Radial boundary values of Poisson integrals on infinite-dimensional balls

Abstract

We consider a Gelfand triple $E'\rightarrow H\rightarrow E$, so that E is a separable complex Banach space with dual $E'$, and H is its dense Hilbert subspace. We investigate the problem of analytic extensions on an open ball $\mathcal{Q}\subset E'$ and their radial boundary values in the Hardy spaces $\mathcal{H}_{\mu}^{p}$ ($1\le p\le\infty$) using the Poisson integrals on the unitary group $U(\infty)$ over H endowed with an invariant probability measure μ. For this purpose, we construct a Poisson-type kernel with the help of the symmetric Fock space Γ generated by H and prove that the set of radial boundary values of these analytic functions entirely coincides with $\mathcal{H}_{\mu}^{p}$.

1 Introduction

A goal of the current work is to describe a certain type of complex-valued Poisson kernels generated by symmetric Fock spaces and associated Poisson integrals in the case of Hardy spaces in infinite-dimensional settings. This allows us to get a solution of the radial boundary problem for the corresponding analytic extensions.

The main results of the paper are as follows. We consider a Gelfand triple $E'\rightarrow H\rightarrow E$ consisting of a separable complex Banach space E with dual $E'$ and a densely embedded Hilbert subspace H. In Section 2 we investigate the space $\mathcal{H}^{2}$ of analytic functions on an open ball $\mathcal{Q}$ in $E'$, which is conjugate-linearly isometric to the symmetric Fock space Γ generated by H. Its orthogonal polynomial basis is described in Section 3.

In Section 4 we introduce an invariant probability Wiener-type measure μ on the infinite-dimensional unitary group $U(\infty)=\bigcup U(j)$, irreducibly acting in H, where $U(j)$ are subgroups of unitary ${(j\times j)}$-matrices. This measure is defined as the projective limit of probability Haar measures $\mu_{j}$ on $U(j)$ and is a group analog of probability Wiener measures on Banach spaces, which were introduced by Gross [1]. Its description substantially uses the theory of invariant measures over infinite-dimensional unitary groups developed by Neretin [2] and Olshanski [3].

Using the known Prokhorov criterion and the Schwartz theorem, we show in Theorem 4.1 that μ is invariant under the right actions of $U^{2}(\infty)$ over $U(\infty)$ and that μ is a weak limit of a subsequence $(\mu_{j_{k}})$. In Theorem 4.3 a concentration property of the sequence $(\mu_{j})$ is established.

The Hardy spaces $\mathcal{H}_{\mu}^{p}$ ($1\le p\le\infty$) of $L^{p}_{\mu}$-integrable complex-valued functions are described in Section 5. An orthogonal polynomial basis in the Hilbert space $\mathcal{H}^{2}_{\mu}$ is given by Theorem 5.1. Integral formulas for analytic extensions to the open ball ${\mathcal {Q}\subset E'}$ by means of a group generalization of the Paley-Wiener map associated with μ are established in Theorems 6.2 and 8.1.

The tools are applied in Section 8 to describe the radial boundary values of functions defined by the integral Poisson formula. In the space $\mathcal{H}_{\mu}^{p}$ with ${1\le p<\infty}$ this problem is described by Theorem 8.3. The existence of weak radial boundary values in $\mathcal{H}_{\mu}^{\infty}$ is established in Theorem 8.4.

Note that the Hardy spaces $\mathcal{H}^{p}_{\mu}$ of analytic functions on infinite-dimensional polydiscs were considered in the works of Cole and Gamelin [4] and Ørted and Neeb [5]. Similar spaces on more general infinite-dimensional domains that are not necessarily polydiscs were investigated by Pinasco and Zalduendo [6], Carando et al. [7], and others.

2 On analyticity associated with Gelfand triples

Let $(E,\|\cdot\|)$ be a complex separable Banach space, and $E'$ be its normed dual. Consider a complex separable Hilbert space H with scalar product ${\langle\cdot\mid\cdot\rangle}$ and norm $\|\cdot\| _{H}=\langle\cdot\mid\cdot\rangle^{1/2}$ such that the sequence of linear mappings $E'\rightarrow H\stackrel {J}{\looparrowright}E$ forms a Gelfand triple with a continuous dense embedding J.

Denote $B:= \{h\in{H}\colon\|h\|_{H}<1 \}$ and $S:= \{h\in {H}\colon\|h\|_{H}=1 \}$. The Hermitian dual $H^{*}$ of H is identified with H via the conjugate-linear isomorphism ${{}^{*}\colon H^{*}\rightarrow H^{**}=H}$ such that $\eta(h)={\langle h\mid\eta^{*}\rangle}$ for all ${h\in H}$, ${\eta\in H^{*}}$.

Since the embedding J is dense and H is reflexive, the transpose mapping ${J^{t} \colon E'\rightarrow H^{*}}$ is injective continuous and has the dense range $\mathscr{R}(J^{t})$.

Fix an orthonormal basis $(e_{j})_{j\in\mathbb{N}}$ in H so that every functional $e_{j}^{*}={\langle\cdot\mid e_{j}\rangle}$ belongs to $\mathscr{R}(J^{t})$. Following [6], we define the involution ${}^{\dagger}\colon h\mapsto h^{\dagger}:=\sum\bar{e}_{j}^{*}(h)e_{j}$ for any $h={\sum e_{j}^{*}(h)e_{j}\in H}$. If ${\eta\in H^{*}}$, then $\eta^{\dagger}$ is defined so that $(\eta^{\dagger})^{*}=(\eta^{*})^{\dagger}$, that is, $\eta(h^{\dagger})=\bar{\eta}^{\dagger}(h)$. These involutions in H and $H^{*}$ are isometric and depend on the basis chosen.

