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

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

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

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

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,

$$\begin{aligned} s^\lambda _\imath ({\mathfrak {u}}):=\frac{\mathop {\text {det}} \big [\phi _{\imath _i}^{\lambda _j+\eta -j}({\mathfrak {u}})\big ]_{1\le i,j\le \eta }}{\prod _{1\le i<j\le \eta }[\phi _{\imath _i}({\mathfrak {u}})-\phi _{\imath _j}({\mathfrak {u}})]} \quad \text {and}\quad {\phi }^\lambda _\imath := {\phi }^{\lambda _1}_{\imath _1}\ldots {\phi }^{\lambda _\eta }_{\imath _\eta }. \end{aligned}$$

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

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

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

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

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

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

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

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


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

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

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

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

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]

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

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 \({\mathfrak {e}}^{{\mathbb {Y}}_n}\)

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

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 \({\mathfrak {e}}^{\mathbb {Y}}\)

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

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

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

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

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

Lemma 2

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

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

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

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


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

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

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

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

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

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

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,

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

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

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


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

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

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

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

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

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


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.


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,

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

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 \({\mathfrak {e}}^{*{\mathbb {Y}}}\)

$$\begin{aligned} \psi ^*(h)=\left\langle \exp (h)\mid \psi \right\rangle _\beta =\sum _{n\ge 0}\frac{1}{n!}\sum _{\imath ^\lambda \vdash n}\frac{n!}{\lambda !} {\mathfrak {e}}^{*\lambda }_\imath (h) \big \langle {\mathfrak {e}}^{\odot \lambda }_\imath \mid \psi _n\big \rangle _\beta \end{aligned}$$

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

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

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

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


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

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

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


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

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

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

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


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

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


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 \(*={F}\circ \varPhi \).


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

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

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 \(t_\imath ={f}_\imath \,{{\mathfrak {e}}}_\imath ^*(h)\).


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,

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

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

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

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 \(m\in {\mathbb {N}}_0\),

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

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 \(m\in {\mathbb {N}}_0\),

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

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

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

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 \(a,b\in {H}\).


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

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

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

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

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 \(\psi _n\in {H}^{\odot {n}}_\beta \),

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

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

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

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

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

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,

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

on the dense subspace \({\mathfrak {D}}({{\bar{\phi }}}_a^2)\cap {\mathfrak {D}}(\partial _b^{\dagger 2})\subset {L}^2_\chi \).


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

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

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

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

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

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

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

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


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),

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

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

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

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 \({P}_\beta (H)\).