## 1 Introduction

An aim of this work is to investigate irreducible Weyl–Schrödinger representations of the complexified Heisenberg group $${\mathcal {H}}_\mathbb {C}$$ (see [17, n.9]), consisting of matrix elements X(abt) with any $$a,b\in {H}$$ and $$t\in {\mathbb {C}}$$ such that

\begin{aligned}&X(a,b,t)=\begin{bmatrix} 1 &{}\quad a&{}\quad t \\ 0 &{}\quad \mathbb {1}&{}\quad b \\ 0&{}\quad 0&{}\quad 1 \end{bmatrix},\quad \nonumber \\&X(a,b,t)\cdot X(a',b',t')=\begin{bmatrix} 1&{}\quad a+a'&{}\quad t+t'+\langle {a}\mid {b'}\rangle \\ 0 &{}\quad \mathbb {1}&{}\quad b+ b'\\ 0&{}\quad 0&{}\quad 1 \end{bmatrix} \end{aligned}
(1)

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

The group $${\mathcal {H}}_\mathbb {C}$$ 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 [23, 27]).

In [21, 24] 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 $$\pi _i^{i+1}:{U(i+1)\rightarrow U(i)}$$ 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 [5]) 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 [25, 34, 35].

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, 2]. 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 [19] by using the reproducing kernel Hilbert spaces. The Schrödinger representation of such groups using Gaussian measures over a real Hilbert space was described in [3]. Since the group $${\mathcal {H}}_\mathbb {C}$$ 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 $$\phi :H\ni {h}\longmapsto \phi _h\in L^2_\chi$$ defined by Paley–Wiener maps

\begin{aligned} {\phi }_h({\mathfrak {u}}):= \sum {\phi }_i({\mathfrak {u}})\,{\mathfrak {e}}_i^*(h) \quad \text {with}\quad {\phi }_i({\mathfrak {u}}):=\left\langle u_i({\mathfrak {e}}_i)\mid {\mathfrak {e}}_i\right\rangle , \quad u_i=\pi _i({\mathfrak {u}}), \end{aligned}
(2)

where $${\mathfrak {e}}^*_i(\cdot ):=\left\langle \cdot \mid {\mathfrak {e}}_i\right\rangle$$ and the projections $$\pi _i:{\mathfrak {U}}\ni {\mathfrak {u}} \rightarrow u_i\in U(i)$$ are uniquely defined by $$\pi _i^{i+1}$$. Every function $$\phi _h$$ of variable $${{\mathfrak {u}}\in {\mathfrak {U}}}$$ satisfies the equality (Corollary 3)

\begin{aligned} \int \exp \big \{\mathop {\text {Re}}\phi _h\big \}\,d\chi = \exp \Bigg \{\frac{1}{4}\Vert h\Vert ^2\Bigg \},\quad {h\in H}. \end{aligned}

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,

\begin{aligned} s^\lambda _\imath ({\mathfrak {u}}):=\frac{\mathop {\text {det}} \big [\phi _{\imath _i}^{\lambda _j+\eta -j}({\mathfrak {u}})\big ]_{1\le i,j\le \eta }}{\prod _{1\le i<j\le \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 }. \end{aligned}
(3)

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 (Theorem 1) where $${L}_\chi ^{2,n}$$ are formed by n-homogeneous polynomials $${\phi }^\lambda _\imath$$, normed as follows

\begin{aligned} \Vert {\phi }^\lambda _\imath \Vert _\chi ^2=\int |{\phi }^\lambda _\imath |^2d\chi =\beta _\lambda \lambda !,\qquad \beta _\lambda :=\dfrac{(\eta -1)!}{(\eta -1+n)!}, \quad \lambda !:=\lambda _1!\ldots \lambda _\eta !. \end{aligned}

It is also shown that the surjective linear isometry $$\varPsi :{H}^2_\beta \ni \psi ^*_f\longmapsto {f}\in {L}^2_\chi$$ holds (Lemma 5), 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 $$\Vert {\mathfrak {e}}^{*\lambda }_\imath \Vert _{{H}^2_\beta }= \big (\beta _\lambda {\lambda !}\big )^{1/2}$$.

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 $$\varGamma _\beta (H)$$ is normed as $$\Vert {\mathfrak {e}}^{\odot \lambda }_\imath \Vert _{\varGamma _\beta }= \Vert {\mathfrak {e}}^{*\lambda }_\imath \Vert _{{H}^2_\beta }$$ then each function $$f\in {L}^2_\chi$$ admits the superposition

\begin{aligned} f=\varPsi \circ \psi ^*_f,\qquad \psi ^*_f(h) =\sum _{n\ge 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 _{\varGamma _\beta }, \quad {h\in {H}}, \end{aligned}

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 $$\psi _f\in \varGamma _\beta (H)$$.

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

\begin{aligned} {\hat{f}}(h)=\int \exp ({{\bar{\phi }}}_h)f\,d\chi \quad \text {where}\quad F=\varPsi ^{-1}:{L}^2_\chi \ni {f}\longmapsto {\hat{f}}:=\psi ^*_f\in {H}^2_\beta . \end{aligned}

The Fourier transform F acts isometrically on the Hardy space of analytic functions $${H}^2_\beta$$ (Theorem 2). 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 (Lemma 7). Using this we show (in Theorem 3) that an irreducible representation of the Heisenberg group $${\mathcal {H}}_{\mathbb {C}}$$ can be realized on $${L}^2_\chi$$ in the Weyl–Schrödinger form

\begin{aligned} 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 _bM^\dagger _{a^*} \end{aligned}

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

\begin{aligned} {W}^\dagger (a,b){W}^\dagger (a',b') =\exp \left\{ - \big [{\mathfrak {p}}_{a,b}^\dagger ,{\mathfrak {p}}_{a',b'}^\dagger \big ]\right\} {W}^\dagger (a',b'){W}^\dagger (a,b) \end{aligned}

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 (Theorem 4) that the following Cauchy problem with $$\partial _i^\dagger :=\partial _{{\mathfrak {e}}_i}^\dagger$$,

\begin{aligned} \frac{dw(r)}{dr}=-\sum \big (\partial _i^\dagger +{{\bar{\phi }}}_i\big )^2w(r), \quad w(0)=f,\quad r>0, \end{aligned}

has the unique solution $$w(r)={\mathfrak {G}}^\dagger _rf$$ 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

\begin{aligned} {\mathfrak {G}}^\dagger _rf&=\frac{1}{\sqrt{4\pi r}}\int _{c_0} \exp \Bigg \{-\frac{\Vert \tau \Vert _{w_0}^2}{4r}\Bigg \}{W}^\dagger _\tau {f}\,d{\mathfrak {w}}(\tau ),\\ {W}^\dagger _\tau {f}&:=\lim _{n\rightarrow \infty } \exp \Bigg \{-\frac{\Vert {p_n^\sim (\tau )}\Vert _{w_0}^2}{2}\Bigg \} \prod _{i=1}^{n} T^\dagger _{\mathbb {i}\tau _i{\mathfrak {e}}_i} M^\dagger _{-\mathbb {i}\tau _i{\mathfrak {e}}_i^*}. \end{aligned}

Here $$\tau =(\tau _i)$$ belongs to the abstract Wiener space $$\{w_0,\Vert \cdot \Vert _{w_0}\}$$ 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 [16, 17]. 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 $$m\in {\mathbb {N}}_0={\mathbb {N}}\cup \{0\}$$, $$\mathbb {1}=U(0)$$, irreducibly acting on a separable Hilbert space H, where subgroups U(m) are identified with ranges of injections $$U(m)\ni u_m\longmapsto {\begin{bmatrix} u_m &{}\quad 0\\ 0 &{}\quad \mathbb {1}\\ \end{bmatrix}\in U(\infty )}$$. Following to [21, 24], we use the Livšic transforms $$\pi ^{m+1}_m:U(m+1)\rightarrow U(m)$$ of the form

\begin{aligned}&\pi ^{m+1}_m:{u_{m+1}}:=\begin{bmatrix} z_m &{}\quad a \\ b &{}\quad t \\ \end{bmatrix}\longmapsto u_m:=\left\{ \begin{array}{clc} z_m-[a(1+t)^{-1}b]&{}:&{}\quad t\ne -1 \\ z_m&{}:&{}\quad t=-1 \\ \end{array}\right. \end{aligned}
(4)

with $$z_m\in U(m)$$ defined by excluding $$x_1=y_1\in {\mathbb {C}}$$ from $$\begin{bmatrix} y_m \\ y_1\\ \end{bmatrix}=\begin{bmatrix} z_m &{}\quad a\\ -b &{}\quad -t\\ \end{bmatrix} \begin{bmatrix} x_m \\ x_1\\ \end{bmatrix}$$ for $$x_m,y_m\in {\mathbb {C}}^m$$ and $$a,b\in {\mathbb {C}}$$ [24, Lem. 3.1]. It is surjective (not continuous) Borel mapping [24, Lem. 3.11].

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

\begin{aligned} \pi _m:{{\mathfrak {U}}\ni {\mathfrak {u}}\longmapsto u_m\in U(m)}\quad \text {such that}\quad \pi _m={\pi ^{m+1}_m\circ \pi _{m+1}}. \end{aligned}

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

\begin{aligned} {\mathfrak {U}}\ni {\mathfrak {u}}\longmapsto {\mathfrak {u}}.g\in {\mathfrak {U}}\quad \text {with}\quad g={(v,w)\in {U(\infty )\times U(\infty )}} \end{aligned}

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 [24, n.4]) which assigns the stabilized sequence $${\mathfrak {u}}=(u_k)$$ to each $${u_m\in U(m)}$$ such that

\begin{aligned} \begin{aligned} U(m)&\ni u_m\longmapsto (u_k)\in {\mathfrak {U}},\\ u_k&=\left\{ \begin{array}{ll} \pi ^m_k(u_m)=(\pi ^{k+1}_k\circ \ldots \circ \pi ^m_{m-1})(u_m)&{}:k<m,\\ u_m &{}: k\ge m. \end{array}\right. \end{aligned} \end{aligned}
(5)

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. [24, Lem.4.8], [27, Thm 2.2], [30, Cor.4.2]), we determine the probability measure on $${\mathfrak {U}}$$ to be the projective limit

\begin{aligned} \chi :=\varprojlim \chi _m\quad \text {under}\quad \chi _m=\pi _m^{m+1}(\chi _{m+1}) \end{aligned}

where $$\pi _m^{m+1}(\chi _{m+1})$$ means an image-measure and $${\chi _0=1}$$. As is known [30, 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 $$f:{\mathfrak {U}}\rightarrow {\mathbb {C}}$$ with the following norm and inner product

\begin{aligned} \Vert f\Vert _\chi =\langle f\mid f\rangle _\chi ^{1/2},\quad \langle f_1\mid f_2\rangle _\chi :=\int f_1{{\bar{f}}}_2\,d\chi . \end{aligned}

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

### Lemma 1

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

\begin{aligned} \int f\,d\chi =\lim \int f\,d(\chi _m\circ \pi _m)=\lim \int (f\circ \pi _m^{-1})\,d\chi _m. \end{aligned}
(6)

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

\begin{aligned} \int f({\mathfrak {u}}.g)\,d\chi ({\mathfrak {u}})&=\int f({\mathfrak {u}})\,d\chi ({\mathfrak {u}}), \quad g\in U(\infty )\times U(\infty ), \end{aligned}
(7)
\begin{aligned} \int f\,d\chi&=\int d\chi ({\mathfrak {u}}) \int f({\mathfrak {u}}.g)\,d(\chi _m\otimes \chi _m)(g). \end{aligned}
(8)

### 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

\begin{aligned} \begin{aligned} (\chi _m\circ \pi _m)({\mathcal {K}})&=\pi ^{m+1}_m(\chi _{m+1})\left[ \pi _m({\mathcal {K}})\right] \\&=\chi _{m+1}\left[ (\pi ^{m+1}_m)^{-1}[\pi _m({\mathcal {K}})]\right] \ge (\chi _{m+1}\circ \pi _{m+1})({\mathcal {K}}). \end{aligned} \end{aligned}
(9)

This ensures that the necessary and sufficient conditions of the Prokhorov theorem [4, Thm IX.52] and its modification from [30, Thm 4.2] are satisfied.

Indeed, let $${\check{U}}(m)\subset U(m)$$ be the set of matrices with no eigenvalue $$\{-1\}$$ for $${m\ge 1}$$. As is known [24, n.3], $${\check{U}}(m)$$ is open in U(m) and $${\chi _m(U(m){\setminus }{\check{U}}(m))}=0$$. In virtue of [24, Lem. 3.11] the restrictions $$\pi ^{m+1}_m:{\check{U}}(m+1)\rightarrow {\check{U}}(m)$$ are continuous and surjective. The projective limit $$\varprojlim {\check{U}}(m)$$ under these restrictions has continuous surjective projections $$\pi _m:\varprojlim {\check{U}}(m)\rightarrow {\check{U}}(m)$$. Restrict $$\chi _m$$ to $${\check{U}}(m)$$. By [30, 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

\begin{aligned} {(\chi _m\circ \pi _m)({\mathcal {K}})}\ge {1-\varepsilon }\quad \text {for all}\quad {m\in {\mathbb {N}}}. \end{aligned}

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

\begin{aligned} {\check{\chi }}({\mathcal {K}})=\inf (\chi _m\circ \pi _m)({\mathcal {K}})\quad \text {for all}\quad {\mathcal {K}}\subset \varprojlim {\check{U}}(m). \end{aligned}
(10)

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

\begin{aligned} \pi ^{m+1}_m(K_{m+1})\subset K_m\quad \text {and}\quad \chi _{m+1}(K_{m+1})>1-\varepsilon \quad \text {for all}\quad {m\ge 1}. \end{aligned}

Assume that $$K_1,\ldots ,K_m$$ are constructed. Since $$\chi _m=\pi ^{m+1}_m(\chi _{m+1})$$, we get

\begin{aligned} \chi _m(K_m)=\chi _{m+1}[(\pi ^{m+1}_m)^{-1}(K_m)]>1-\varepsilon . \end{aligned}

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

\begin{aligned} K_{m+1}\subset (\pi ^{m+1}_m)^{-1}(K_m)\quad \text {such that}\quad \chi _{m+1}(K_{m+1})>1-\varepsilon . \end{aligned}

The induction is complete. Then $${\mathcal {K}}=\varprojlim K_m$$ with $$K_0=\mathbb {1}$$ is compact. By virtue of (10), we have $${{\check{\chi }}}({\mathcal {K}})\ge 1-\varepsilon$$. 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 (9) and (10) we get

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

By the known Portmanteau theorem [14, Thm 13.16] it follows that the limit (6) exists. Whereas, the property (7) is a consequence of the equalities

\begin{aligned} \chi ({\mathcal {K}}.g)=\lim \chi _m(K_m.g)=\lim \chi _m(K_m)=\chi ({\mathcal {K}}) \end{aligned}

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

\begin{aligned} \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}}) \end{aligned}

by the Fubini theorem. It yields (8) since the internal integral on the right-hand side is independent of g by (7) and $${\int \,d(\chi _m\otimes \chi _m)(g)=1}.$$ The proof is complete. $$\square$$

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

\begin{aligned} d_{HS}(u,v)=\sqrt{m^{-1}\mathop {\textsf {tr}}|u-v|_{HS}}\quad \text {where}\quad |u-v|_{HS}=\sqrt{(u-v)^\star (u-v)}. \end{aligned}

As is well known (see [9, 31]), $$(U(m),d_{HB},\chi _m)$$ is a Lévy family. Namely, the following sequence of isoperimetric constants dependent on $$\varepsilon >0$$

\begin{aligned} \alpha (U(m),\varepsilon )=1-\inf \big \{\chi _m[(\varOmega _m)_\varepsilon ]:\varOmega _m \text { be Borel set in } U(m), \chi _m(\varOmega _m)>1/2\big \} \end{aligned}

with $$(\varOmega _m)_\varepsilon =\left\{ u_m\in U(m):d_{HS}\left( u_m,\varOmega _m\right) <\varepsilon \right\}$$ is such that

\begin{aligned} \alpha (U(m),\varepsilon )\rightarrow 0\quad \text {as}\quad {m\rightarrow \infty }. \end{aligned}

Taking into account the Lemma 1, we can formulate the following conclusion.