Thus, we have the Gelfand triple $E'\stackrel{J^{*}}{\rightarrow}H\stackrel {J}{\looparrowright}E$ with an injective covariance operator ${J\circ{J}^{*} \in\mathscr {L}(E',E)}$ such that ${J}^{*}:={{}^{*}\circ{}^{\dagger}\circ{J}^{t}}$, where the injective mapping $J^{*}$ is continuous and has the dense range $\mathscr{R}(J^{*})$. The unbounded inverse $A=({J}\circ{J}^{*})^{-1}$ is defined on the dense domain $\mathscr{D}(A)=H$ in E. Denote by

$$\mathcal{Q}:={ \bigl\{ z\in E'\colon h={J}^{*}z\in B \bigr\} } $$

the inverse image of the open unit ball B with respect to the injective mapping ${J^{*}\colon E'\rightarrow H}$. Clearly, the set $\mathcal{Q}$ is the open unit ball in the dual space $E'$ endowed with the norm $\|z\|_{J^{*}}:=\|{J}^{*}z\|_{H}$ induced from H.

It is important to note that the set $\mathcal{Q}$ is also open with respect to the norm topology in $E'$ because this topology is stronger than that induced by $J^{*}$, so it contains all open sets induced from H.

Let $H^{\otimes n}$ be the complete nth tensor power of H endowed with the scalar product ${\langle\psi_{n}\mid\psi'_{n}\rangle} ={\langle h_{1}\mid h'_{1}\rangle\cdots\langle h_{n}\mid h'_{n}\rangle}$ for all $\psi_{n}={h_{1}\otimes\cdots\otimes h_{n}}$, $\psi'_{n}={h_{1}'\otimes\cdots \otimes h_{n}'}\in H^{\otimes n}$ and ${h_{i}, h'_{i}\in{H}}$ ($i=1,\ldots,n$).

As $\sigma\colon\{1,\ldots,n\}\mapsto\{\sigma(1),\ldots,\sigma(n)\}$ runs through all n-element permutations, the complete symmetric nth tensor power $H^{\odot n}$ is defined as the range of $H^{\otimes n}$ under the orthogonal projector $S_{n}\colon\psi_{n}\mapsto {h_{1}\odot\cdots\odot h_{n}}:=(n!)^{-1}\sum_{\sigma}{h_{\sigma(1)}\otimes \cdots\otimes h_{\sigma(n)}}$.

As usual, the symmetric Fock space is defined to be the orthogonal sum

$$\varGamma=\bigoplus_{n\in\mathbb{Z}_{+}}{H}^{\odot n},\qquad H^{\odot 0}=\mathbb{C}, $$

of all series $\psi=\bigoplus_{n}\psi_{n}$ convergent with respect to the norm $\Vert \cdot \Vert _{\varGamma}= \langle\cdot\mid\cdot \rangle^{1/2}$ defined by the scalar product $\langle\psi\mid\psi'\rangle=\sum\langle\psi_{n}\mid \psi_{n}'\rangle$.

The set of elements $h^{\otimes n}:={h\otimes\cdots\otimes h}={h\odot \cdots\odot h}:=h^{\odot n}$ with any ${h\in H}$ is total in ${H}^{\odot n}$ by virtue of the polarization formula for symmetric tensor products ${h_{1}\odot\cdots\odot h_{n}}=(2^{n}n!)^{-1}\sum_{\theta_{1},\ldots,\theta _{n}=\pm1} {\theta_{1}\cdots\theta_{n} h^{\otimes n}}$ with $h=\sum_{k=1}^{n}\theta_{k} h_{k}$ for any ${h_{1},\ldots, h_{n}\in H}$ (see, e.g., [8], Section 1.5).

Let us consider the Γ-valued function with a total range

$$\mathcal{Q}\ni z\mapsto \bigl(1-{J}^{*}z \bigr)^{-\otimes1}:=\sum _{n\in \mathbb{Z}_{+}} h^{\otimes n},\quad h={J}^{*}z\in B, \qquad h^{\otimes0}=1, $$

which is analytic because ${\|(1-h)^{-\otimes1}\|^{2}_{\varGamma}}={\sum\| h\|^{2n}_{H}}={(1-\|h\|^{2}_{H})^{-1}<\infty}$. Using this function, we define the Hilbert space of analytic complex-valued functions in the variable ${z\in\mathcal{Q}}$, associated with the symmetric Fock space Γ, as

$$\mathcal{H}^{2}:= \bigl\{ \psi^{\star}(z)= \bigl\langle \bigl(1-{J}^{*}z \bigr)^{-\otimes1}\mid\psi \bigr\rangle \colon\psi\in\varGamma \bigr\} ,\qquad \bigl\langle \psi^{\star}\mid\varphi^{\star}\bigr\rangle _{\mathcal {H}^{2}}:= \langle\varphi\mid\psi \rangle. $$

The space $\mathcal{H}^{2}$ is endowed with the Hilbertian norm $\Vert \psi^{\star} \Vert _{\mathcal{H}^{2}}:=\Vert \psi \Vert _{\varGamma}$. Note that $\psi^{\star}(z)=(\psi^{\star}\circ A)(h)$ for all ${h={J}^{*}z\in B }$. The mapping $\psi\mapsto\psi^{\star}$ is a conjugate-linear isometry from Γ on $\mathcal{H}^{2}$.

