e.g., [21, Prop.1]) the appropriative reproducing kernel Hilbert space can be determined as $\overline{\mathscr{R}(Q)}=l_{2}$.

Consider the abstract Wiener space defined by $\mathcal{I}\colon l_{2}\looparrowright c_{0}$. Given $\xi_{1}^{*},\ldots,\xi_{n}^{*}\in l^{1}=c_{0}^{*}$, we assign the family of cylinder sets $\Omega_{n}^{c}=\left\{\xi\in c_{0}\colon(\xi_{1}^{*}(\xi),\ldots,\xi_{n}^{*}(\xi))\in\Omega_{n}\right\}$ with any Borel $\Omega_{n}\subset\mathbb{R}^{n}$ that are not a $\sigma$-field. Define the $\sigma$-additive extension $\mathfrak{w}$ of the Gaussian measure $\gamma$ onto the Borel $\sigma$-algebra $\mathscr{B}(c_{0})$, called futhure the Wiener measure, such that

[TABLE]

By Gross’ theorem [10] there exists a smaller abstract Wiener space $\{w_{0},\|\cdot\|_{w_{0}}\}$ such that injections ${l_{2}\looparrowright{w}_{0}\looparrowright c_{0}}$ are continuous and the increasing sequence of orthogonal projectors $p_{n}\colon{l}_{2}\to\mathbb{R}^{n}$ has the extension $(p^{\sim}_{n})$ on $w_{0}$ that is convergent to the identity operator on $w_{0}$ and ${\mathfrak{w}(w_{0})=1}$. The integral of any cylinder function ${\upsilon\colon c_{0}\to\mathbb{R}}$ such that $\upsilon=\rho\circ p_{n}^{\sim}$ is defined to be $\int_{\Omega_{n}^{c}}\upsilon\,d\mathfrak{w}=\int_{\Omega_{n}}\rho\,d\gamma$. The Fernique theorem [5],[14, Thm 3.1] implies that these exist $\varepsilon,\eta>0$ such that $\|\cdot\|_{w_{0}}$ satisfies the following conditions with a sufficiently large $K>0$,

[TABLE]

Let us go back to the Weyl system ${W}^{\dagger}$. Consider in $L^{2}_{\chi}$ the dense subspace ${L}^{+2}_{\chi}:=\bigcup_{n\geq 0}\bigoplus_{m=0}^{n}L^{2,m}_{\chi}$. Let ${a}={b}=\mathbbm{i}\xi_{m}\mathfrak{e}_{m}$ with ${\xi_{m}\in\mathbb{R}}$. Then by Theorem 8.1

[TABLE]

Theorem 9.1**.**

For any $f\in{L}^{+2}_{\chi}$ and $\xi=(\xi_{m})\in c_{0}$ there exists the limit

[TABLE]

$\mathfrak{w}$-almost everywhere on $c_{0}$ such that the $1$-parameter Gaussian semigroup

[TABLE]

on the space ${L}^{+2}_{\chi}$ is generated by -\sum\big{(}\partial_{m}^{\dagger}+\bar{\phi}_{m}\big{)}^{2} with $\partial_{m}^{\dagger}:=\partial_{\mathfrak{e}_{m}}^{\dagger}$. As a consequence, $w(r)=\mathfrak{G}^{\dagger}_{r}f$ is unique solution of the Cauchy problem

[TABLE]

Proof.

Note that $(M_{b^{*}}T_{a})^{*}=T_{a}^{*}M_{b^{*}}^{*}=M_{a^{*}}T_{b}$. Hence, $(\partial_{a}^{\dagger}+\bar{\phi}_{a})^{*}=\partial_{a}^{\dagger}+\bar{\phi}_{a}$ is self-adjoint for $a=b$, as a generator of the group ${W}^{\dagger}(\tau{a},\tau{a})=\exp\left\{{\|\tau a\|^{2}}/{2}\right\}T_{\tau{a}}^{\dagger}M_{\tau{a}^{*}}^{\dagger}$ with ${\tau\in\mathbb{R}}$. Replacing $a=b$ by $\mathbbm{i}\tau a$ with ${\tau\in\mathbb{R}}$, we obtain that

[TABLE]

with self-adjoint $\partial_{a}^{\dagger}+\bar{\phi}_{a}$. By relations (7.4), ${W}^{\dagger}(\mathbbm{i}\tau{a},\mathbbm{i}\tau{a})$ is unitary.