### Corollary 1

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

\begin{aligned} {\chi (\varOmega _\varepsilon )=\lim _{m\rightarrow \infty }\chi _m\left[ (\varOmega _m)_\varepsilon \right] =1} \end{aligned}

holds. Consequently, all Borel sets $${\mathfrak {U}}{\setminus }\varOmega _\varepsilon$$ with $${\chi _m(\varOmega _m)>1/2}$$ 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 $${\mathscr {I}}_\eta :=\big \{\imath =\big ({\imath _1},\ldots ,{\imath _\eta }\big )\in {\mathbb {N}}^\eta :\imath _1<\imath _2<\ldots <\imath _\eta \big \}$$ 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 $${\lambda _1\ge \lambda _2\ge \ldots \ge \lambda _\eta }$$ be a partition of an n-letter word $$\imath ^\lambda =\big \{\Box _{ij}:{1\le i\le \eta }, j=1,\ldots ,\lambda _i\big \}$$ 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 $$[\imath ^\lambda ]=\begin{array}{c@{\quad }c@{\quad }c@{\quad }c} \Box _{11} &{} \dots &{} \dots &{}\Box _{1\lambda _1}\\ \vdots &{} \vdots &{} \ddots &{} \\ \Box _{\eta 1} &{} \dots &{}\Box _{\eta \lambda _\eta } &{} \end{array}\!\!\!\!\!$$ with n nonzero entries in some way without repetitions. So, each $$\lambda$$-tableau $$[\imath ^\lambda ]$$ can be identified with a bijection $$[\imath ^\lambda ]\rightarrow \imath ^\lambda$$. 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 $${\mathbb {Y}}_0=\left\{ \emptyset \in {\mathbb {Y}}:|\emptyset |=0\right\}$$ and $$\eta (\emptyset )=0$$.

As before, $$\left\{ H,\langle \cdot \mid \cdot \rangle \right\}$$ is a separable complex Hilbert space with an orthonormal basis $$\left\{ {\mathfrak {e}}_i:i\in {\mathbb {N}}\right\}$$ and $${\Vert \cdot \Vert ={\langle \cdot \mid \cdot \rangle ^{1/2}}}$$. For its adjoint space $$H^*$$ the conjugate-linear isometry $${*:H^*\rightarrow H^{**}=H}$$ 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 $$\Vert \psi _n\Vert ={\left\langle \psi _n\mid \psi _n\right\rangle }^{1/2}$$ where $${\left\langle \psi _n\mid \psi _n'\right\rangle }:= {\langle h_1\mid h'_1\rangle \ldots \langle h_n\mid h'_n\rangle }$$.

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 $${{\mathcal {S}}_n:{H}^{\otimes n}\ni \psi _n\longmapsto {h_1\odot \ldots \odot h_n}:=(n!)^{-1}{\sum }_{\sigma \in S_n}\sigma (\psi _n)}$$. 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 $$\left\{ h^{\otimes n}:{h}\in {H}\right\} \subset {H}^{\odot n}$$ is total by the polarization formula [7, n.1.5]

\begin{aligned} {h}_1\odot \ldots \odot {h}_n=\frac{1}{2^nn!} \sum _{\theta _1,\ldots ,\theta _n=\pm 1} \theta _1\dots \theta _n{h}^{\otimes n},\quad {h}=\sum _{i=1}^n\theta _ih_i. \end{aligned}
(11)

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 $${\sigma '\in {S}_n}$$ such that $${\sigma '\big ({\mathfrak {e}}^{\otimes \lambda _1}_{\imath _1}\otimes \ldots \otimes {\mathfrak {e}}^{\otimes \lambda _\eta }_{\imath _\eta }\big )}= {{\mathfrak {e}}^{\otimes \lambda _1}_{\imath _1}\odot \ldots \odot {\mathfrak {e}}^{\otimes \lambda _\eta }_{\imath _\eta }}\in H_\eta ^{\odot n}$$. Taking all $$\imath \in {\mathscr {I}}$$, we conclude that the system indexed by semistandard $$\lambda$$-tabloids

\begin{aligned} \begin{aligned} {\mathfrak {e}}^{{\mathbb {Y}}_n}&=\left\{ {\mathfrak {e}}^{\odot \lambda }_\imath :={\mathfrak {e}}^{\otimes \lambda _1}_{\imath _1}\odot \ldots \odot {\mathfrak {e}}^{\otimes \lambda _\eta }_{\imath _\eta } :\imath ^\lambda \vdash n, \ \lambda \in {\mathbb {Y}}_n, \ \imath \in {\mathscr {I}}\right\} , \quad {{\mathfrak {e}}^{\odot \emptyset }_\imath =1}\\ \text {where}&\quad \langle {\mathfrak {e}}^{\odot \lambda }_\imath \mid {\mathfrak {e}}^{\odot \lambda '}_{\imath '}\rangle =\left\{ \begin{array}{clcl} {\lambda !}/{n!} &{}: \lambda =\lambda &{}\text { and }&{}\imath =\imath ' \\ 0 &{}: \lambda \ne \lambda '&{}\text { or }&{}\imath \ne \imath ' \end{array}\right. \end{aligned} \end{aligned}

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

The system $$\big \{{\mathfrak {e}}_\imath ^{\otimes \lambda }:= {\mathcal {S}}_n\big ({\mathfrak {e}}^{\otimes \lambda _1}_{\imath _1}\otimes \ldots \otimes {\mathfrak {e}}^{\otimes \lambda _\eta }_{\imath _\eta }\big ):\imath ^\lambda \vdash n, \ \lambda \in {\mathbb {Y}}_n, \ \imath \in {\mathscr {I}}\big \}$$, 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 $$\varGamma (H)={\bigoplus }_{n\ge 0} H^{\odot n}$$ with the orthogonal basis $${\mathfrak {e}}^{{\mathbb {Y}}}:={\bigcup \big \{{\mathfrak {e}}^{{\mathbb {Y}}_n}:}{n}\in {\mathbb {N}}_0\big \}$$ of elements $$\psi =\bigoplus \psi _n$$ with $$\psi _n\in H^{\odot n}$$ endowed with the inner product and norm

\begin{aligned} \langle \psi \mid \psi '\rangle _\varGamma =\sum {n!}\langle \psi _n\mid \psi _n'\rangle ,\quad \Vert \psi \Vert _\varGamma =\langle \psi \mid \psi \rangle ^{1/2}_\varGamma . \end{aligned}

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

\begin{aligned} h^{\otimes n}=\sum _{\imath ^\lambda \vdash n} \frac{n!}{\lambda !}{\mathfrak {e}}^{\odot \lambda }_\imath \,{\mathfrak {e}}^{*\lambda }_\imath (h),\quad \Vert h^{\otimes n}\Vert ^2 = \sum _{\imath ^\lambda \vdash n} \frac{n!}{\lambda !}|{\mathfrak {e}}^{*\lambda }_\imath (h)|^2,\quad {\mathfrak {e}}^{*\lambda }_\imath := {\mathfrak {e}}^{*\lambda _1}_{\imath _1}\ldots {\mathfrak {e}}^{*\lambda _\eta }_{\imath _\eta },\nonumber \\ \end{aligned}
(12)

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

\begin{aligned} \exp (h):=\bigoplus _{n\ge 0}\frac{h^{\otimes n}}{n!} =\bigoplus _{n\ge 0}\frac{1}{n!}\Bigg (\sum _{i\ge 0}{\mathfrak {e}}_i\,{\mathfrak {e}}_i^*(h)\Bigg )^{\otimes n}= \bigoplus _{n\ge 0}\frac{1}{n!}\sum _{\imath ^\lambda \vdash n} \frac{n!}{\lambda !}{\mathfrak {e}}^{\odot \lambda }_\imath \,{\mathfrak {e}}^{*\lambda }_\imath (h)\nonumber \\ \end{aligned}
(13)

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

\begin{aligned} \begin{aligned} \Vert \exp (h)\Vert ^2_\varGamma&=\sum _{n\ge 0}\frac{1}{n!}\sum _{\imath ^\lambda \vdash n} \Big (\frac{n!}{\lambda !}\Big )^2\Vert {\mathfrak {e}}^{\odot \lambda }_\imath \Vert ^2 |{\mathfrak {e}}^{*\lambda }_\imath (h)|^2=\sum _{n\ge 0}\frac{1}{n!}\sum _{\imath ^\lambda \vdash n} \frac{n!}{\lambda !}|{\mathfrak {e}}^{*\lambda }_\imath (h)|^2\\&=\sum \frac{1}{n!}\left( \sum |{\mathfrak {e}}^*_i(h)|^2\right) ^n=\sum \frac{1}{n!}\Vert h\Vert ^{2n}=\exp \Vert h\Vert ^2. \end{aligned} \end{aligned}
(14)

### Definition 1

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

\begin{aligned} {\phi }_h({\mathfrak {u}}):= \sum {\phi }_i({\mathfrak {u}})\,{\mathfrak {e}}_i^*(h) \quad \text {with}\quad {\phi }_i({\mathfrak {u}}):=\left\langle u_i({\mathfrak {e}}_i)\mid {\mathfrak {e}}_i\right\rangle , \quad u_i=\pi _i({\mathfrak {u}}) \end{aligned}

where projections $$\pi _i:{\mathfrak {U}}\ni {\mathfrak {u}} \rightarrow u_i\in U(i)$$ 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. [12, n.4.4], [29]), the Paley–Wiener maps uniquely determine the embedding $$\phi :H\ni {h}\longmapsto \phi _h\in L^2_\chi$$.

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 $$\left| \left\langle u_i({\mathfrak {e}}_i)\mid {\mathfrak {e}}_i\right\rangle \right| \le 1$$. 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

\begin{aligned} \beta _\lambda :=\dfrac{(\eta -1)!}{(\eta -1+|\lambda |)!}\le 1,\quad \eta =\eta (\lambda ). \end{aligned}
(15)

### Lemma 2

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

\begin{aligned} {\phi }^\lambda _\imath ({\mathfrak {u}}):= {\phi }^{\lambda _1}_{\imath _1}({\mathfrak {u}})\ldots {\phi }^{\lambda _\eta }_{\imath _\eta }({\mathfrak {u}}), \quad {\phi }^\emptyset _\imath \equiv 1 \end{aligned}
(16)

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

\begin{aligned} \phi ^{\mathbb {Y}}:=\bigcup \big \{{\phi }^{{\mathbb {Y}}_n}:{n}\in {\mathbb {N}}_0\big \} \quad \text {with}\quad {\phi }^{{\mathbb {Y}}_n}:=\bigcup \big \{{\phi }^\lambda _\imath :\imath ^\lambda \vdash n, \imath \in {\mathscr {I}}_\eta \big \} \end{aligned}

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

\begin{aligned} \Vert {\phi }^\lambda _\imath \Vert _\chi ^2= \int |{\phi }^\lambda _\imath |^2d\chi =\lambda !\beta _\lambda , \quad \imath ^\lambda \vdash n,\quad \lambda !:=\lambda _1!\ldots \lambda _\eta !. \end{aligned}

### Proof

According to (4), 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 $$t\ne -1$$ for any integer $$l\ge 1$$. This means that $${({\phi }_k\circ \pi _m^{-1})(u_m)}= \left\langle u_m({\mathfrak {e}}_m) \mid {\mathfrak {e}}_k\right\rangle {\equiv \!\!\!\!\!\!/} 0$$ for all $$k\le m$$ and that

\begin{aligned} \begin{aligned} {(\phi _m\circ \pi _{m+l}^{-1})(u_{m+l})}&= \left\langle u_m({\mathfrak {e}}_m) \mid {\mathfrak {e}}_m\right\rangle \quad \text {for}\quad t=-1, \\ {(\phi _m\circ \pi _{m+l}^{-1})(u_{m+l})}&={\left\langle u_m({\mathfrak {e}}_m) \mid {\mathfrak {e}}_m\right\rangle }- {a(1+t)^{-1}b\left\langle {\mathfrak {e}}_m\mid {\mathfrak {e}}_m\right\rangle }\quad \text {for}\quad t\ne -1. \end{aligned}\nonumber \\ \end{aligned}
(17)