Functions $\psi^{\star}\in\mathcal{H}^{2}$ are analytic in the variable $z\in\mathcal{Q}$, as a composition of the analytic Γ-valued function $z\mapsto (1-{J}^{*}z )^{-\otimes1}$ and the linear continuous functional ${\psi^{*}= \langle\cdot\mid\psi \rangle}$ (see, e.g., [9], Proposition 2.4.2).

3 Orthogonal homogenous polynomials

Denote by $\lambda=(\lambda_{1},\ldots,\lambda_{j})\in\mathbb{N}^{j}$ with ${\lambda _{1}\ge\lambda_{2}\ge\cdots\ge\lambda_{j}>0}$ a partition of ${n\in\mathbb{N}}$, that is, $n=|\lambda|:={\lambda _{1}+\cdots+\lambda_{j}}$. Any λ may be identified with a Young diagram of length $\ell (\lambda)=j$. Let $\mathbb{Y}$ denote all Young diagrams, and $\mathbb{Y}_{n}:= \{ \lambda\in\mathbb{Y}\colon|\lambda|=n \}$. Assume that $\mathbb{Y}$ includes the empty partition $\emptyset= (0, 0, \ldots)$.

Let $\mathbb{N}^{\ell(\lambda)}_{*}:= \{\imath= ({\imath_{1}},\ldots,{\imath_{\ell(\lambda)}} )\in \mathbb{N}^{\ell(\lambda)}\colon\imath_{j}\neq \imath_{k}, \forall j\neq k \}$. An orthogonal basis in $H^{\odot n}$ is formed by the system of symmetric tensor products

$$e^{\odot\mathbb{Y}_{n}}=\bigcup \bigl\{ e^{\odot\lambda}_{\imath}:= e^{\otimes\lambda_{1}}_{\imath_{1}}\odot\cdots\odot{e}^{\otimes\lambda_{\ell (\lambda)}}_{\imath_{\ell(\lambda)}} \colon (\lambda,\imath)\in\mathbb{Y}_{n}\times\mathbb{N}^{\ell(\lambda)}_{\ast}\bigr\} , \qquad {e^{\odot\emptyset}_{\imath}=1,} $$

with the norm (see [10], Section 2.2.2)

$$ \bigl\Vert e^{\odot\lambda}_{\imath}\bigr\Vert _{\varGamma}=\sqrt{\lambda!/n!}, \quad \text{where } \lambda!:=\lambda_{1}! \cdot\ldots\cdot\lambda_{\ell(\lambda)}!. $$

(3.1)

Then $e^{\odot\mathbb{Y}}:=\bigcup \{e^{\odot\mathbb{Y}_{n}}\colon {n}\in\mathbb{Z}_{+} \}$ forms an orthogonal basis in Γ.

Throughout the paper we assume that there exists a unique sequence $(z_{j}) \subset E'$ such that the elements ${{J}^{*}z_{j}=e_{j}}$ form an orthonormal basis of $H^{*}$ dual to $(e_{j})$. To any index pair $(\lambda,\imath)\in\mathbb{Y}_{n}\times\mathbb{N}^{\ell (\lambda)}_{\ast}$, we uniquely assign the n-homogenous polynomial

$$\zeta^{\lambda}_{\imath}(z):=\prod_{k=1}^{\ell(\lambda)} \zeta^{\lambda _{k}}_{\imath_{k}}(z)= \bigl\langle h^{\otimes n} \mid{e}^{\odot\lambda}_{\imath}\bigr\rangle ,\quad h={J}^{*}z\in H,\qquad \zeta^{\emptyset}_{\imath}\equiv1, $$

considered as a function in the variable ${z\in E'}$ and defined via the Fourier coefficients $\zeta_{j}(z):={\langle{J}^{*}z\mid {e}_{j}\rangle}$ of an element ${h={J}^{*}z\in H}$. In other words, $\zeta^{\lambda}_{\imath}(z)=(\zeta^{\lambda}_{\imath}\circ A)(h)$ where $\zeta_{j}(z)={\langle{h}\mid{e}_{j}\rangle}$.

Lemma 3.1

The system of n-homogeneous polynomials in the variable ${z\in E'}$,

$$\zeta^{\mathbb{Y}}= \bigl\{ \zeta^{\lambda}_{\imath}(z)\bigl\Vert e^{\odot\lambda }_{\imath}\bigr\Vert _{\varGamma}^{-1} \colon(\lambda,\imath)\in\mathbb {Y}\times\mathbb{N}^{\ell(\lambda)}_{\ast}\bigr\} $$

with norms $\|\zeta^{\lambda}_{\imath}\|_{\mathcal{H}^{2}}=\|e^{\odot\lambda }_{\imath}\|_{\varGamma}$ forms an orthonormal basis in $\mathcal{H}^{2}$. Every function ${\psi^{\star}\in\mathcal{H}^{2}}$ for any ${z\in\mathcal{Q}}$ has the following Fourier expansion with respect to $\zeta^{\mathbb{Y}}$:

$$ \psi^{\star}(z)= \sum_{(\lambda,\imath)\in\mathbb{Y}\times\mathbb{N}^{\ell(\lambda)}_{\ast}} \tilde{\psi}^{\star}{(\lambda,\imath)}\zeta^{\lambda}_{\imath}(z), \qquad \tilde{\psi}^{\star}{(\lambda,\imath)}:=\bigl\| e^{\odot\lambda}_{\imath}\bigr\| _{\varGamma}^{-2} \bigl\langle \psi^{\star}\mid \zeta^{\lambda}_{\imath}\bigr\rangle _{\mathcal{H}^{2}}. $$