Lemma 7.1 implies that $[M_{-\mathbbm{i}\xi_{m}\mathfrak{e}_{m}^{*}}^{\dagger},T_{\mathbbm{i}\xi_{k}\mathfrak{e}_{k}}^{\dagger}]=0$ and ${[M_{-\mathbbm{i}\xi_{m}\mathfrak{e}_{m}^{*}}^{\dagger},M_{-\mathbbm{i}\xi_{k}\mathfrak{e}_{k}^{*}}^{\dagger}]=0}$, as well as, $[T_{\mathbbm{i}\xi_{m}\mathfrak{e}_{m}}^{\dagger},T_{\mathbbm{i}\xi_{k}\mathfrak{e}_{k}}^{\dagger}]=0$ for any $m\operatorname{=/\ }k$. In view of the relations (7.4),

[TABLE]

Check that (9.1) holds. Denote ${W}^{\dagger}_{p_{n}^{\sim}(\xi)}:=\prod_{m=1}^{n}{W}^{\dagger}(\mathbbm{i}\xi_{m}\mathfrak{e}_{m},\mathbbm{i}\xi_{m}\mathfrak{e}_{m})$ and $T^{\dagger}_{p_{n}^{\sim}(\xi)}:=\prod_{m=1}^{n}T^{\dagger}_{\mathbbm{i}\xi_{m}\mathfrak{e}_{m}}$, as well as, $M^{\dagger}_{p_{n}^{\sim}(\xi)}:=\prod_{m=1}^{n}M^{\dagger}_{-\mathbbm{i}\xi_{m}\mathfrak{e}_{m}^{*}}$ with ${\xi=(\xi_{m})\in{w}_{0}}$. Using (7.1) with the operator norm over $H^{2}_{\beta}$, we get the inequality

[TABLE]

The relation $T_{\mathbbm{i}\xi_{m}\mathfrak{e}_{m}}^{\dagger}=\varPsi{T}_{\mathbbm{i}\xi_{m}\mathfrak{e}_{m}}\varPsi^{*}$ implies that the left-hand side term above can be changed by $\ln\prod_{m=1}^{n}\|T_{\mathbbm{i}\xi_{m}\mathfrak{e}_{m}}^{\dagger}\|_{\mathscr{L}({L}^{2}_{\chi})}^{2}$. For $M^{\dagger}_{p_{n}^{\sim}(\xi)}=\prod_{m=1}^{n}M^{\dagger}_{-\mathbbm{i}\xi_{m}\mathfrak{e}_{m}^{*}}$ similarly.

Using the unitarity of groups ${W}^{\dagger}(\mathbbm{i}\xi_{m}\mathfrak{e}_{m},\mathbbm{i}\xi_{m}\mathfrak{e}_{m})$, we find by virtue of (9.3) that their product ${W}^{\dagger}_{p_{n}^{\sim}(\xi)}=\exp\left\{-\|{p_{n}^{\sim}(\xi)}\|_{l_{2}}^{2}/2\right\}T^{\dagger}_{p_{n}^{\sim}(\xi)}M^{\dagger}_{p_{n}^{\sim}(\xi)}$ is also unitary. Taking into account the continuity of ${\mathcal{I}_{0}\colon l_{2}\looparrowright{w}_{0}}$ and that $p_{n}^{\sim}$ converges to the identity mapping on $w_{0}$, as well as, that ${\mathfrak{w}(w_{0})=1}$, we obtain for all ${f\in{L}^{+2}_{\chi}}$, $n\geq 0$,

[TABLE]

The Lebesgue dominated convergence theorem implies that there exists $\lim\|{W}^{\dagger}_{p_{n}^{\sim}(\xi)}f\|_{\chi}$ $\mathfrak{w}$-almost everywhere in variable ${\xi\in{w}_{0}}$ for all ${f\in L^{2,m}_{\chi}}$ and ${m>0}$. By completeness of $L^{2,m}_{\chi}$, the limit ${W}^{\dagger}_{\xi}{f}$ is well defined $\mathfrak{w}$-almost everywhere and

[TABLE]

The $\|\cdot\|_{\chi}$-norm of integrant in (9.1) is bounded by $\exp\left\{\varepsilon\|\xi\|_{w_{0}}^{2}\right\}$ with any $\varepsilon>0$. By Fernique’s theorem and (9.4), the integral (9.1) with the Wiener measure $\mathfrak{w}$ exists for all $f\in{L}^{+2}_{\chi}$. The equality $\mathfrak{w}(w_{0})=1$ implies that the integral (9.1) is absolutely convergent uniformly in variables $r>0$ on the whole space $c_{0}$. It provides the $C_{0}$-property of $\mathfrak{G}_{r}$ in variables $r>0$ on any finite sum $\bigoplus_{m=0}^{n}L^{2,m}_{\chi}$.