Let $$U(\eta )$$ with $$\eta =\eta (\lambda )$$ be the unitary group acting over the linear complex $$\mathop {\text {span}}\left\{ {\mathfrak {e}}_{\imath _1},\ldots ,{\mathfrak {e}}_{\imath _\eta }\right\}$$ in H. Let $$\chi _\eta$$ be the probability Haar measure on $$U(\eta )$$ and $$\pi _\eta :{\mathfrak {U}}\rightarrow U(\eta )$$ be the corresponding projector. Using (6) and (17), we obtain

\begin{aligned} \begin{aligned} \int |{\phi }^\lambda _\imath ({\mathfrak {u}})|^2d\chi ({\mathfrak {u}})&=\lim \int |({\phi }^\lambda _\imath \circ \pi _m^{-1})(u_m)|^2d\chi _m(u_m) \\&=\lim \int |({\phi }^{\lambda _1}_{\imath _1}\circ \pi _m^{-1})(u_m)\ldots ({\phi }^{\lambda _\eta }_{\imath _\eta }\circ \pi _m^{-1})(u_m)|^2d\chi _m(u_m)\\&=\int |({\phi }^{\lambda _1}_{\imath _1}\circ \pi _\eta ^{-1})(u_\eta )\ldots ({\phi }^{\lambda _\eta }_{\imath _\eta }\circ \pi _\eta ^{-1})(u_\eta )|^2d\chi _\eta (u_\eta ). \end{aligned}\nonumber \\ \end{aligned}
(18)

By (18) and the known integral formula for unitary groups $$U(\eta )$$ [28, 1.4.9], we get

\begin{aligned} \int |{\phi }^\lambda _\imath |^2d\chi =\int \prod _{k=1}^{\eta (\lambda )}\left| \left\langle u_\eta ({\mathfrak {e}}_\eta )\mid {\mathfrak {e}}_{\imath _k}\right\rangle \right| ^2d\chi _\eta (u_\eta )= \frac{(\eta (\lambda )-1)!\lambda !}{(\eta (\lambda )-1+|\lambda |)!}. \end{aligned}

On the other hand, the invariant property (8) provides the formula

\begin{aligned} \int f\,d\chi =\frac{1}{2\pi }\int \!d\chi ({\mathfrak {u}}) \int _{-\pi }^{\pi }f\left[ \exp (\mathbb {i}\vartheta ){\mathfrak {u}}\right] d\vartheta , \qquad f\in L_\chi ^\infty . \end{aligned}
(19)

From (19) it follows the orthogonality relations $${\phi ^{\lambda '}_\jmath \perp \phi ^\lambda _\imath }$$ with $${|\lambda '|\ne |\lambda |}$$, since

\begin{aligned} \int {\phi }^{\lambda '}_\jmath \bar{{\phi }}^\lambda _\imath \,d\chi = \frac{1}{2\pi }\int \phi ^{\lambda '}_\jmath \bar{{\phi }}^\lambda _\imath \,d\chi \int _{-\pi }^\pi {\exp \left[ \mathbb {i}\big (|\lambda '|-|\lambda |\big )\vartheta \right] }\,d\vartheta =0 \end{aligned}

for any $$\lambda ',\lambda \in {\mathbb {Y}}{\setminus }\{\emptyset \}$$. Let $$|\lambda '|=|\lambda |$$ and $$\eta (\lambda ')>\eta (\lambda )$$ for definiteness. Then there exists an index k with a nonzero integer $$\lambda '_k$$ in $$\lambda '=\big (\lambda '_1,\ldots ,\lambda '_k,\ldots ,\lambda '_{\eta (\lambda ')}\big )\in {\mathbb {Y}}{\setminus }\{\emptyset \}$$ such that $$\eta (\lambda )<k\le \eta (\lambda ')$$. In this case $${{\phi }^{\lambda '}_\jmath \perp {\phi }^\lambda _\imath }$$ because (19) yields

\begin{aligned} \int {\phi }^{\lambda '}_\jmath \bar{{\phi }^\lambda _\imath }\,d\chi = \frac{1}{2\pi }\int {\phi }^{\lambda '}_\jmath \bar{{\phi }}^\lambda _\imath \,d\chi \int _{-\pi }^{\pi }\exp \left( \mathbb {i}\lambda '_k\vartheta \right) d\vartheta =0. \end{aligned}

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

\begin{aligned} \int {\phi }^{\lambda '}_\jmath \bar{{\phi }}^\lambda _\imath \,d\chi = \frac{1}{2\pi }\int {\phi }^{\lambda '}_\jmath \bar{{\phi }}^\lambda _\imath \,d\chi \int _{-\pi }^\pi \exp \left[ \mathbb {i}(\lambda '_k-\lambda _k)\vartheta \right] d\vartheta =0. \end{aligned}

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

## 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. [18]) to be

\begin{aligned} s^\lambda _\imath (t_\imath ):={D_\lambda (t_\imath )}/{\varDelta (t_\imath )}\quad \text {where}\quad D_\lambda (t_\imath )=\mathop {\text {det}}\big [t_{\imath _i}^{\lambda _j+\eta -j}\big ] \text { with }\lambda _j=0 \text { for }j>\eta , \end{aligned}

$$\varDelta (t_\imath )={\prod _{1\le i<j\le \eta }(t_{\imath _i}-t_{\imath _j})}$$ 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 [8, 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 (3). Denote

\begin{aligned} s^{\mathbb {Y}}_n:=\bigcup \big \{s^\lambda _\imath :\imath ^\lambda \vdash n\big \}, \quad s^{\mathbb {Y}}:=\bigcup \big \{s^{\mathbb {Y}}_n:{n}\in {\mathbb {N}}_0\big \}\quad \text {with} \quad {s_0=s^\emptyset _\imath \equiv 1}. \end{aligned}

### Theorem 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,

\begin{aligned} L^2_\chi ={\mathbb {C}}\oplus L_\chi ^{2,1}\oplus L_\chi ^{2,2}\oplus \ldots . \end{aligned}
(20)

For any $${h\in {H}}$$ the equality (2) uniquely defines the conjugate-linear embedding

\begin{aligned} {\phi }:{H}\ni {h}\longmapsto {\phi }_h\in L^2_\chi \quad \text {such that}\quad \Vert {\phi }_h\Vert _\chi =\Vert h\Vert . \end{aligned}
(21)

### Proof

Let $$U(\eta )$$ be the unitary group over the linear complex $$\mathop {\text {span}}\left\{ {\mathfrak {e}}_{\imath _1},\ldots ,{\mathfrak {e}}_{\imath _\eta }\right\}$$ with $$\eta =\eta (\lambda )$$. Taking into account (17) similarly as (18), we obtain

\begin{aligned} \int {s}_\imath ^\lambda {\bar{s}}_\imath ^\mu \,d\chi =\int {s}_\imath ^\lambda (z_\eta ) \,{\bar{s}}_\imath ^\mu (z_\eta )\,d\chi _\eta (z_\eta )=\delta _{\lambda \mu } \end{aligned}

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 $$\big \{s^\lambda _\imath :\lambda \in {\mathbb {N}}^\eta \big \}$$ are characters of the group $$U(\eta )$$. Hence, by the Weyl integration formula, the right-hand side integral is equal to Kronecker’s delta $$\delta _{\lambda \mu }$$ [26, Thm 8.3.2 & Thm 11.9.1].

The family of finite alphabets $${\imath \in {\mathscr {I}}}$$ is directed and for any $$\imath ,\imath '$$ there exists $$\imath ''$$ such that $$\imath \cup \imath '\subset \imath ''$$. 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 $${|\mu |\ne |\lambda |}$$ for any $${\imath ,\jmath \in {\mathscr {I}}}$$ follows from (19), since

\begin{aligned} \int {s}_\jmath ^\mu \bar{s}_\imath ^\lambda \,d\chi =\frac{1}{2\pi }\int {s}_\jmath ^\mu \bar{s}_\imath ^\lambda \,d\chi \int _{-\pi }^\pi {\exp \big (\mathbb {i}(|\mu |-|\lambda |)\vartheta \big )}\,d\vartheta =0 \end{aligned}

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, (20) is proved.

By Lemma 2 the subsystem $$\phi _k=s^1_k$$ is orthonormal in $$L^2_\chi$$, hence by Definition 1 it instantly follows that $$\Vert {\phi }_h\Vert _\chi ^2=\sum |{\mathfrak {e}}_k^*(h)|^2\int |\phi _k|^2d\chi =\Vert h\Vert ^2.$$ It follows the isometric embedding (21).

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

\begin{aligned} {\tilde{\pi }}^{m+1}_m:{\tilde{U}}(m+1)\longrightarrow {U}(m). \end{aligned}

This fact follows from [33, 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 $${{\tilde{\pi }}}_m:\tilde{{\mathfrak {U}}}\rightarrow {U}(m)$$.

Since $$U(\infty )$$ on H acts irreducibly, for any $${\mathfrak {u}}'\ne {\mathfrak {u}}''$$ there is m such that

\begin{aligned} \phi _m({\mathfrak {u}}')=\left\langle \pi _m({\mathfrak {u}}')({\mathfrak {e}}_m)\mid {\mathfrak {e}}_m\right\rangle \ne \left\langle \pi _m({\mathfrak {u}}'')({\mathfrak {e}}_m)\mid {\mathfrak {e}}_m\right\rangle =\phi _m({\mathfrak {u}}''), \end{aligned}

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 $${{\bar{\phi }}}_m({\mathfrak {u}})= \left\langle {\mathfrak {e}}_m\mid \pi _m({\mathfrak {u}})\right. \left. ({\mathfrak {e}}_m)\right\rangle = \left\langle \pi _m({\mathfrak {u}}^\star )({\mathfrak {e}}_m)\mid {\mathfrak {e}}_m\right\rangle$$ 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 1 it inductively was shown that for every $$\varepsilon >0$$ there exists a compact set $$\varprojlim K_m\subset \check{{\mathfrak {U}}}$$ such that

\begin{aligned} {\tilde{\chi }}_m(K_m)\ge 1-\varepsilon \quad \text {for all}\quad m \end{aligned}

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

\begin{aligned} {{\tilde{\chi }}}(\varOmega )=\inf {\tilde{\chi }}_m(\varOmega )=\inf \chi _m(\varOmega )=\varprojlim \chi _m(\varOmega )= \chi (\varOmega ) \end{aligned}

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

\begin{aligned} {{\tilde{\chi }}}|_{\check{{\mathfrak {U}}}}= \chi |_{\check{{\mathfrak {U}}}}=\chi |_{\check{{\mathfrak {U}}}\bigsqcup ({\mathfrak {U}}{\setminus }\check{{\mathfrak {U}}})} =\chi |_{\mathfrak {U}}\quad \text {since}\quad \chi ({\mathfrak {U}}{\setminus }\check{{\mathfrak {U}}})=0. \end{aligned}

In particular, $${\tilde{\chi }}=\varprojlim {\tilde{\chi }}_m$$ is regular on $$\tilde{{\mathfrak {U}}}$$ by the Riesz–Markov theorem [20, 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 $${(f_n)\subset \mathop {\text {span}}(s^{\mathbb {Y}})}$$ such that $${\int |f-f_n|^2d\chi }{\rightarrow 0}$$. 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 $$n\ne m$$. This yields (20).

$$\square$$

## 5 Unitarily-Weighted Symmetric Fock Space

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

\begin{aligned} \langle {\mathfrak {e}}^{\otimes \lambda }_\imath \mid {\mathfrak {e}}^{\otimes \lambda '}_{\imath '}\rangle _{H^{\otimes n}_\beta } =\left\{ \begin{array}{clcl} \dfrac{(\eta -1)!}{(\eta -1+n)!}&{}: \lambda =\lambda '&{}\text {and}&{}\imath =\imath ' \\ 0 &{}: \lambda \ne \lambda '&{}\text {or}&{}\imath \ne \imath '. \end{array}\right. \end{aligned}
(22)

Here $${\mathfrak {e}}^{\otimes \lambda }_\imath :=\sigma '({\mathfrak {e}}^{\otimes \lambda _1}_{\imath _1}\otimes \ldots \otimes {\mathfrak {e}}^{\otimes \lambda _\eta }_{\imath _\eta })$$ with $$\eta =\eta (\lambda )$$ and $$\sigma '\in S_n$$ is fixed. Let $${H}^{\otimes n}_\beta$$ be the completion of $$\big \{{H}^{\otimes n},\Vert \cdot \Vert _{H^{\otimes n}_\beta }\big \}$$. Its closed subspace, defined by the projection

\begin{aligned} {{\mathcal {S}}_n:{H}^{\otimes n}_\beta \ni {\mathfrak {e}}^{\otimes \lambda }_\imath \longmapsto {\mathfrak {e}}^{\odot \lambda }_\imath =(n!)^{-1}\sum \limits _{\sigma \in S_n}\sigma ({\mathfrak {e}}^{\otimes \lambda }_\imath )} \end{aligned}

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

\begin{aligned} \langle {\mathfrak {e}}^{\odot \lambda }_\imath \mid {\mathfrak {e}}^{\odot \lambda '}_{\imath '}\rangle _{H^{\otimes n}_\beta } =\left\{ \begin{array}{clcl} \dfrac{\lambda !}{n!}\dfrac{(\eta -1)!}{(\eta -1+n)!}&{}: \lambda =\lambda &{}\text {and}&{}\imath =\imath ' \\ 0 &{}: \lambda \ne \lambda '&{}\text {or}&{}\imath \ne \imath '. \end{array}\right. \end{aligned}
(23)

### Definition 2

The unitarily-weighted symmetric Fock space is defined to be the Hilbertian orthogonal sum $${\varGamma _{\!\beta }(H)=\bigoplus _{n\ge 0}{H}_\beta ^{\odot n}}$$ of elements $$\psi =\bigoplus \psi _n$$, $$\psi _n\in {H}_\beta ^{\odot n}$$ with the orthogonal basis $${\mathfrak {e}}^{{\mathbb {Y}}}= \bigcup \big \{{\mathfrak {e}}^{{\mathbb {Y}}_n}:{n}\in {\mathbb {N}}_0\big \}$$ and the following inner product and norm

\begin{aligned} \langle \psi \mid \psi '\rangle _\beta =\sum {n!}\langle \psi _n\mid \psi _n'\rangle _{H^{\otimes n}_\beta },\quad \Vert \psi \Vert _\beta =\langle \psi \mid \psi \rangle _\beta ^{1/2}. \end{aligned}

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

### Lemma 3

The set of coherent states $$\left\{ \exp (h):{h}\in {H}\right\}$$ is total in $$\varGamma _\beta (H)$$ and the expansion (13) is convergent in $$\varGamma _\beta (H)$$. The injections

\begin{aligned} \varGamma (H)\looparrowright \varGamma _\beta (H)\quad \text {and}\quad {{H}^{\odot n} \looparrowright {H}^{\odot n}_\beta } \end{aligned}