(3.2)

Proof

It suffices to observe that the following orthogonality relation holds:

$$\bigl\langle \zeta^{\lambda}_{\imath}\mid\zeta^{\mu}_{\jmath}\bigr\rangle _{\mathcal{H}^{2}} = \bigl\langle {e}^{\odot\mu}_{\jmath}\mid{e}^{\odot\lambda}_{\imath}\bigr\rangle = \left \{ \textstyle\begin{array}{l@{\quad}l} \|e^{\odot\lambda}_{\imath}\|^{2}_{\varGamma}: &\imath=\jmath , \lambda=\mu, \\ 0: &\imath \neq \jmath \text{ or } \lambda \neq \mu. \end{array}\displaystyle \right . $$

 □

Taking into account that ${J}^{*}z=\sum\zeta_{j}(z)e_{j}$ and using the tensor multinomial theorem and (3.1), we obtain the following Fourier decomposition with respect to the basis ${e}^{\odot \mathbb{Y}}$ in Γ:

$$\begin{aligned} \bigl(1-{J}^{*}z \bigr)^{-\otimes1}&=\sum _{n\in\mathbb {Z}_{+}}\bigl({J}^{*}z\bigr)^{\otimes n} \\ &=\sum_{n\in\mathbb{Z}_{+}} \biggl(\sum _{k\in\mathbb{N}}\zeta_{k}(z){e}_{k} \biggr)^{\otimes n} =\sum_{(\lambda,\imath)\in\mathbb{Y}\times\mathbb{N}^{\ell(\lambda )}_{\ast}} \frac{\zeta^{\lambda}_{\imath}(z){e}^{\odot\lambda}_{\imath}}{\|{e}^{\odot \lambda}_{\imath}\|^{2}_{\varGamma}} \end{aligned}$$

(3.3)

for all ${z\in\mathcal{Q}}$. Applying this, we conclude that every analytic function $\psi^{\star}\in \mathcal{H}^{2}$ with $\psi=\bigoplus_{n}\psi_{n}\in\varGamma$ ($\psi_{n}\in{H}^{\odot n}$) has the Taylor expansion at zero

$$ \psi^{\star}(z)=\sum_{n\in\mathbb{Z}_{+}} \bigl\langle \bigl({J}^{*}z\bigr)^{\otimes n}\mid\psi_{n} \bigr\rangle , \quad z\in \mathcal{Q}, $$

where

$$ \bigl\langle \bigl({J}^{*}z\bigr)^{\otimes n}\mid\psi_{n} \bigr\rangle =\sum_{(\lambda,\imath)\in\mathbb{Y}_{n}\times\mathbb{N}^{\ell(\lambda )}_{\ast}} \frac{\langle{e}^{\odot\lambda}_{\imath}\mid\psi_{n}\rangle}{ \|{e}^{\odot\lambda}_{\imath}\|^{2}_{\varGamma}}\zeta^{\lambda}_{\imath}(z) $$

are Hilbert-Schmidt polynomials in the variable ${h={J}^{*}z\in H}$ with any ${z\in E'}$.

Lemma 3.2

Each analytic function $\psi^{\star}\in\mathcal {H}^{2}$ can be uniquely written as

$$ \psi^{\star}(z)= \bigl\langle \psi^{\star}(\cdot)\mid \mathcal{C}(\cdot ,z) \bigr\rangle _{\mathcal{H}^{2}} = \bigl\langle \psi^{\star}(\cdot)\mid\mathcal{P}(\cdot,z) \bigr\rangle _{\mathcal{H}^{2}}, \quad z,z'\in\mathcal{Q}, $$

(3.4)

where $\mathcal{C}(z',z)= \langle (1-{J}^{*}z' )^{-\otimes1}\mid (1-{J}^{*}z )^{-\otimes1} \rangle$ and $\mathcal{P}(z',z)=\vert \mathcal{C}(z',z)\vert ^{2}/\mathcal{C}(z,z)$.

Proof

From (3.3) it follows that the complex-valued function $\mathcal{C}(z',z)$ in the variable ${z\in\mathcal{Q}}$ with fixed ${z'\in\mathcal{Q}}$ belongs to $\mathcal{H}^{2}$. Using that ${J}^{*}z=\sum\zeta_{j}(z)e_{j}$, we obtain

$$\begin{aligned} \mathcal{C}\bigl(z',z\bigr)&=\sum_{n\in\mathbb{Z}_{+}} \bigl\langle \bigl({J}^{*}z'\bigr)^{\otimes n}\mid\bigl({J}^{*}z \bigr)^{\otimes n} \bigr\rangle =\frac{1}{1-\langle{J}^{*} z'\mid{J}^{*} z\rangle} \\ &=\sum_{n\in\mathbb{Z}_{+}} \biggl(\sum _{j\in\mathbb{N}}\zeta_{j}\bigl(z'\bigr)\bar{\zeta} _{j}(z) \biggr)^{n} =\sum_{(\lambda,\imath)\in\mathbb{Y}\times\mathbb{N}^{\ell(\lambda )}_{\ast}} \frac{\zeta^{\lambda}_{\imath}(z')\bar{\zeta}^{\lambda}_{\imath}(z)}{\|e^{\odot \lambda}_{\imath}\|^{2}_{\varGamma}}. \end{aligned}$$