Prove that the semigroup $\mathfrak{G}_{r}$ is generated by $\sum\mathfrak{p}_{m}^{\dagger 2}$ with $\mathfrak{p}^{\dagger}_{m}:={\mathbbm{i}(\partial_{m}^{\dagger}+\bar{\phi}_{m})}$. By differentiation of ${W}^{\dagger}(\mathbbm{i}\xi_{m}{a},\mathbbm{i}\xi_{m}{a})$ at ${\xi_{m}=0}$, we get that its generator coincides with $\mathfrak{p}^{\dagger}_{m}$. In fact, {W}^{\dagger}(\mathbbm{i}\xi_{m}{a},\mathbbm{i}\xi_{m}{a}){f}=\exp\big{\{}\xi_{m}\mathfrak{p}_{m}^{\dagger}\big{\}}f for all ${f\in\phi^{\mathbb{Y}}}$. Applying the next formula for Gamma functions with $\alpha=(\alpha_{1},\ldots,\alpha_{n})\in\mathbb{N}_{0}^{n}$

[TABLE]

we find that for any ${L}^{+2}_{\chi}$-valued cylinder function $h_{n}=({W}^{\dagger}_{\xi}{f})\circ p_{n}^{\sim}$ we have

[TABLE]

Using (9.4), we obtain that $0\leq r\longmapsto\mathfrak{G}_{r}^{\dagger}$ is the $1$-parameter $C_{0}$-semigroup on any finite sum $\bigoplus_{m=0}^{n}L^{2,m}_{\chi}$ with densely defined closed generator $\sum_{m=1}^{n}\mathfrak{p}_{m}^{\dagger 2}$. Applying the known relation [34] between the initial problem (9.2) and the $1$-parameter $C_{0}$-semigroup $\mathfrak{G}_{r}^{\dagger}$, we obtain that the function $w_{n}(r)=\mathfrak{G}^{\dagger}_{r}f_{n}$ for any ${n\in\mathbb{N}}$ solves this problem in the sense that $d\mathfrak{G}^{\dagger}_{r}f_{n}/dr|_{r=0}=\sum_{m=1}^{n}\mathfrak{p}_{m}^{\dagger 2}f_{n}$ for all ${f_{n}\in}\bigoplus_{m=0}^{n}L^{2,m}_{\chi}$. The theorem is proved. ∎

Taking into account the isometries ${H}^{2}_{\beta}\stackrel{{\scriptstyle\varPsi}}{{\simeq}}{L}^{2}_{\chi}$ and ${P}_{\beta}^{n}(H)\stackrel{{\scriptstyle\varPsi}}{{\simeq}}{L}^{2,n}_{\chi}$ from (6.3), defined by linearization, we can rewrite the Cauchy problem in polynomial form.

Consider the Weyl system ${W(a,b)=\exp\left\{\langle{a}\mid{b}\rangle/2\right\}M_{b^{*}}T_{a}}$ defined by (8.4) on the dense subspace of polynomials ${P}_{\beta}(H):={\sum}_{n\geq 0}{P}_{\beta}^{n}(H)$ in ${H}^{2}_{\beta},$ consisting of all finite sums of $n$-homogenous polynomials $\psi^{*}(h)=\sum\psi_{n}^{*}(h)$ of variable ${h\in H}$ with components $\psi_{n}^{*}=\mathcal{P}\circ\psi_{n}\in{P}_{\beta}^{n}(H)$. Replacing $a$ by $\tau a$ and $b$ by $\tau b$ with real ${\tau\in\mathbb{R}}$, we get that $T_{\tau a}$ and $M_{\tau b^{*}}$ are generated by closed generators on ${P}_{\beta}(H)$,

[TABLE]

As a consequence, the $1$-parameter Weyl system ${W}(\tau{a},\tau{b})$ has the generator

[TABLE]

densely defined on ${P}_{\beta}(H)$ such that $(\tau b)^{*}+\partial_{\tau a}^{*}=\tau(b^{*}+\partial_{a}^{*})$ for real $\tau$. Let ${W}_{p^{\sim}_{n}(\xi)}=\prod_{m=1}^{n}{W}(\mathbbm{i}\xi_{m}\mathfrak{e}_{m},\mathbbm{i}\xi_{m}\mathfrak{e}_{m})$, $T_{p^{\sim}_{n}(\xi)}=\prod_{m=1}^{n}T_{\mathbbm{i}\xi_{m}\mathfrak{e}_{m}}$, $M_{p^{\sim}_{n}(\xi)}=\prod_{m=1}^{n}M_{-\mathbbm{i}\xi_{m}\mathfrak{e}_{m}^{*}}$.

Corollary 9.2**.**

For all $\psi^{*}\in{P}_{\beta}(H)$ and $\xi=(\xi_{m})\in c_{0}$ there exists the limit

[TABLE]

$\mathfrak{w}$-almost everywhere on $c_{0}$ such that the $1$-parameter Gaussian semigroup

[TABLE]

is generated by $-\sum(\mathfrak{e}_{m}^{*}+\partial_{m}^{*})^{2}$. Thus, $w(r)=\mathfrak{G}_{r}\psi^{*}$ is unique solution of the problem

[TABLE]

in the space of Hilbert–Schmidt polynomials ${P}_{\beta}(H)$.