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

\begin{aligned} {\mathcal {T}}_a\exp (h):=\exp (h+a) =\exp (\partial _a)\exp (h)\quad \text {with}\quad \partial _a\exp (h)=\frac{d\exp (h+za)}{dz}{\Big |}_{z=0} \end{aligned}

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

\begin{aligned} \Vert {\mathcal {T}}_{a}\psi \Vert ^2_\beta \le \exp \big (\Vert a\Vert ^2\big )\Vert \psi \Vert ^2_\beta \quad \text {and}\quad {\mathcal {T}}_{a+b}= {\mathcal {T}}_a{\mathcal {T}}_b={\mathcal {T}}_b{\mathcal {T}}_a,\quad {a,b\in H}. \end{aligned}
(24)

### Proof

Taking into account that $$\beta _\lambda \le 1$$, we get the following inequalities

\begin{aligned} \Vert h^{\otimes n}\Vert ^2_{H^{\otimes n}_\beta }&\!=\!\sum _{\imath ^\lambda \vdash n}\! \Big (\frac{n!}{\lambda !}\Big )^2\Vert {\mathfrak {e}}^{\odot \lambda }_\imath \Vert ^2_{H^{\otimes n}_\beta } |{\mathfrak {e}}^{*\lambda }_\imath (h)|^2\!=\!\sum _{\imath ^\lambda \vdash n}\beta _\lambda \frac{n!}{\lambda !}|{\mathfrak {e}}^{*\lambda }_\imath (h)|^2\! \le \Vert h^{\otimes n}\Vert ^2 =\Vert h\Vert ^{2n},\\&\Vert \exp (h)\Vert ^2_\beta =\sum _{n\ge 0}\frac{1}{n!}\sum _{\imath ^\lambda \vdash n}\beta _\lambda \frac{n!}{\lambda !} |{\mathfrak {e}}^{*\lambda }_\imath (h)|^2 {\mathop {\le }\limits ^{(15)}}\exp \Vert h\Vert ^2{\mathop {=}\limits ^{(14)}}\Vert \exp (h)\Vert ^2_\varGamma . \end{aligned}

Hence, (12), (13) are convergent in $$\varGamma _\beta (H)$$. This implies that $${h\mapsto \exp (h)}$$ is analytic and inclusions $$\varGamma (H)\looparrowright \varGamma _\beta (H)$$ and $${{H}^{\odot n}\looparrowright {H}_\beta ^{\odot n}}$$ are contractive. By the polarization formula (11) their ranges are dense.

Using the binomial formula $${(h+za)^{\otimes n}}= \bigoplus _{m=0}^{n}\left( {\begin{array}{c}n\\ m\end{array}}\right) (za)^{\otimes m}\odot h^{\otimes (n-m)}$$, we find

\begin{aligned} \partial _{a}^m\exp (h)= \frac{d^m\exp (h+za)}{dz^m}{\mathrel {\Big |}}_{z=0} =\bigoplus _{n\ge m}\frac{{\mathcal {S}}_{n/m}[a^{\otimes m}\otimes h^{\otimes (n-m)}]}{(n-m)!},\quad {z\in {\mathbb {C}}} \end{aligned}

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 $$\Vert {\mathcal {S}}_{n/m}\Vert \le 1$$.

Applying the expansions (12) to $$a^{\otimes m}$$ and $$h^{\otimes (n-m)}$$, by (22), we get

\begin{aligned} \Vert a^{\otimes m}\otimes h^{\otimes (n-m)} \Vert ^2_{H^{\otimes n}_\beta }=\!\!\sum _{ \begin{array}{c} \imath ^\lambda \vdash m \\ \jmath ^\mu \vdash (n-m) \end{array}}\!\! \Big (\frac{m!}{\lambda !}\frac{(n-m)!}{\mu !}\Big )^2 \Vert {\mathfrak {e}}^{\odot \lambda }_\imath \otimes {\mathfrak {e}}^{\odot \mu }_\jmath \Vert ^2_{H^{\otimes n}_\beta } |{\mathfrak {e}}^{*\lambda }_\imath (a)|^2|{\mathfrak {e}}^{*\mu }_\jmath (h)|^2 \end{aligned}

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 $$\eta (\lambda ,\mu )\ge \max \{\eta (\lambda ),\eta (\mu )\}$$ and so

\begin{aligned} \Vert {\mathfrak {e}}^{\odot \lambda }_\imath \otimes {\mathfrak {e}}^{\odot \mu }_\jmath \Vert ^2 _{H^{\otimes n}_\beta }=\frac{(\eta (\lambda ,\mu )-1)!}{(\eta (\lambda ,\mu )-1+n)!}\le \min \{\beta _\lambda ,\beta _\mu \}, \end{aligned}

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

\begin{aligned} \Vert a^{\otimes m}\otimes h^{\otimes (n-m)} \Vert ^2_{H^{\otimes n}_\beta }&\le \sum _{ \begin{array}{c} \imath ^\lambda \vdash m \\ \jmath ^\mu \vdash (n-m) \end{array}} \Big (\frac{m!}{\lambda !}\frac{(n-m)!}{\mu !}\Big )^2\min \{\beta _\lambda ,\beta _\mu \} |{\mathfrak {e}}^{*\lambda }_\imath (a)|^2|{\mathfrak {e}}^{*\mu }_\jmath (h)|^2\\&=\Vert a^{\otimes m}\Vert ^2\Vert h^{\otimes (n-m)} \Vert ^2_{H^{\otimes (n-m)}_\beta }=\Vert a\Vert ^{2m}\Vert h^{\otimes (n-m)} \Vert ^2_{H^{\otimes (n-m)}_\beta } \end{aligned}

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

\begin{aligned} \begin{aligned} \Vert \partial _{a}^m\exp (h)\Vert ^2_\beta&=\!\sum _{n\ge m}\!\! \frac{\Vert {\mathcal {S}}_{n/m}[a^{\otimes m}\otimes h^{\otimes (n-m)}]\Vert ^2_\beta }{(n-m)!} \le \!\sum _{n\ge m}\!\!\frac{\Vert {\mathcal {S}}_{n/m}\Vert ^2\Vert a^{\otimes m}\otimes h^{\otimes (n-m)}\Vert ^2_\beta }{(n-m)!}\\&\le \Vert {a}^{\otimes m}\Vert ^2\sum _{n\ge m} \frac{\Vert {\mathcal {S}}_{n/m}\Vert ^2\Vert {h}^{\otimes (n-m)}\Vert ^2_\beta }{(n-m)!} \le \Vert {a}\Vert ^{2m}\Vert \exp (h)\Vert ^2_\beta . \end{aligned} \end{aligned}

Summing with coefficients 1/m!, we get $$\Vert {\mathcal {T}}_{a}\exp (h)\Vert ^2_\beta \le \exp \big (\Vert a\Vert ^2\big )\Vert \exp (h)\Vert ^2_\beta$$. This inequality and totality of $$\left\{ \exp (x):h\in {H}\right\}$$ in $$\varGamma _\beta (H)$$ yield the required inequality (24). It also follows that $$\varGamma _\beta (H)$$ is invariant under $${\mathcal {T}}_{a}$$ and that the group property (24) holds, since $$\partial _{a+b}=\partial _a+\partial _b$$ for all $$a,b\in H$$ by linearity.

$$\square$$

### Lemma 4

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

\begin{aligned} \varPhi :{\mathcal {T}}_{a}\exp (h)\longmapsto \sum _{n\ge 0}\frac{1}{n!}\sum _{\imath ^\lambda \vdash n} \frac{n!}{\lambda !}\phi _\imath ^\lambda {\mathfrak {e}}^{*\lambda }_\imath (h+a),\quad a\in H, \end{aligned}

has the unique isometric conjugate-linear extension

\begin{aligned} {\varPhi :\varGamma _\beta (H)\ni \psi \longmapsto \varPhi \psi \in L_\chi ^2 }\quad \text { with the adjoint mapping}\quad {\varPhi ^*:L_\chi ^2 \rightarrow \varGamma _\beta (H)} \end{aligned}

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

\begin{aligned} \varPhi :{\mathfrak {e}}^{\odot \lambda }_\imath /\Vert {\mathfrak {e}}^{\odot \lambda }_\imath \Vert _\beta \longmapsto \phi _\imath ^\lambda /\Vert {\phi }^\lambda _\imath \Vert _\chi \quad \text {for all}\quad \lambda \in {\mathbb {Y}}, \ \imath \in {\mathscr {I}}_{\eta (\lambda )}. \end{aligned}

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

### Proof

By Lemma 3 the $$\varGamma _\beta (H)$$-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 2 and Theorem 1, $$\phi :H\ni {h}\longmapsto \phi _h\in L_\chi ^2$$ may be extended to $$\varPhi$$ in following way

\begin{aligned} \begin{aligned} \varPhi {\mathcal {T}}_{a}\exp (h)&=\sum _{n\ge 0}\frac{1}{n!}\sum _{\imath ^\lambda \vdash n} \frac{n!}{\lambda !}\phi _\imath ^\lambda {\mathfrak {e}}^{*\lambda }_\imath (h+a) =\prod _{i\ge 0}\sum _{n\ge 0}\frac{\phi _i^n}{n!}{\mathfrak {e}}^{*n}_i(h+a)\\&=\prod \exp \left( \phi _i{\mathfrak {e}}^*_i(h+a)\right) = \exp \left( \phi _{h+a}\right) \quad \text {where}\\ \varPhi [(h+a)^{\odot n}]&=\phi _{h+a}^n=\sum \limits _{\imath ^\lambda \vdash n} \frac{n!}{\lambda !}\phi _\imath ^\lambda {\mathfrak {e}}^{*\lambda }_\imath (h+a),\quad a\in H \end{aligned} \end{aligned}

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

\begin{aligned} \begin{aligned} \Vert \exp (\phi _{h+a})\Vert _\chi ^2&=\sum _{n\ge 0}\frac{1}{n!^2} \sum _{\imath ^\lambda \vdash n}\Vert \phi _\imath ^\lambda \Vert _\chi ^2 \frac{n!^2}{\lambda !^2}|{\mathfrak {e}}^{*\lambda }_\imath (h+a)|^2\\&=\sum _{n\ge 0}\frac{1}{n!^2}\sum _{\imath ^\lambda \vdash n} \frac{n!^2}{\lambda !}\beta _\lambda |{\mathfrak {e}}^{*\lambda }_\imath (h+a)|^2\le \sum _{n\ge 0}\frac{1}{n!}\sum _{\imath ^\lambda \vdash n} \frac{n!}{\lambda !}|{\mathfrak {e}}^{*\lambda }_\imath (h+a)|^2\\&=\prod \exp |{\mathfrak {e}}^*_i(h+a)|^2=\exp \Vert h+a\Vert ^2. \end{aligned} \end{aligned}

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 [8, I.4.3]) holds,

\begin{aligned} \hbar _\lambda :={n!}\Big (\prod \limits _{i\le \lambda _j}h(i,j)\Big )^{-1}\quad \text {where}\quad \hbar _\lambda =\mathop {\text {dim}}S^\lambda _\imath , \end{aligned}
(25)

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

\begin{aligned} \left( {\phi }_{\imath _1}({\mathfrak {u}}){{\mathfrak {e}}}^*_{\imath _1}(h),\ldots , {\phi }_{\imath _\eta }({\mathfrak {u}}){{\mathfrak {e}}}^*_{\imath _\eta }(h)\right) :=t_\imath ({\mathfrak {u}},h). \end{aligned}

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 [18, I.7] and taking into account (2), (3), (25), we obtain

\begin{aligned} \phi _h^n({\mathfrak {u}})= \sum \limits _{\imath ^\lambda \vdash n}\hbar _\lambda s^\lambda _\imath ({\mathfrak {u}},h),\quad h\in H \end{aligned}

where $$s^\lambda _\imath =0$$ if $$\lambda ^\intercal _1>l_\lambda$$ and the summation is over all standard tabloids. Hence, $$\big \{\phi _h^n:h\in H\big \}$$ is total in $$L_\chi ^{2,n}$$ by Theorem 1. In consequence, $$\left\{ \exp (\phi _h):h\in H\right\}$$ is total in $$L_\chi ^2$$. This yields surjectivity of $$\varPhi$$ and of all its restrictions to $${H}^{\odot n}_\beta$$.

$$\square$$

### Corollary 2

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

## 6 Fourier Analysis on Virtual Unitary Matrices

Consider the isometry $${H}^{*\odot n}_\beta {\mathop {\simeq }\limits ^{{\mathcal {P}}}}{P}_\beta ^n(H)$$ (see e.g., [7, 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 $$\Vert \psi _n^*\Vert _{P_\beta ^n}=\Vert \psi _n\Vert _{H^{\otimes n}_\beta }$$ where

\begin{aligned} \psi _n^*(h):=\langle h^{\otimes n}\mid \psi _n\rangle _{H^{\otimes n}_\beta }\simeq \left\langle ({h},\ldots , {h})\mid {\mathcal {P}}\circ \psi _n\right\rangle , \quad {\psi _n\in H^{\odot n}_\beta }. \end{aligned}

Let $${H}^2_\beta =\sum _{n\ge 0}{P}_\beta ^n(H)$$ 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 $$\psi =\sum \psi _n\in \varGamma _\beta (H)$$. Since the set $$\{\exp (h):h\in H\}$$ is total in $$\varGamma _\beta (H)$$, elements of $${H}^2_\beta$$ can be written as

\begin{aligned} {H}^2_\beta =\left\{ \psi ^*(h)= \left\langle \exp (h)\mid \psi \right\rangle _\beta :\psi =\sum \psi _n\in \varGamma _\beta (H)\right\} . \end{aligned}

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

### Definition 3

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

\begin{aligned} {\langle \psi ^*(\cdot )\mid \varphi ^*(\cdot )\rangle _{{H}^2_\beta }}:={\left\langle \varphi \mid \psi \right\rangle _\beta }\quad \text {where}\quad \Vert \psi ^*\Vert _{{H}^2_\beta }^2=\langle \psi ^*(\cdot )\mid \psi ^*(\cdot )\rangle _{{H}^2_\beta }= \sum n!\Vert \psi _n^*\Vert ^2_{P_\beta ^n}. \end{aligned}

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

\begin{aligned} {*:\varGamma _\beta (H)\ni \psi \longmapsto \psi ^*\in {H}^2_\beta },\quad \psi =\sum \psi _n. \end{aligned}