Expanding any $\psi^{\star}\in\mathcal{H}^{2}$ in the orthogonal series with respect to $\zeta^{\mathbb{Y}}$, we obtain (3.2). Substituting (3.2) into formula (3.4) and applying Lemma 3.1, we get

$$\begin{aligned} \bigl\langle \psi^{\star}\bigl(z'\bigr)\mid\mathcal{C} \bigl(z',z\bigr) \bigr\rangle _{\mathcal {H}^{2}}&= \biggl\langle \sum _{(\lambda,\imath)} \frac{\zeta^{\lambda}_{\imath}(z')\langle\psi^{\star}\mid\zeta^{\lambda}_{\imath}\rangle_{\mathcal{H}^{2}}}{\|e^{\odot\lambda}_{\imath}\|^{2}_{\varGamma}} \Bigm|\sum _{(\lambda,\imath)}\frac{\zeta^{\lambda}_{\imath}(z')\bar{\zeta}^{\lambda}_{\imath}(z)}{\|e^{\odot\lambda}_{\imath}\|^{2}_{\varGamma}} \biggr\rangle \\ &=\sum_{(\lambda,\imath)} \frac{\zeta^{\lambda}_{\imath}(z)\langle\psi^{\star}\mid\zeta^{\lambda}_{\imath}\rangle_{\mathcal{H}^{2}}}{\|e^{\odot\lambda}_{\imath}\|^{2}_{\varGamma}}. \end{aligned}$$

So, the first equality in (3.4) holds. If $\omega^{\star}(z'):= \langle\psi^{\star}(\cdot)\mid\mathcal{C}(z',\cdot)[\mathcal {C}(z',z')]^{-1}\mathcal{C}(\cdot,z') \rangle_{\mathcal{H}^{2}}$, then $\omega^{\star}(z)=\psi^{\star}(z)$ for all ${z\in\mathcal{Q}}$. As a result, we obtain

$$\begin{aligned} \psi^{\star}(z)&= \bigl\langle \omega^{\star}(\cdot)\mid \mathcal{C}(\cdot ,z) \bigr\rangle _{\mathcal{H}^{2}} \\ &= \bigl\langle \mathcal{C}(z,\cdot)\bigl[\mathcal{C}(z,z)\bigr]^{-1} \psi^{\star}(z)\mid\mathcal{C}(\cdot,z) \bigr\rangle _{\mathcal{H}^{2}}= \bigl\langle \psi^{\star}(\cdot)\mid\mathcal{P}(\cdot,z) \bigr\rangle _{\mathcal{H}^{2}}. \end{aligned}$$

Hence, the second equality in (3.4) holds. Finally, the totality in Γ of elements $(1-{J}^{*}z )^{-\otimes1}$ with any ${z\in\mathcal{Q}}$ yields the uniqueness of these representations. □

4 Invariant Wiener measures on $U(\infty)$

We still assume that the orthonormal basis $(e_{j})$ of H lies in the range of ${{J}^{*}\colon E'\rightarrow H}$, that is, there exist $(z_{j}) \subset E'$ such that ${{J}^{*}z_{j}=e_{j}}$.

Let $U(\infty)=\bigcup U(j)$ be the infinite-dimensional unitary matrix group with unit $\mathbb {1}$. The group $U(\infty)$ acts irreducibly on H. Denote $U^{2}(\infty):= {U(\infty)\times U(\infty)}$ and $U^{2}(j):= U(j)\times U(j)$. The right action on $U(\infty)$ (similarly, on $U(j)$) is defined as

$$ u\cdot g=w^{-1}uv \quad \text{for all } {u\in U( \infty)},\ {g=(v, w)\in U^{2}(\infty)}. $$

(4.1)

Following [2, 3], we write every $u_{j}\in U(j)$ with $j>1$ in the block matrix form $u_{j}=\bigl [ {\scriptsize\begin{matrix}{} v_{j-1} & a \cr b & t\end{matrix}} \bigr ]$ with $t\in\mathbb{C}$ corresponding to the partition $j=(j-1)+1$ so that $v_{j-1}$ is a $(j-1)\times(j-1)$-matrix. Consider the projective limit $\varprojlim U(j)$ taken with respect to the Livšic-type mapping (which is not a group homomorphism)

$$ \pi^{j}_{j-1}\colon{u_{j}}= \begin{bmatrix} v_{j-1} & a \\ b & t \end{bmatrix} \mapsto u_{j-1}=\left \{ \textstyle\begin{array}{l@{\quad}l} v_{j-1}-[a(1+t)^{-1}b]:& t\neq -1, \\ v_{j-1}:& t=-1, \end{array}\displaystyle \right . $$

from $U(j)$ on $U(j-1)$, which is Borel and surjective and is commuted with the right action of $U^{2}(j-1)$ (see [2], Proposition 0.1, [3], Lemma 3.1). In particular, it follows that $\pi^{j}_{j-1}\colon \bigl [{\scriptsize\begin{matrix}{} v_{j-1} & 0\cr 0 &1 \end{matrix}} \bigr ] \mapsto v_{j-1}$ for all $v_{j-1}\in U(j-1)$.

Let $\pi_{j}\colon\varprojlim U(j)\ni(u_{j})\mapsto u_{j}\in U(j)$ be the projection, so that $\pi_{j-1}={\pi_{j-1}^{j}\circ\pi_{j}}$.