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

\begin{aligned} \varPsi :{H}^2_\beta \ni {\mathfrak {e}}^{*\lambda }_\imath /\Vert {\mathfrak {e}}^{\odot \lambda }_\imath \Vert _\beta \longmapsto \phi _\imath ^\lambda /\Vert {\phi }^\lambda _\imath \Vert _\chi \in L^2_\chi \end{aligned}

defines the surjective isometry

\begin{aligned} \varPsi :{H}^2_\beta \longrightarrow L^2_\chi \quad \text {and its adjoint}\quad {\varPsi ^*:L^2_\chi \longrightarrow {H}^2_\beta }. \end{aligned}

### Lemma 5

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

\begin{aligned} {\mathfrak {e}}^{*{\mathbb {Y}}_n}:=\bigcup \big \{{\mathfrak {e}}^{*\lambda }_\imath :\imath ^\lambda \vdash n, \imath \in {\mathscr {I}}\big \} \quad \text {and}\quad {\mathfrak {e}}^{*{\mathbb {Y}}}:=\bigcup \big \{{\mathfrak {e}}^{*{\mathbb {Y}}_n}:{n}\in {\mathbb {N}}_0\big \} \end{aligned}

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

\begin{aligned} \Vert {\mathfrak {e}}^{*\lambda }_\imath \Vert _{P_\beta ^n}^2=\beta _\lambda \Vert {\mathfrak {e}}^{\odot \lambda }_\imath \Vert ^2= \dfrac{(\eta (\lambda )-1)!}{(\eta (\lambda )-1+n)!}\frac{\lambda !}{n!}, \quad \imath ^\lambda \vdash n. \end{aligned}

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

\begin{aligned} \psi ^*(h)=\left\langle \exp (h)\mid \psi \right\rangle _\beta =\sum _{n\ge 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 _n\big \rangle _\beta \end{aligned}
(26)

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

\begin{aligned} \begin{aligned}&\psi ^*(h)=\big \langle \psi ^*(\cdot )\mid \exp \langle \cdot \mid h\rangle \big \rangle _{H^2_\beta } =\big \langle \psi ^*(\cdot )\mid {E}(\cdot ,h)\big \rangle _{{H}^2_\beta },\quad {\psi \in \varGamma _\beta (H)}\\ \text {where}\quad&E(h',h):=|\exp \langle h'\mid h\rangle |^2\!/\exp \langle h\mid h\rangle \quad \text {for all}\quad h\in {H}. \end{aligned} \end{aligned}
(27)

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

\begin{aligned} {H}^2_\beta {\mathop {\simeq }\limits ^{\varPsi }}{L}^2_\chi ,\quad {P}_\beta ^n(H){\mathop {\simeq }\limits ^{\varPsi }}{L}^{2,n}_\chi . \end{aligned}
(28)

### Proof

Taking into account (13) and (23), we conclude that every $$\psi ^*\in {H}^2_\beta$$ such that $$\psi =\bigoplus \psi _n\in \varGamma _\beta (H)$$ with $${\psi _n\in {H}^{\odot n}_\beta }$$ has the following expansion

\begin{aligned} \psi ^*(h) =\sum _{n\ge 0}\frac{1}{n!}\sum _{\imath ^\lambda \vdash n}\frac{n!}{\lambda !}{\mathfrak {e}}^{*\lambda }_\imath (h) \langle {\mathfrak {e}}^{\odot \lambda }_\imath \mid \psi _n\rangle _\beta \quad \text {where} \quad \psi =\bigoplus _{n\ge 0}\sum _{\imath ^\lambda \vdash n} \frac{\langle {\mathfrak {e}}^{\odot \lambda }_\imath \mid \psi _n\rangle _\beta }{\Vert {\mathfrak {e}}^{\odot \lambda }_\imath \Vert ^2_\beta }{\mathfrak {e}}^{\odot \lambda }_\imath . \end{aligned}

On the other hand, in relative to the inner product $$\langle \cdot \mid \cdot \rangle _\varGamma$$, we have

\begin{aligned} \exp \langle h'\mid h\rangle =\bigoplus _{n\ge 0}\frac{1}{n!}\sum _{\imath ^\lambda \vdash n} \frac{n!}{\lambda !}{\mathfrak {e}}^{*\lambda }_\imath (h')\,\bar{{\mathfrak {e}}}^{*\lambda }_\imath (h)= \sum _{n\ge 0}\frac{1}{n!}\sum _{\imath ^\lambda \vdash n} \frac{{\mathfrak {e}}^{*\lambda }_\imath (h')\bar{{\mathfrak {e}}}^{*\lambda }_\imath (h)}{\Vert {\mathfrak {e}}^{\odot \lambda }_\imath \Vert ^2}. \end{aligned}

Verify the first equality in (27) by substituting (26) into the formula (27). We get

\begin{aligned} \psi ^*(h)&=\bigg \langle \sum _{n\ge 0}\sum _{\imath ^\lambda \vdash n} \frac{\langle {\mathfrak {e}}^{\odot \lambda }_\imath \mid \psi _n\rangle _\beta }{\Vert {\mathfrak {e}}^{\odot \lambda }_\imath \Vert ^2_\beta }{\mathfrak {e}}^{*\lambda }_\imath (h')\mid \sum _{n\ge 0}\frac{1}{n!}\sum _{\imath ^\lambda \vdash n} \frac{{\mathfrak {e}}^{*\lambda }_\imath (h')\bar{{\mathfrak {e}}}^{*\lambda }_\imath (h)}{\Vert {\mathfrak {e}}^{\odot \lambda }_\imath \Vert ^2}\bigg \rangle _{\!\!H^2_\beta }\\&=\sum _{n\ge 0}\frac{1}{n!}\sum _{\imath ^\lambda \vdash n}\frac{n!}{\lambda !} {\mathfrak {e}}^{*\lambda }_\imath (h) \langle {\mathfrak {e}}^{\odot \lambda }_\imath \mid \psi _n\rangle _\beta =\left\langle \exp (h)\mid \psi \right\rangle _\beta . \end{aligned}

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

\begin{aligned} \begin{aligned} \psi ^*(h)&=\omega ^*(h)=\left\langle \omega ^*\mid \exp (\cdot \mid h) \right\rangle _{H^2_\beta }\\&=\big \langle \psi ^*(\cdot )\mid \exp (h\mid \cdot )[\exp (h\mid h)]^{-1}\exp (\cdot \mid h) \big \rangle _{H^2_\beta } =\big \langle \psi ^*(\cdot )\mid {E}(\cdot ,h)\big \rangle _{H^2_\beta }. \end{aligned} \end{aligned}

Hence, the second equality in (27) holds. Lemma 4 yields (28). $$\square$$

### Remark 1

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 (21) coincides with $$L_\chi ^{2,1}$$.

### Lemma 6

Denote $$\exp \langle h'\mid h\rangle :=K(h',h)$$. The functions

\begin{aligned} H\ni {h}\longmapsto (\varPsi \circ {K})({\mathfrak {u}},h)\quad \text {and}\quad H\ni {h}\longmapsto (\varPsi \circ {E})({\mathfrak {u}},h) \end{aligned}

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

\begin{aligned} (\varPsi \circ {K})({\mathfrak {u}},h)=\exp \left( \phi _h({\mathfrak {u}})\right) ,\qquad (\varPsi \circ {E})({\mathfrak {u}},h)= \exp \big (2\mathop {\text {Re}}\phi _h({\mathfrak {u}})-\Vert h\Vert ^2\big ) \end{aligned}

where the last exponential function has the power series expansion

\begin{aligned} \begin{aligned} \exp \left\{ 2\mathop {\text {Re}}\phi _h-\Vert h\Vert ^2\right\}&= \sum _{m,n\ge 0}\frac{\Vert h\Vert ^{m+n}}{m!n!} {\mathfrak {h}}_{n,m}\left( \phi _{h/\Vert h\Vert },{{\bar{\phi }}}_{h/\Vert h\Vert }\right) \\ {\mathfrak {h}}_{n,m}(z,{\bar{z}})&=\sum ^{m\wedge n}_{k=0}(-1)^kk!\left( {\begin{array}{c}m\\ k\end{array}}\right) \left( {\begin{array}{c}n\\ k\end{array}}\right) {z}^{m-k}{\bar{z}}^{n-k} \end{aligned} \end{aligned}
(29)

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 $${K}(h',h)$$ in variable $${h'\in {H}}$$, we obtain

\begin{aligned} (\varPsi \circ {K})({\mathfrak {u}},h)&=\!\sum _{n\ge 0}\frac{1}{n!} \sum \limits _{\imath ^\lambda \vdash n}\frac{n!}{\lambda !} \phi ^\lambda _\imath ({\mathfrak {u}}){{\mathfrak {e}}}^{*\lambda }_\imath (h) =\!\sum _{n\ge 0}\frac{1}{n!}\Big (\sum _{i\ge 0}\phi _i({\mathfrak {u}}){{\mathfrak {e}}}^*_i(h)\Big )^{\!n}\!\! =\exp \big (\phi _h({\mathfrak {u}})\big ). \end{aligned}

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

\begin{aligned} (\varPsi \circ {E})({\mathfrak {u}},h)&=\Big |\sum _{n\ge 0}\frac{1}{n!}\sum _{\imath ^\lambda \vdash n} \frac{n!}{\lambda !}\phi ^\lambda _\imath ({\mathfrak {u}}){{\mathfrak {e}}}^{*\lambda }_\imath (h)\Big |^2 \Bigg (\sum _{n\ge 0}\frac{1}{n!}\sum _{\imath ^\lambda \vdash n}\frac{n!}{\lambda !} |{\mathfrak {e}}^{*\lambda }_\imath (h)|^2\Bigg )^{-1}\\&=\exp \big (2\mathop {\text {Re}}\phi _h({\mathfrak {u}})-\Vert h\Vert ^2\big ). \end{aligned}

By Lemma 4, $$(\varPsi \circ {K})(\cdot ,h)$$ and $$(\varPsi \circ {E})(\cdot ,h)$$ with $${h\in {H}}$$ take values in $$L_\chi ^2$$. The expansion (29) follows from [13, n.12] where polynomials $${\mathfrak {h}}_{n,m}(z,{\bar{z}})$$ were introduced. $$\square$$

### Theorem 2

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

\begin{aligned} {\hat{f}}(h):={\left\langle \exp (h)\mid \varPhi ^* f\right\rangle _\beta }\quad \text {of variable}\quad {h\in H} \end{aligned}

and its Taylor coefficients at zero $$d^n_0{\hat{f}}$$ have the integral representations

\begin{aligned} \begin{aligned} {\hat{f}}(h)&=\int \exp ({{\bar{\phi }}}_h)f\,d\chi = \int \exp \big (2\mathop {\text {Re}}\phi _h-\Vert h\Vert ^2\big )f\,d\chi ,\\ d^n_0{\hat{f}}(h)&=\int {{\bar{\phi }}}_h^nf_n\,d\chi , \end{aligned} \end{aligned}
(30)

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

\begin{aligned} {L^2_\chi {\mathop {\simeq }\limits ^{F}}{H}^2_\beta }\quad \text {and}\quad L^{2,n}_\chi {\mathop {\simeq }\limits ^{F}}{P}_\beta ^n(H). \end{aligned}

### Proof

Since $$\varPsi =\varPhi \circ *^{-1}$$, we obtain $$\varPsi ^*=*\circ \varPhi ^*$$. From (27) 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,

\begin{aligned} {\hat{f}}(h)&={\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 }}\\&={\big \langle f(\cdot )\mid (\varPsi \circ E)(\cdot ,h)\big \rangle _\chi } ={\int \exp \big (2\mathop {\text {Re}}\phi _h-\Vert h\Vert ^2_H\big )f\,d\chi } \end{aligned}

by Lemma 6. On the other hand, according to the same claim

\begin{aligned} {\hat{f}}(h)=\big \langle (\varPsi ^*\circ f)(\cdot )\mid {K}(\cdot ,h)\big \rangle _{{H}^2_\beta }= \big \langle f(\cdot )\mid (\varPsi \circ K)(\cdot ,h)\big \rangle _\chi =\int \exp \left( {{\bar{\phi }}}_h\right) f\,d\chi . \end{aligned}

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

\begin{aligned} {\hat{f}}\left( \alpha {x}\right) = \int \exp \left( {{\bar{\phi }}}_{\alpha x}\right) f\,d\chi = \sum \alpha ^n\!\int \frac{{{\bar{\phi }}}_{x}^n}{n!}f_n\,d\chi , \quad {\alpha \in {\mathbb {C}}}. \end{aligned}

Using the n-homogeneity of derivatives, we find

\begin{aligned} d^n_0{\hat{f}}(\alpha {x})=\frac{d^n}{d\alpha ^n}\! \sum \alpha ^n\int \frac{{{\bar{\phi }}}_{x}^n}{n!}f_n\,d\chi \mid _{\alpha =0}=\int {{\bar{\phi }}}_x^nf_n\,d\chi . \end{aligned}

Finally, we notice that the isometry $${L^2_\chi {\mathop {\simeq }\limits ^{F}}{H}^2_\beta }$$ holds, since the isometry $$\varPhi ^*$$ is surjective by Lemma 5. Similarly, we get $$L^{2,n}_\chi {\mathop {\simeq }\limits ^{F}}{P}_\beta ^n(H)$$. $$\square$$

### Corollary 3

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

\begin{aligned} \int \exp \big \{\mathop {\text {Re}}\phi _h\big \}\,d\chi = \exp \Big \{\frac{1}{4}\Vert h\Vert ^2\Big \}. \end{aligned}

### Proof

It is enough to put $$f\equiv 1$$ and to replace h by h/2 in the formula (30).

$$\square$$

### Corollary 4

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

### Proof

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

$$\square$$

### Corollary 5

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

\begin{aligned} {\hat{f}}(h)=\sum \frac{1}{n!} {d}^n_0{\hat{f}}(h)\quad \text {with}\quad f={\sum f_n\in L^{2}_\chi },\quad {f_n\in L^{2,n}_\chi } \end{aligned}

has the coefficients

\begin{aligned} d^n_0{\hat{f}}(h)=\int f_n{{\bar{\phi }}}_h^n\,d\chi = \sum _{\imath ^\lambda \vdash n}\hbar _\lambda {s}^\lambda _\imath [{f}_\imath \,{{\mathfrak {e}}}_\imath ^*(h)],\quad {f}_\imath :=\int {f}{\bar{\phi }}_\imath \,d\chi \end{aligned}
(31)

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 [18, 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 (3), whereas $$\hbar _\lambda$$ by (25). Thus,

\begin{aligned} \exp \phi _h({\mathfrak {u}})=\sum _{n\ge 0}\frac{1}{n!} \sum _{\imath ^\lambda \vdash n}\hbar _\lambda s^\lambda _\imath ({\mathfrak {u}},h)= \sum _{n\ge 0}\frac{1}{n!} \sum _{\imath ^\lambda \vdash n}\frac{n!}{\lambda !} \phi ^\lambda _\imath ({\mathfrak {u}}){\mathfrak {e}}^{*\lambda }_\imath (h). \end{aligned}
(32)

Using (32) in combination with Theorem 1, we find

\begin{aligned} {\hat{f}}(h)=\int f({\mathfrak {u}})\exp {{\bar{\phi }}}_h({\mathfrak {u}})\,d\chi ({\mathfrak {u}}) =\sum _{n\ge 0}\frac{1}{n!}\sum _{\imath ^\lambda \vdash n} \hbar _\lambda {\bar{s}}^\lambda _\imath [{f}_\imath \,{{\mathfrak {e}}}_\imath ^*(h)] \end{aligned}

where the derivative at zero may be defined as

\begin{aligned} d^n_0{\hat{f}}(h)=\sum \limits _{\imath ^\lambda \vdash n} \hbar _\lambda {s}^\lambda _\imath [{f}_\imath \,{{\mathfrak {e}}}_\imath ^*(h)]\quad \text {with}\quad {s}^\lambda _\imath [{f}_\imath \,{{\mathfrak {e}}}_\imath ^*(h)]:= \int f({\mathfrak {u}}){\bar{s}}^\lambda _\imath ({\mathfrak {u}},h)\,d\chi ({\mathfrak {u}}). \end{aligned}

In fact, for zh with $${z\in {\mathbb {C}}}$$ and $$\imath ^\lambda \vdash n$$ with $$\lambda ^\intercal _1>\eta (\lambda )$$ we find

\begin{aligned} {s}^\lambda _\imath [{f}_\imath \,{\mathfrak {e}}^*_\imath (zh)]=z^n {s}^\lambda _\imath [{f}_\imath \,{\mathfrak {e}}^*_\imath (h)]. \end{aligned}

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 1 yield the first equality in (31). By Lemmas 5 and 6 the second formula in (31) also holds. $$\square$$

### Remark 2

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 $$\big \{{\mathfrak {e}}^{*\lambda }={\mathfrak {e}}^{*\lambda _1}_1\ldots {\mathfrak {e}}^{*\lambda _m}_m :{\lambda =(\lambda _1,\ldots ,\lambda _m)\in {\mathbb {Y}}}\big \}$$. The Fourier transform

\begin{aligned} {\hat{f}}(h) =\int \exp ({{\bar{\phi }}}_h)f\,d\chi _m= \int \exp \big (2\mathop {\text {Re}}\phi _h-\Vert h\Vert ^2\big )f\,d\chi _m,\quad h\in {\mathbb {C}}^m \end{aligned}

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

\begin{aligned} {F:{\mathfrak {e}}^{*\lambda }\mapsto \phi ^\lambda }\quad \text {such that}\quad \Vert {\mathfrak {e}}^{*\lambda }\Vert _{H^2_\beta }^2= \Vert {\phi }^\lambda \Vert _{\chi _m}^2=\frac{(m-1)!\lambda !}{(m-1+|\lambda |)!} \end{aligned}

where the space $$L_{\chi _m}^2$$ with the Haar measure $$\chi _m$$ on U(m) has the orthogonal basis $$\big \{\phi ^\lambda = {\phi }^{\lambda _1}_1\circ \pi _m^{-1}\ldots {\phi }^{\lambda _m}_m\circ \pi _m^{-1}:\lambda \in {\mathbb {Y}}\big \}$$.

## 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 $$\psi \in \varGamma _\beta (H)$$, $${a,h\in H}$$. By (27), $$\left\langle {\mathcal {T}}_a\exp (h)\mid \psi \right\rangle _\beta =T_a\psi ^*(h)=\big \langle T_a\psi ^*(\cdot )\mid \exp \langle \cdot \mid h\rangle \big \rangle _{H^2_\beta }$$. Hence,

\begin{aligned} T_a\psi ^*(h)=\left\langle {\mathcal {T}}_a\exp (h)\mid \psi \right\rangle _\beta \!= \left\langle \psi ^*(\cdot )\mid \exp \langle \cdot \mid h+a\rangle \right\rangle _{H^2_\beta }\!= \left\langle \psi ^*(\cdot )\mid {M}_{a^*}\exp \langle \cdot \mid h\rangle \right\rangle _{H^2_\beta } \end{aligned}

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 $$\{{\exp \langle \cdot \mid h\rangle }:h\in H\}$$ 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, $$\Vert {T}_a\psi ^*\Vert _{H^2_\beta } =\Vert {M}_{a^*}\psi ^*\Vert _{H^2_\beta }$$. The isometry $$H^2_\beta \simeq \varGamma _\beta (H)$$ yields $$\Vert {T}_a\psi ^*\Vert _{H^2_\beta }=\Vert {\mathcal {T}}_a\psi \Vert _\beta$$. According to (24), we have

\begin{aligned} \begin{aligned}&\Vert T_a\psi ^*\Vert ^2_{H^2_\beta }\le \exp \big (\Vert a\Vert ^2\big ) \Vert \psi ^*\Vert ^2_{H^2_\beta } \quad \text {and} \quad {T}_{a+b}={T}_a{T}_b={T}_b{T}_a \\&\Vert M_{a^*}\psi ^*\Vert ^2_{H^2_\beta }\le \exp \big (\Vert a\Vert ^2\big ) \Vert \psi ^*\Vert ^2_{H^2_\beta } \quad \text {and} \quad {M}_{a^*+b^*}={M}_{a^*}{M}_{b^*}={M}_{b^*}{M}_{a^*} \end{aligned}\nonumber \\ \end{aligned}
(33)

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

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

\begin{aligned} {{\mathbb {C}}\ni z\longmapsto M^\dagger _{z{a^{*}}}},\quad dM^\dagger _{z{a^{*}}}/dz\mid _{z=0}={{\bar{\phi }}}_a \quad \text {with}\quad {{\bar{\phi }}}_a=\varPsi {a^{*}}\varPsi ^* \end{aligned}

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

The group $$(H,+)$$ on $$L^2_\chi$$ can be also represented by $$T_a^\dagger :=\varPsi {T}_a\varPsi ^*:L^2_\chi \rightarrow L^2_\chi$$. From Lemmas 3 and 5 it follows that the generator of strongly continuous subgroup

\begin{aligned} {{\mathbb {C}}\ni z\longmapsto T^\dagger _{z{\mathfrak {a}}}},\quad dT^\dagger _{za}/dz\mid _{z=0}=\partial _a^\dagger \quad \text {with}\quad \partial _a^\dagger :=\varPsi \partial _a^*\varPsi ^* \end{aligned}

has the dense domain $${\mathfrak {D}}(\partial _a^\dagger )={\big \{f\in L^2_\chi :\partial _a^\dagger f\in L^2_\chi \big \}}$$ and is closed, since $$\partial _a^*$$ is closed. By (27) $${\hat{f}}(h)=\left\langle \exp (h)\mid \varPhi ^*f\right\rangle _\beta = \big \langle (\varPsi ^*\circ f)(\cdot )\mid \exp \langle \cdot \mid h\rangle \big \rangle _{H^2_\beta }$$. Hence, by Lemma 6,

\begin{aligned} T_a^\dagger {\hat{f}}(h)=\big \langle (\varPsi ^*\circ f)(\cdot )\mid T_a\exp \langle \cdot \mid h\rangle \big \rangle _{H^2_\beta }=\int f\exp \left( {{\bar{\phi }}}_{h+a}\right) \,d\chi . \end{aligned}

### Lemma 7

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 $$f\in {\mathfrak {D}}({{\bar{\phi }}}_a^m)={\big \{f\in L^2_\chi :{{\bar{\phi }}}_a^m{f}}{\in L^2_\chi \big \}}$$ with $$m\in {\mathbb {N}}_0$$,

\begin{aligned} \partial _a^{*m}T_a{F}(f)= {F}\big ({{\bar{\phi }}}_a^mM^\dagger _{a^{*}}f\big ),\quad a\in {H}. \end{aligned}
(34)

For every $$f\in {\mathfrak {D}}(\partial _a^{\dagger m})={\big \{f\in L^2_\chi :\partial _a^{\dagger m}{f}\in L^2_\chi \big \}}$$ with $$m\in {\mathbb {N}}_0$$,

\begin{aligned} a^{* m}M_{a^{*}}{F}(f)={F}\big (\partial _a^{\dagger m}T_a^\dagger f\big ),\quad {a}\in {H}. \end{aligned}
(35)

As a conclusion, $$\partial _{\mathbb {i}a}^\dagger =-\mathbb {i}\partial _a^\dagger$$. Moreover, the following commutation relations hold,

\begin{aligned} M_{a^*}^\dagger T_b^\dagger = \exp \langle {a}\mid {b}\rangle T_b^\dagger M_{a^*}^\dagger , \qquad \big ({{\bar{\phi }}}_{a}\partial _b^\dagger - \partial _b^\dagger {{\bar{\phi }}}_a\big )f={\langle {a}\mid {b}\rangle }f, \end{aligned}
(36)

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

\begin{aligned} \partial _a^{*m}T_a{\hat{f}}(h) =\int \frac{d^mM_{za^{*}}^\dagger f}{dz^m}\Big |_{z=0}\exp {{\bar{\phi }}}_h\,d\chi =\int ({{\bar{\phi }}}_a^mf)\exp {{\bar{\phi }}}_h\,d\chi ,\quad m\ge 0 \end{aligned}

for all $${f\in L^2_\chi }$$. This gives (34). 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

\begin{aligned} \begin{aligned} a^{* m}M_{a^{*}}{\hat{f}}(h)&= \frac{d^mM_{za^{*}}{\hat{f}}(h)}{dz^m}\Big |_{z=0} =\int \frac{d^mT_{za}^\dagger f}{dz^m}\Big |_{z=0}\exp {{\bar{\phi }}}_h\,d\chi \\&=\int (\partial _a^{\dagger m}f)\exp {{\bar{\phi }}}_h\,d\chi \quad \text {with}\quad f\in {\mathfrak {D}}(\partial _a^{\dagger m}),\quad \psi ^*=\varPsi ^*f. \end{aligned} \end{aligned}
(37)

This together with the group property by applying F and $${F}^{-1}$$ yields (35).

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

\begin{aligned} M_{b^{*}}{T}_a{\hat{f}}(h)&= \exp \langle {h}\mid {b}\rangle {\hat{f}}(h+a),\\ T_a{M}_{b^{*}}{\hat{f}}(h)&=\exp \langle {h+a}\mid {b}\rangle {\hat{f}}(h+a) =\exp \langle {a}\mid {b}\rangle {M}_{b^{*}}{T}_a{\hat{f}}(h). \end{aligned}

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

\begin{aligned} \big (d^2/dt^2\big )T_{ta}M_{tb^{*}}{\hat{f}}\mid _{t=0}= \big (\partial _a^{*2}+2\partial _a^*b^{*}+b^{*2}\big ){\hat{f}}. \end{aligned}
(38)

Subsequently, taking into account (38) 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

\begin{aligned} \big (\partial _a^{*2}+2\partial _a^*b^{*}+b^{*2}\big ){\hat{f}}&=(d/dt)\big [(d/dt)\exp \langle ta\mid {\bar{t}}b\rangle M_{tb^{*}}T_{ta}{\hat{f}}\big ]_{t=0}\\&=2{ \langle {a}\mid {b}\rangle }{\hat{f}}+ \big (\partial _a^{*2}+2b^{*}\partial _a^*+b^{*2}\big ){\hat{f}}. \end{aligned}

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 $$\varPsi ^*(f)=\bigoplus \psi _n\in \varGamma _\beta (H)$$ with $$\psi _n\in {H}^{\odot {n}}_\beta$$,

\begin{aligned} T_aM_{b^{*}}&= \exp \langle {a}\mid {b}\rangle M_{b^{*}}T_a,\quad \left( \partial _a^*b^{*}-b^{*}\partial _a^*\right) {\hat{f}}={ \langle {a}\mid {b}\rangle }{\hat{f}}. \end{aligned}
(39)

Corollary 4 yields $${F}=*\circ \varPhi ^*$$ and $${F}^{-1}=\varPhi \circ *^{-1}$$. The equality (37) 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,

\begin{aligned}&M_{a^*}^\dagger T_b^\dagger ={F}^{-1}T_aM_{b^{*}}\,{F} =\exp \langle {a}\mid {b}\rangle {F}^{-1}M_{b^{*}}T_a\,{F} =\exp \langle {a}\mid {b}\rangle T_b^\dagger M_{a^*}^\dagger ,\\&\big ({{\bar{\phi }}}_a\partial _b^\dagger -\partial _b^\dagger {{\bar{\phi }}}_a\big )f= {F}^{-1}\left( \partial _a^*b^{*}-b^{*}\partial _a^*\right) {F}f=\langle {a}\mid {b}\rangle {f} \end{aligned}

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 $$\psi _n\in {H}^{\odot {n}}_\beta$$. $$\square$$

## 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). 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 $${h\in H}$$

\begin{aligned} {W}^\dagger (h):={W}^\dagger (a,b)= \exp \Big \{\frac{1}{2}\langle {a}\mid {b}\rangle \Big \}T^\dagger _bM^\dagger _{a^*}. \end{aligned}

For convenience, we will use the quaternion algebra $$\mathbb {H}={\mathbb {C}}\oplus {\mathbb {C}}\mathbb {j}$$ of numbers $$\zeta ={{(\alpha _1+\alpha _2\mathbb {i})}+{(\alpha '_1+\alpha '_2\mathbb {i})}\mathbb {j}} ={\alpha +\alpha '\mathbb {j}}$$ such that $$\mathbb {i}^2=\mathbb {j}^2=\mathbb {k}^2= \mathbb {i}\mathbb {j}\mathbb {k}=-1$$, $$\mathbb {k}=\mathbb {i}\mathbb {j}=-\mathbb {j}\mathbb {i}$$, $$\mathbb {k}\mathbb {i} = -\mathbb {i}\mathbb {k}= \mathbb {j}$$, where $$(\alpha ,\alpha ')\in {\mathbb {C}}^2$$ with $$\alpha ={\alpha _1+\alpha _2\mathbb {i}},\alpha '={\alpha '_1+\alpha '_2\mathbb {i}\in {\mathbb {C}}}$$ and $$\alpha _\imath ,\alpha '_\imath \in {\mathbb {R}}$$$$(\imath =1,2)$$ [26, 5.5.2]. Let us denote $$\alpha ':=\mathfrak {I}{\zeta }$$ for all $$\zeta =\alpha +\alpha '\mathbb {j}\in \mathbb {H}$$.