In what follows, every $U(j)$ is identified with its range under the natural inclusion $U(j)\looparrowright U(\infty)$ that assigns to any $u_{j}\in U(j)$ the block matrix $\bigl [{\scriptsize\begin{matrix}{} u_{j} & 0\cr 0 &\mathbb {1} \end{matrix}} \bigr ] \in U(\infty)$, and let $U(\infty)$ be endowed with the topology of inductive limit under the natural inclusions $U(j-1)\looparrowright U(j)$. Accordingly, $\pi^{j}_{j-1}$ are defined over $U(\infty)$ as block matrices transformations. Let $\pi^{k}_{j}:=\pi^{j+1}_{j}\circ\cdots\circ\pi^{k}_{k-1}$ for ${j< k}$ and $\pi^{k}_{j}$ for ${j= k}$ be the identical mapping over $U(\infty)$.

Let us consider the dense injective mapping $\tau\colon U(\infty )\looparrowright\varprojlim U(j)$ that to any ${u_{k}\in U(k)}$ assigns the unique stabilized sequence $(u_{j})$ such that (see [3], n. 4)

$$ \tau\colon U(k)\ni u_{k}\mapsto(u_{j})\in \varprojlim U(j),\qquad u_{j}=\left \{ \textstyle\begin{array}{l@{\quad}l} \pi^{k}_{j}(u_{k}):&j< k, \\ u_{k}:& j=k, \\ \bigl[ {\scriptsize\begin{matrix}{}u_{k} & 0\cr 0 &\mathbb {1}\end{matrix}} \bigr] :& j>k. \end{array}\displaystyle \right . $$

(4.2)

Denote by $U_{\tau}(\infty)$ the group $U(\infty)$ endowed with the induced topology under the mapping $\tau\colon U(\infty )\looparrowright\varprojlim U(j)$. From (4.2) it follows that the identical mapping $U(\infty )\mapsto U_{\tau}(\infty)$ is continuous.

We equip every group $U(j)$ with the probability Haar measure $\mu_{j}$. As is well known [2], Theorem 1.6, the image measure $\pi _{j-1}^{j}(\mu_{j})$ is equal to $\mu_{j-1}$. In other words, $\mu_{j-1}(\Omega)={[\mu_{j}\circ(\pi _{j-1}^{j})^{-1}](\Omega)}$ for all Borel sets Ω in $U(j-1)$. Following [3], Lemma 4.8 and [2], n. 3.1, with the help of the Kolmogorov consistency theorem, we uniquely define on $\varprojlim U(j)$ the probability Radon measure $\overleftarrow{\mu}$ as the projective limit of the sequence $(\mu_{j})$ under the mappings $\pi^{j}_{j-1}$:

$$\overleftarrow{\mu}:=\varprojlim\mu_{j} \quad \text{so that}\quad \mu_{j}=\pi_{j}(\overleftarrow{\mu}) \quad \text{for all } {j \in\mathbb{N}}, $$

where the image $\pi_{j}(\overleftarrow{\mu})$ is such that $\mu_{j}(\Omega)=(\overleftarrow{\mu}\circ\pi_{j}^{-1})(\Omega)$ for all Borel sets Ω in $U(j)$.

Theorem 4.1

There exists a unique probability Radon measure μ on $U(\infty)$ such that $\overleftarrow{\mu}(\Omega)=(\mu\circ\tau^{-1})(\Omega)$ for all Borel sets $\Omega\subset\varprojlim U(j)$ and

$$ \int f(u\cdot g) \,d\mu(u)= \int f(u) \,d\mu(u),\quad g\in U^{2}(\infty),\ f\in C_{b} \bigl(U(\infty) \bigr), $$

(4.3)

where $C_{b} (U(\infty) )$ is the algebra of bounded continuous complex-valued functions on $U(\infty)$. Moreover, there exists a subsequence of Haar measures $(\mu_{j_{k}})$ that weakly converges to μ in the sense that

$$ \lim_{k\to\infty} \int f \,d\mu_{j_{k}}= \int f \,d\mu\quad \textit{for all } f\in C_{b} \bigl(U_{\tau}(\infty) \bigr), $$

(4.4)

where $C_{b} (U_{\tau}(\infty) )$ is the subalgebra in $C_{b} (U(\infty) )$ of continuous functions on $U_{\tau}(\infty)$.

Proof

Let $\check{U}(j)\subset U(j)$ be the set of matrices for which $\{-1\}$ is not an eigenvalue. As is known [3], n. 3, $\check{U}(j)$ is open in $U(j)$, and ${\mu_{j}(U(j)\setminus\check{U}(j))}=0$. In virtue of [3], Lemma 3.11, the restrictions $\pi_{j-1}^{j}\colon\check{U}(j)\rightarrow\check {U}(j-1)$ are continuous and surjective. Define the projective limit $\varprojlim\check{U}(j)$ under these continuous mappings. Note that $\pi_{j}\colon\varprojlim\check{U}(j)\rightarrow\check{U}(j)$ are also continuous and surjective.