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

\begin{aligned} \langle {h}\mid {h}'\rangle&=\langle {a}+{b}\mathbb {j}\mid {a}'+{b}'\mathbb {j}\rangle =\langle {a}\mid {a}'\rangle +\langle {b}\mid {b}'\rangle + \left[ \langle {a}'\mid {b}\rangle -\langle {a}\mid {b}'\rangle \right] \mathbb {j} \end{aligned}

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

\begin{aligned} \mathfrak {I}\langle {h}\mid {h}'\rangle =\langle {a}'\mid {b}\rangle -\langle {a}\mid {b}'\rangle ,\qquad \mathfrak {I}\langle {h}\mid {h}\rangle =0. \end{aligned}

### Theorem 3

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

\begin{aligned} S^\dagger :{\mathcal {H}}_{\mathbb {C}}\ni X(a,b,t)\longmapsto \exp (t){W}^\dagger (h),\quad {h}=a+b\mathbb {j} \end{aligned}

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

\begin{aligned} {W}^\dagger (h+h')=\exp \Big \{-\frac{\mathfrak {I}\langle {h}\mid {h}'\rangle }{2}\Big \} {W}^\dagger (h){W}^\dagger (h') \end{aligned}
(40)

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

\begin{aligned} {W}^\dagger \left( (\tau +\tau '){h}\right) ={W}^\dagger (\tau {h}){W}^\dagger (\tau '{h})= {W}^\dagger (\tau '{h}){W}(\tau {h}) \end{aligned}
(41)

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,

\begin{aligned} {W}^\dagger ({h}){W}^\dagger ({h}')&= \exp \big \{\mathfrak {I}\left\langle {h}\mid {h}'\right\rangle \big \} {W}^\dagger ({h}'){W}^\dagger ({h})\quad \text {where}\nonumber \\ \mathfrak {I}\left\langle {h}\mid {h}'\right\rangle&=- \big [{\mathfrak {p}}_{h}^\dagger ,{\mathfrak {p}}_{h'}^\dagger \big ] \quad \text {with}\quad \big [{\mathfrak {p}}_{h}^\dagger ,{\mathfrak {p}}_{h'}^\dagger \big ]:= {\mathfrak {p}}_{h}^\dagger {\mathfrak {p}}_{h'}^\dagger - {\mathfrak {p}}_{{h}'}^\dagger {\mathfrak {p}}_{h}^\dagger \end{aligned}
(42)

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 $${\mathbb {C}}\times (H\oplus {H}\mathbb {j})$$ with multiplication $$(t,{h})(t',{h}')= \left( t+t'-\frac{1}{2}\mathfrak {I}\langle {h}\mid {h}'\rangle ,\,{h}+{h}'\right)$$ for all $${h}={a}+{b}\mathbb {j}$$, $${h}'={a}'+{b}'\mathbb {j}\in {H} \oplus {H} \mathbb {j}$$. The mapping $${G}:X({a},{b},t)\longmapsto \left( t-\frac{1}{2}\langle {a}\mid {b}\rangle , \,{a}+{b}\mathbb {j}\right)$$ is a group isomorphism, since

\begin{aligned}&{G}\big (X({a},{b},t)X({a}',{b}',t')\big ) ={G}\big (X({a}+{a}',{b}+{b}', t+t'+\langle {a}\mid {b}'\rangle )\big )\\&\quad =\Big (t+t'+\langle {a}\mid {b}'\rangle -\frac{1}{2}\big ( \langle {a}+{a}'\mid {b}+{b}'\rangle \big ), ({a}+{a}')+ ({b}+{b}')\mathbb {j}\Big )\\&\quad =\Big (t+t'-\frac{1}{2}\big (\langle {a}\mid {b}\rangle + \langle {a}'\mid {b}'\rangle \big ) +\frac{1}{2}\big (\langle {a}\mid {b}'\rangle - \langle {a}'\mid {b}\rangle \big ),({a}+{a})+ ({b}+{b}')\mathbb {j}\Big )\\&\quad =\Big (t-\frac{1}{2}\langle {a}\mid {b}\rangle , \, {a}+{b}\mathbb {j}\Big ) \Big (t'-\frac{1}{2}\langle {a}'\mid {b}'\rangle , \, {a}'+{b}'\mathbb {j}\Big )={G}\left( X({a},{b},t)\right) {G}\left( X({a}',{b}',t')\right) . \end{aligned}

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

\begin{aligned} {W}({h})= \exp \Big \{\frac{1}{2}\langle {a}\mid {b}\rangle \Big \} M_{{b}^*}T_{{a}}, \quad {h}={a}+{b}\mathbb {j}. \end{aligned}
(43)

Using group properties and the commutation relation (39), we obtain

\begin{aligned}&\exp \Big \{-\frac{\mathfrak {I}\langle {h}\mid {h}'\rangle }{2}\Big \} {W}({h}){W}({h}') =\exp \Big \{\frac{\langle {a}\mid {b}'\rangle }{2} -\frac{\langle {a}'\mid {b}\rangle }{2}\Big \} {W}({h}){W}({h}')\nonumber \\&\quad =\exp \Big \{\frac{\langle {a}\mid {b}\rangle }{2} +\frac{\langle {a}'\mid {b}'\rangle }{2}\Big \} \exp \Big \{\frac{\langle {a}\mid {b}'\rangle }{2} -\frac{\langle {a}'\mid {b}\rangle }{2}\Big \} M_{{b}^*}T_{a}M_{{{b}'}^*}T_{{a}'}\nonumber \\&\quad =\exp \Big \{\frac{1}{2}\langle {a} +{a}'\mid {b}+{b}'\rangle \Big \} M_{{b}^*+{{b}'}^*}T_{{a}+{a}'} ={W}({h}+{h}'). \end{aligned}
(44)

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

\begin{aligned} S:{\mathcal {H}}_\mathbb {C} \ni X({a},{b},t)\longmapsto \exp (t){W}({h})=\exp \Big \{t+\frac{1}{2}\langle {a}\mid {b}\rangle \Big \} M_{{b}^*}T_{{a}} \end{aligned}

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 $${h}_0\ne 0$$ in H and an integer $${n>0}$$ such that

\begin{aligned} \exp \Big \{t+\frac{1}{2}\langle {a}\mid {b}\rangle \Big \} \exp {\langle {c}\mid {a}\rangle } \langle {c}+{b}\mid {h}_0\rangle ^n=0\quad \text {for all}\quad {a},{b},{c}\in {H} . \end{aligned}

But, this is only possible for $${h}_0=0$$. It gives a contradiction. Finally, using that

\begin{aligned} \exp \Big \{t+\frac{1}{2}\langle {a}\mid {b}\rangle \Big \} T_{b}^\dagger M_{a^*}^\dagger = {F}^{-1}\Big (\exp \Big \{t+\frac{1}{2}\langle {a}\mid {b}\rangle \Big \} M_{b^*}T_a\Big ){F}, \end{aligned}

we obtain that $$S^\dagger ={F}^{-1}S\,{F}$$ is irreducible. Applying F, $${F}^{-1}$$ to (44) we get (40).

Consider the Weyl system $${W}^\dagger$$ on the space $$L^2_\chi$$. By (40) we obtain the equality

\begin{aligned} {W}^\dagger (h){W}^\dagger (h')&= \exp \Big \{\frac{\mathfrak {I}\langle {h}\mid {h}'\rangle }{2}\Big \} {W}^\dagger ({h}+{h}')=\exp \Big \{-\frac{\mathfrak {I}\langle {h}'\mid {h}\rangle }{2}\Big \}{W}^\dagger ({h}'+{h})\\&=\exp \Big \{\!\!-{\mathfrak {I}\langle {h}'\mid {h}\rangle }\Big \} \exp \Big \{\frac{\mathfrak {I}\langle {h}'\mid {h}\rangle }{2}\Big \}{W}^\dagger ({h}'+{h})\\&=\exp \Big \{\!\!-{\mathfrak {I}\langle {h}'\mid {h}\rangle }\Big \} {W}^\dagger (h'){W}^\dagger ({h}). \end{aligned}

Using this equality, we get (41) for any fixed $${h}={a}+{b}\mathbb {j}\in {H}\oplus {H}\mathbb {j}$$. 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

\begin{aligned} {\mathfrak {p}}_{{a},{b}}^\dagger =\frac{d}{d\tau }{W}^\dagger (\tau {h})\Big |_{\tau =0} =\frac{d}{d\tau }\exp \Big \{\frac{1}{2}\langle {\tau {a}}\mid {\tau {b}}\rangle \Big \} T^\dagger _{\tau {b}}M^\dagger _{\tau {a}^*}\Big |_{\tau =0}=\partial _{b}^\dagger +{{\bar{\phi }}}_{a}. \end{aligned}

Taking into account the inequalities (33) and that F is isometric, we get

\begin{aligned} \Vert {W}^\dagger (\tau {a},\tau {b})f\Vert ^2_\chi \le \exp \big (\Vert \tau a\Vert ^2+\Vert \tau b\Vert ^2\big )\Vert f\Vert _\chi ^2,\quad f\in L^2_\chi . \end{aligned}

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 $${\mathfrak {D}}({\mathfrak {p}}_{h}^\dagger ) ={\big \{f\in L^2_\chi :{\mathfrak {p}}^\dagger _{h}f\in L^2_\chi \big \}}$$. Moreover, its generator $${\mathfrak {p}}^\dagger _{h}$$ is closed (see, e.g., [32]). Note also that $${\mathfrak {p}}^\dagger _{\tau {h}}=\tau {\mathfrak {p}}^\dagger _{h}$$ for $$\tau \in {\mathbb {R}}$$.

Finally, applying the commutation relation (36) 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

\begin{aligned} -\mathfrak {I}\langle {h}\mid {h}'\rangle&=\left\langle {a}\mid {b}'\right\rangle - \left\langle {a}'\mid {b}\right\rangle ={{\bar{\phi }}}_{a}\partial _{{b}'}^\dagger - {{\bar{\phi }}}_{{a}'}\partial _{b}^\dagger + \partial _{b}^\dagger {{\bar{\phi }}}_{{a}'}- \partial _{{b}'}^\dagger {{\bar{\phi }}}_{a}\\&=(\partial _{b}^\dagger +{{\bar{\phi }}}_{a}) (\partial _{{b}'}^\dagger +{{\bar{\phi }}}_{{a}'})- (\partial _{{b}'}^\dagger +{{\bar{\phi }}}_{{a}'}) (\partial _{b}^\dagger +{{\bar{\phi }}}_{a})= \big [{\mathfrak {p}}_{h}^\dagger ,{\mathfrak {p}}_{{h}'}^\dagger \big ]. \end{aligned}

$$\square$$

## 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 $${\mathcal {I}}:{l}_2\looparrowright c_0$$ is continuous, the Gelfand triple $$l_1 {\mathop {\longrightarrow }\limits ^{{\mathcal {I}}^*}}l_2 \looparrowright c_0$$ with adjoint $${\mathcal {I}}^*$$ holds. The mapping $$Q:{l}_1\rightarrow c_0$$ with $$Q:={\mathcal {I}}\circ {\mathcal {I}}^*$$ is positive and $$\langle Q\xi ^*\mid Q\xi ^*\rangle _{l_2}:=\xi ^*(Q\xi ^*)=\sum \xi _n^2=\Vert \xi \Vert ^2_{l_2}$$ where $$\xi =Q\xi ^*\in {\mathscr {R}}(Q)$$ and $$\xi ^*\in {l}_1=c^*_0$$. By the Aronszajn-Kolmogorov decomposition theorem (see e.g., [22, 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 $${\mathcal {I}}:l_2\looparrowright c_0$$. Given $$\xi _1^*,\ldots ,\xi _n^*\in l^1=c_0^*$$, we assign the family of cylinder sets $$\varOmega _n^c=\left\{ \xi \in c_0:(\xi _1^*(\xi ),\ldots ,\right. \left. \xi _n^*(\xi ))\in \varOmega _n\right\}$$ with any Borel $$\varOmega _n\subset {\mathbb {R}}^n$$ 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

\begin{aligned} {\mathfrak {w}}(\varOmega _n^c):=\gamma (\varOmega _n)\quad \text {with}\quad \gamma (\varOmega _n):=(2\pi )^{-n/2}\int _{\varOmega _n} \exp \big \{-\Vert \omega \Vert _{l_2}^2/2\big \}\,d\omega . \end{aligned}

By Gross’ theorem [10] there exists a smaller abstract Wiener space $$\{w_0,\Vert \cdot \Vert _{w_0}\}$$ such that injections $${l_2\looparrowright {w}_0\looparrowright c_0}$$ are continuous and the increasing sequence of orthogonal projectors $$p_n:{l}_2\rightarrow {\mathbb {R}}^n$$ 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 $${\upsilon :c_0\rightarrow {\mathbb {R}}}$$ such that $$\upsilon =\rho \circ p_n^\sim$$ is defined to be $$\int _{\varOmega _n^c}\upsilon \,d{\mathfrak {w}}=\int _{\varOmega _n}\rho \,d\gamma$$. The Fernique theorem [6, 15, Thm 3.1] implies that these exist $$\varepsilon ,\eta >0$$ such that $$\Vert \cdot \Vert _{w_0}$$ satisfies the following conditions with a sufficiently large $$K>0$$,

\begin{aligned} \int _{c_0}\exp \big \{\varepsilon \Vert \xi \Vert ^2_{w_0}\big \}d{\mathfrak {w}}(\xi )<\infty ,\quad {\mathfrak {w}}\big (\Vert \xi \Vert _{w_0}\ge K\big )\le \exp \big \{-\eta K^2\big \}. \end{aligned}

Let us go back to the Weyl system $${W}^\dagger$$. Consider in $$L^2_\chi$$ the dense subspace $${L}^{+2}_\chi :=\bigcup _{n\ge 0}\bigoplus _{m=0}^nL^{2,m}_\chi$$. Let $${a}={b}=\mathbb {i}\xi _m{\mathfrak {e}}_m$$ with $${\xi _m\in {\mathbb {R}}}$$. Then by Theorem 3

\begin{aligned} {W}^\dagger (\mathbb {i}\xi _m{\mathfrak {e}}_m,\mathbb {i}\xi _m{\mathfrak {e}}_m) = \exp \big \{{-\xi _m^2}/{2}\big \} T^\dagger _{\mathbb {i}\xi {\mathfrak {e}}_m}M^\dagger _{-\mathbb {i}\xi {\mathfrak {e}}_m^*}. \end{aligned}