As is well known (see, e.g., [11], Theorem 6), by the Prokhorov criterion there exists a Radon probability measure μ̌ on $\varprojlim\check{U}(j)$ such that $\pi_{j}(\check{\mu})=\mu_{j}$ for all ${j\in\mathbb{N}}$ iff for every $\varepsilon>0$, there exists a compact set $\mathcal{K}$ in $\varprojlim\check{U}(j)$ such that ${(\mu_{j}\circ\pi_{j})(\mathcal{K})}\ge{1-\varepsilon}$ for all ${j\in\mathbb{N}}$. In this case, μ̌ is uniquely determined by the formula

$$\check{\mu}(\mathcal{K})=\inf_{j}(\mu_{j}\circ \pi_{j}) (\mathcal{K}). $$

Apply this criterion. Since $\mu_{k}({U(k)\setminus\check{U}(k)})=0$, ${\sup_{K_{k}\subset\check{U}(k)}\mu_{k}(K_{k})=1}$ as $K_{k}$ runs over all compact sets in $\check{U}(k)$. It follows that for every $\varepsilon>0$, there exists a compact set ${K_{k}\subset \check{U}(k)}$ such that

$$ \mu_{k}(K_{k})\ge1-\varepsilon. $$

(4.5)

In accordance with (4.2), we put $K_{j}:=\pi_{j}^{k}(K_{k})$ for ${j< k}$ and $K_{j}:= \bigl [ {\scriptsize\begin{matrix}{}K_{k} & 0\cr 0 &\mathbb {1} \end{matrix}} \bigr ]$ for ${j\ge k}$. Taking into account the definition of image measures, we have

$$ \mu_{j}(K_{j})=\left \{ \textstyle\begin{array}{l@{\quad}l} \mu_{k}(K_{k})=[\mu_{k}\circ(\pi_{j}^{k})^{-1}](K_{j}):&j< k, \\ \mu_{k}(K_{k}):& j\ge k \end{array}\displaystyle \right .\quad \text{for all } {j\in\mathbb{N}}. $$

(4.6)

Thus, for any compact set $\mathcal{K}=(K_{j})\subset\varprojlim\check {U}(j)$ such that condition (4.5) for $K_{k}=\pi_{k}(\mathcal {K})$ with fixed k is satisfied and $K_{j}=\pi_{j}(\mathcal{K})$ for all other $j\neq k$ are defined in accordance with (4.2), the following condition holds:

$${(\mu_{j}\circ\pi_{j}) (\mathcal{K})}=\mu_{k}(K_{k}) \ge{1-\varepsilon} \quad \text{for all } j\in\mathbb{N}. $$

So, the necessary and sufficient conditions of Prokhorov’s criterion are satisfied. Thus, there exists a unique Radon probability measure μ̌ on $\varprojlim\check{U}(j)$ such that $\pi_{j}(\check{\mu })=\mu_{j}$ for all ${j\in\mathbb{N}}$ and

$$ \check{\mu}(\mathcal{K})=\inf_{j} \mu_{j}(K_{j})=\mu_{k}(K_{k}) $$

(4.7)

because of equalities (4.6). This measure μ̌ can be extended to $\varprojlim{U}(j)\setminus\varprojlim\check{U}(j)$ as zero since $\mu_{k}$ is zero on $U(k)\setminus\check{U}(k)$. Consequently, $\check{\mu}(\mathcal{K}\cdot g)=\inf_{j}\mu_{j}(K_{j}\cdot g)=\mu_{k}(K_{k}\cdot g)$ for all ${g\in U^{2}(k)}$. The invariance property of the Haar measures $\mu_{k}$ yields

$$ \check{\mu}(\mathcal{K}\cdot g)=\mu_{k}(K_{k} \cdot g)=\mu_{k}(K_{k})=\check{\mu}(\mathcal {K}) \quad \text{for all } g\in U^{2}(k). $$

(4.8)

Hence, μ̌ is invariant under the right actions (see also [2], Proposition 3.2). It remains to note that the uniqueness property of the projective limit $\varprojlim\mu_{j}$ implies that $\check{\mu}=\overleftarrow{\mu}$.

The inductive limit $U_{\tau}(\infty)$ is regular because inclusions $U(j)\looparrowright U(j+1)$ are compact. Hence, any compact subset of $U_{\tau}(\infty)$ is contained in a subgroup $U(k)$ with fixed k. In virtue of (4.7) and the equality $\check{\mu}=\overleftarrow{\mu}$, we obtain

$$ \sup_{\mathcal{K}}\overleftarrow{\mu}(\mathcal{K})=1 \quad \Bigl(\text{since } \sup_{K_{k}\subset{U}(k)}\mu_{k}(K_{k})=1 \Bigr), $$

(4.9)

where the supremum is taken over all compact sets $\mathcal{K}=(K_{j})$ in $\varprojlim{U}(j)$ such that $\tau^{-1}(\mathcal{K})$ coincides with $K_{k}=\pi_{k}(\mathcal{K})$. By the known Schwartz theorem (see, e.g., [11], Theorem 5) condition (4.9) is necessary and sufficient for the existence of a unique probability Radon measure μ on $U_{\tau}(\infty)$ such that $\overleftarrow{\mu}(\Omega)=(\mu\circ\tau^{-1})(\Omega)$ for all Borel sets $\Omega\subset\varprojlim U(j)$. In other words, the measure $\overleftarrow{\mu}$ coincides with the image of μ under τ, that is, $\overleftarrow{\mu}=\tau(\mu)$. By (4.8) and the equality $\check{\mu}=\overleftarrow{\mu}$,

$$\mu(K\cdot g)=\mu(K) \quad \text{for all } {K=\tau^{-1}(\Omega)\subset U(\infty)},\ g\in U^{2}(\infty), $$

which directly yields (4.3).