### Theorem 4

For any $$f\in {L}^{+2}_\chi$$ and $$\xi =(\xi _m)\in c_0$$ there exists the limit

\begin{aligned} {W}^\dagger _\xi {f}=\lim _{n\rightarrow \infty }{W}^\dagger _{p_n^\sim (\xi )}{f},\quad {W}^\dagger _{p_n^\sim (\xi )}:= \exp \Bigg \{-\frac{\Vert {p_n^\sim (\xi )}\Vert _{w_0}^2}{2}\Bigg \} \prod _{m=1}^{n} T^\dagger _{\mathbb {i}\xi _m{\mathfrak {e}}_m} M^\dagger _{-\mathbb {i}\xi _m{\mathfrak {e}}_m^*} \end{aligned}

$${\mathfrak {w}}$$-almost everywhere on $$c_0$$ such that the 1-parameter Gaussian semigroup

\begin{aligned} {\mathfrak {G}}^\dagger _rf=\frac{1}{\sqrt{4\pi r}}\int _{c_0} \exp \Big \{-\frac{\Vert \xi \Vert _{w_0}^2}{4r}\Big \}{W}^\dagger _\xi {f}\,d{\mathfrak {w}}(\xi ), \quad r>0 \end{aligned}
(45)

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, $$w(r)={\mathfrak {G}}^\dagger _rf$$ is unique solution of the Cauchy problem

\begin{aligned} \frac{dw(r)}{dr}=-\sum \big (\partial _m^\dagger +{{\bar{\phi }}}_m\big )^2w(r), \quad w(0)=f\in {L}^{+2}_\chi . \end{aligned}
(46)

### 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 $${W}^\dagger (\tau {a},\tau {a})=\exp \left\{ {\Vert \tau a\Vert ^2}/{2}\right\} T_{\tau {a}}^\dagger M_{\tau {a}^*}^\dagger$$ with $${\tau \in {\mathbb {R}}}$$. Replacing $$a=b$$ by $$\mathbb {i}\tau a$$ with $${\tau \in {\mathbb {R}}}$$, we obtain that

\begin{aligned} {W}^\dagger (\mathbb {i}\tau {a},\mathbb {i}\tau {a})= \exp \Big \{-\frac{1}{2}\langle {\tau {a}}\mid {\tau {a}}\rangle \Big \} T^\dagger _{\mathbb {i}\tau {a}}M^\dagger _{-\mathbb {i}\tau {a}^*} \quad \text {has the generator} \quad \mathbb {i}(\partial _{a}^\dagger +{{\bar{\phi }}}_{a}) \end{aligned}

with self-adjoint $$\partial _{a}^\dagger +{{\bar{\phi }}}_{a}$$. By relations (36), $${W}^\dagger (\mathbb {i}\tau {a},\mathbb {i}\tau {a})$$ is unitary.

Lemma 7 implies that $$[M_{-\mathbb {i}\xi _m{\mathfrak {e}}_m^*}^\dagger , T_{\mathbb {i}\xi _k{\mathfrak {e}}_k}^\dagger ]=0$$ and $${[M_{-\mathbb {i}\xi _m{\mathfrak {e}}_m^*}^\dagger , M_{-\mathbb {i}\xi _k{\mathfrak {e}}_k^*}^\dagger ]=0}$$, as well as, $$[T_{\mathbb {i}\xi _m{\mathfrak {e}}_m}^\dagger , T_{\mathbb {i}\xi _k{\mathfrak {e}}_k}^\dagger ]=0$$ for any $$m\ne k$$. In view of the relations (36),

\begin{aligned} \big [{{\bar{\phi }}}_{\mathbb {i}\xi _m{\mathfrak {e}}_m}, \partial _{\mathbb {i}\xi _k{\mathfrak {e}}_k}^\dagger \big ]=0 \quad \text {if}\quad m\ne k \quad \text {and}\quad \big [{{\bar{\phi }}}_{\mathbb {i}\xi _m{\mathfrak {e}}_m}, \partial _{\mathbb {i}\xi _m{\mathfrak {e}}_m}^\dagger \big ]=-\xi ^2_m. \end{aligned}
(47)

Check that (45) holds. Denote $${W}^\dagger _{p_n^\sim (\xi )}:= \prod _{m=1}^{n}{W}^\dagger (\mathbb {i}\xi _m{\mathfrak {e}}_m,\mathbb {i}\xi _m{\mathfrak {e}}_m)$$ and $$T^\dagger _{p_n^\sim (\xi )}:=\prod _{m=1}^{n} T^\dagger _{\mathbb {i}\xi _m{\mathfrak {e}}_m}$$, as well as, $$M^\dagger _{p_n^\sim (\xi )}:=\prod _{m=1}^{n} M^\dagger _{-\mathbb {i}\xi _m{\mathfrak {e}}_m^*}$$ with $${\xi =(\xi _m)}{\in {w}_0}$$. Using (33) with the operator norm over $$H^2_\beta$$, we get the inequality

\begin{aligned} \ln \prod _{m=1}^{n}\Vert T_{\mathbb {i}\xi _m{\mathfrak {e}}_m}\Vert _{{\mathscr {L}}(H^2_\beta )}^2 \le \sum _{m=1}^{n}\langle \xi _m{\mathfrak {e}}_m\mid \xi _m{\mathfrak {e}}_m\rangle ^2 =\sum _{m=1}^{n}\xi _m^2=\Vert p_n^\sim (\xi )\Vert _{l_2}^2. \end{aligned}

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

Using the unitarity of groups $${W}^\dagger (\mathbb {i}\xi _m{\mathfrak {e}}_m,\mathbb {i}\xi _m{\mathfrak {e}}_m)$$, we find by virtue of (47) that their product $${W}^\dagger _{p_n^\sim (\xi )}= \exp \left\{ -\Vert {p_n^\sim (\xi )}\Vert _{l_2}^2/2\right\} T^\dagger _{p_n^\sim (\xi )} M^\dagger _{p_n^\sim (\xi )}$$ is also unitary. Taking into account the continuity of $${{\mathcal {I}}_0:l_2\looparrowright {w}_0}$$ 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 $${f\in {L}^{+2}_\chi }$$, $$n\ge 0$$,

\begin{aligned} \Vert {W}^\dagger _{p_n^\sim (\xi )}f\Vert _\chi \le \exp \big \{-\Vert {p_n^\sim (\xi )}\Vert _{l_2}^2/2\big \}\Vert {f}\Vert _\chi \le \exp \big \{-\Vert {\mathcal {I}}_0\Vert ^2\,\Vert \xi \Vert _{w_0}^2/2\big \}\Vert {f}\Vert _\chi . \end{aligned}

The Lebesgue dominated convergence theorem implies that there exists $$\lim \Vert {W}^\dagger _{p_n^\sim (\xi )}f\Vert _\chi$$$${\mathfrak {w}}$$-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

\begin{aligned} \Vert {W}^\dagger _\xi {f}\Vert _\chi \le \exp \big \{-\Vert {\mathcal {I}}_0\Vert ^2\, \Vert \xi \Vert _{w_0}^2/2\big \}\Vert {f}\Vert _\chi \quad \text {for all} \quad {f\in {L}^{+2}_\chi },\quad {\xi \in {w}_0}. \end{aligned}
(48)

The $$\Vert \cdot \Vert _\chi$$-norm of integrant in (45) is bounded by $$\exp \left\{ \varepsilon \Vert \xi \Vert _{w_0}^2\right\}$$ with any $$\varepsilon >0$$. By Fernique’s theorem and (48), the integral (45) 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 (45) 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 $$\bigoplus _{m=0}^nL^{2,m}_\chi$$.

Prove that the semigroup $${\mathfrak {G}}_r$$ is generated by $$\sum {\mathfrak {p}}_m^{\dagger 2}$$ with $${\mathfrak {p}}^\dagger _m:={\mathbb {i}(\partial _m^\dagger +{{\bar{\phi }}}_m)}$$. By differentiation of $${W}^\dagger (\mathbb {i}\xi _m{a},\mathbb {i}\xi _m{a})$$ at $${\xi _m=0}$$, we get that its generator coincides with $${\mathfrak {p}}^\dagger _m$$. In fact, $${W}^\dagger (\mathbb {i}\xi _m{a},\mathbb {i}\xi _m{a}){f}=\exp \big \{\xi _m{\mathfrak {p}}_m^\dagger \big \}f$$ for all $${f\in \phi ^{\mathbb {Y}}}$$. Applying the next formula for Gamma functions with $$\alpha =(\alpha _1,\ldots ,\alpha _n)\in {\mathbb {N}}_0^n$$

\begin{aligned} \begin{aligned} \prod _{m=1}^{n}\!\frac{1}{\sqrt{4\pi r}} \int \exp \left\{ \frac{-\xi _m^2}{4r}\right\} \xi _m^{2\alpha _m}d\xi _m \Big |_{\xi _m=2\sqrt{r}x_m}\!\!=\!\prod _{m=1}^{n}\!\frac{(2\sqrt{r})^{2}}{\sqrt{\pi }}\! \int \exp \big \{\!-x_m^2\big \}x_m^{{2\alpha _m}}dx_m\\ ={2^{2n}r^n}\!\prod _{m=1}^{n}\!\varGamma \Big (\frac{{2\alpha _m}+1}{2}\Big ) =2^nr^n\frac{({2\alpha }-1)!}{(\alpha -1)!}, \end{aligned} \end{aligned}

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

\begin{aligned} {\mathfrak {G}}^\dagger _rh_n&=\prod _{m=1}^{n}\frac{1}{\sqrt{4\pi r}}\int \exp \Big \{-\frac{\xi ^2_m}{4r}\Big \} \exp \big \{\xi _m{\mathfrak {p}}_m^\dagger \big \}d\xi _mh_n\\&=\sum _{\alpha \in {\mathbb {N}}_0^n} \prod _{m=1}^{n}\frac{{\mathfrak {p}}_m^{\dagger \alpha _m}}{\alpha _m!} \frac{1}{\sqrt{4\pi r}} \int \exp \Big \{-\frac{\xi _m^2}{4r}\Big \}\xi _m^{\alpha _m}\,d\xi _mh_n\\&=\sum _{\alpha \in {\mathbb {N}}_0^n}2^nr^n\prod _{m=1}^{n}\dfrac{(2\alpha _m-1)!}{(\alpha _m-1)!} \dfrac{{\mathfrak {p}}_m^{\dagger 2}}{(2\alpha _m)!}h_n =\exp \Big \{r\sum _{m=1}^{n}{\mathfrak {p}}_m^{\dagger 2}\Big \}h_n. \end{aligned}

Using (48), we obtain that $$0\le r\longmapsto {\mathfrak {G}}_r^\dagger$$ is the 1-parameter $$C_0$$-semigroup on any finite sum $$\bigoplus _{m=0}^nL^{2,m}_\chi$$ with densely defined closed generator $$\sum _{m=1}^{n}{\mathfrak {p}}_m^{\dagger 2}$$. Applying the known relation [32] between the initial problem (46) and the 1-parameter $$C_0$$-semigroup $${\mathfrak {G}}_r^\dagger$$, we obtain that the function $$w_n(r)={\mathfrak {G}}^\dagger _rf_n$$ for any $${n\in {\mathbb {N}}}$$ solves this problem in the sense that $$d{\mathfrak {G}}^\dagger _rf_n/dr|_{r=0}=\sum _{m=1}^{n}{\mathfrak {p}}_m^{\dagger 2}f_n$$ for all $${f_n\in }\bigoplus _{m=0}^nL^{2,m}_\chi$$. The theorem is proved. $$\square$$

Taking into account the isometries $${H}^2_\beta {\mathop {\simeq }\limits ^{\varPsi }}{L}^2_\chi$$ and $${P}_\beta ^n(H){\mathop {\simeq }\limits ^{\varPsi }}{L}^{2,n}_\chi$$ from (28), 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 (43) on the dense subspace of polynomials $${P}_\beta (H):={\sum }_{n\ge 0}{P}_\beta ^n(H)$$ 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 $${P}_\beta (H)$$,

\begin{aligned} \partial ^*_a\psi ^*=\lim _{\tau \rightarrow 0}\left( T_{\tau a}\psi ^*-\psi ^*\right) /\tau \quad \text {and}\quad {a}^*\psi ^*=\lim _{\tau \rightarrow 0}\left( M_{\tau a^*}\psi ^*-\psi ^*\right) /\tau , \quad {a,b\in {H}}. \end{aligned}

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

\begin{aligned} \frac{d}{d\tau }{W}(\tau {a},\tau {b})|_{\tau =0} =\frac{d}{d\tau }\exp \Big \{\frac{1}{2}\langle {a}\mid {b}\rangle \Big \}\Big |_{\tau =0}= b^*+\partial _a^* \end{aligned}

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

### Corollary 6

For all $$\psi ^*\in {P}_\beta (H)$$ and $$\xi =(\xi _m)\in c_0$$ there exists the limit

\begin{aligned} {W}_\xi \psi ^*=\lim _{n\rightarrow \infty }{W}_{p^\sim _n(\xi )}\psi ^*,\quad {W}_{p^\sim _n(\xi )}:= \exp \Big \{-\frac{\Vert {p^\sim _n(\xi )}\Vert _{w_0}^2}{2}\Big \} \prod _{m=1}^{n} M_{-\mathbb {i}\xi _m{\mathfrak {e}}_m^*} T_{\mathbb {i}\xi _m{\mathfrak {e}}_m} \end{aligned}

$${\mathfrak {w}}$$-almost everywhere on $$c_0$$ such that the 1-parameter Gaussian semigroup

\begin{aligned} {\mathfrak {G}}_r\psi ^*=\frac{1}{\sqrt{4\pi r}}\int _{c_0} \exp \Big \{\frac{-\Vert \xi \Vert _{w_0}^2}{4r}\Big \}{W}_\xi \psi ^*d{\mathfrak {w}}(\xi ), \quad r>0 \end{aligned}

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

\begin{aligned} \frac{dw(r)}{dr}=-\sum \big ({\mathfrak {e}}_m^*+\partial _m^*\big )^2w(r), \quad w(0)=\psi ^*\in {P}_\beta (H) \end{aligned}

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