Let $C_{b} (U_{\tau}(\infty) )$ be endowed with the uniform norm. Since $U_{\tau}(\infty)$ is metric, the Prokhorov criterion provides the relative compactness property of the sequence $(\mu_{j})$ in the dual space $C_{b}' (U_{\tau}(\infty) )$ endowed with the weak topology. This gives the equality (4.4) since it holds over the dense subspace $C_{0} (U_{\tau}(\infty) )$ of functions with compact supports. □

Corollary 4.2

The following integral formulas hold:

$$\begin{aligned}& \int f \,d\mu= \int d\mu(u) \int_{U^{2}(j)}f(u\cdot g)\, d(\mu_{j}\otimes\mu _{j}) (g), \end{aligned}$$

(4.10)

$$\begin{aligned}& \int f \,d\mu=\frac{1}{2\pi} \int d\mu(u) \int_{-\pi}^{\pi}f \bigl[\exp (\mathbb {i}\vartheta)u \bigr]\,d\vartheta,\quad f\in C_{b} \bigl(U(\infty) \bigr). \end{aligned}$$

(4.11)

Proof

Applying the invariance property (4.3) and the Fubini theorem, similarly to [12], Lemma 2, we get the integral formulas (4.10)-(4.11). □

Consider a concentration property of a relatively compact sequence of Haar measures $(\mu_{j})$ in the case where the corresponding group $U(j)$ is endowed with the normalized Hilbert-Schmidt metric

$$d_{HS}(u,v)=\sqrt{j^{-1} \mathsf{tr}|u-v|_{HS}}, \quad \text{where } |u-v|_{HS}=\sqrt{(u-v)^{*}(u-v)}. $$

This metric is a standard $\ell^{2}$-distance between matrices $u,v\in U(j)$, viewed as elements of a $j^{2}$-dimensional Hilbert space, which is normalized so as to make the identity ${(j\times j)}$-matrix have norm one. The bi-invariance $d_{HS}(u,v)=d_{HS}(u\cdot g,v\cdot g)$ for all ${g\in U^{2}(j)}$ is a consequence of the trace property $\mathsf{tr}(uv) = \mathsf{tr}(vu)$. We define the ε-neighborhood of ${K_{j}\subset U(j)}$ by

$$(K_{j})_{\varepsilon}:= \bigl\{ u_{j}\in U(j)\colon d_{HS} (u_{j},K_{j} )< \varepsilon \bigr\} . $$

Theorem 4.3

For every $\varepsilon>0$ and closed set $K\subset{U}(\infty)$ such that $\mu_{j}(K_{j})\ge1/2$ where $K_{j}:=K\cap U(j)$ for all ${j\in\mathbb{N}}$, the following equalities hold:

$${\mu(K_{\varepsilon+\eta})=\lim_{j\to\infty}\mu_{j} \bigl[(K_{j})_{\varepsilon}\bigr]=1},\qquad K_{\varepsilon+\eta}:= \bigcup(K_{j})_{\varepsilon+\eta}, \quad \eta>0. $$

Proof

As is well known (see [13]), $(U(j),d_{HS},\mu_{j} )$ forms the Lévy sequence, that is, ${\lim_{j\to\infty}\mu_{j} [(K_{j})_{\varepsilon}]=1}$ for any $\varepsilon>0$ and any closed set $K\subset{U}(\infty)$ such that $\mu_{j}(K_{j})\ge1/2$ for all ${j\in\mathbb{N}}$. The topological space $U_{\tau}(\infty)$ is completely regular. Hence, the closed set $K_{\varepsilon}=\overline{\bigcup(K_{j})}_{\varepsilon}$ can be separated by a continuous function. Taking in (4.4) a function ${f\in C_{b} (U_{\tau}(\infty) )}$ such that ${0\le f\le 1}$ where ${f|_{K_{\varepsilon}}\equiv1}$ and ${f|_{U(\infty)\setminus{K}_{\varepsilon+\eta}}\equiv0}$, we obtain

$$\mu(K_{\varepsilon+\eta})\ge \int f \,d\mu=\lim_{k\to\infty} \int f \,d\mu _{j_{k}}\ge \lim_{k\to\infty} \mu_{j_{k}} \bigl[(K_{j_{k}})_{\varepsilon}\bigr]=1 $$

for a weakly convergent subsequence $(\mu_{j_{k}})$. It follows that $\mu(K_{\varepsilon+\eta})=1$ because $1=\mu (U(\infty) )\ge\mu(K_{\varepsilon+\eta})$. □

5 Hardy spaces $\mathcal{H}_{\mu}^{p}$ ($1\le p\le\infty$)

In what follows, the space of complex functions f on $U(\infty)$ endowed with the norm

$$\|f\|_{L^{p}_{\mu}}= \textstyle\begin{cases} \sqrt[p]{\int|f|^{p} \,d\mu},&1\le p< \infty, \\ \operatorname{ess} \sup_{u\in U(\infty)}|f(u)|,&p=\infty, \end{cases} $$

is denoted by $L^{p}_{\mu}$. It is clear that $L^{\infty}_{\mu}\looparrowright L^{p}_{\mu}$ and $\|f\|_{L^{p}_{\mu}}\le\|f\|_{L^{\infty}_{\mu}}$ for all $f\in{L}^{\infty}_{\mu}$.

We still assume that for any basis element $e_{j}$ in H, there exist $z_{j}\in E'$ such that ${{J}^{*}z_{j}=e_{j}}$. By transitivity the orbits $\{u(e)\colon{u}\in U(\infty) \} \subset S$ do not depend on the choice of ${e\in S\cap\mathscr{R}({J}^{*})}$. Fix an arbitrary ${e\in S\cap\mathscr {R}({J}^{*})}$.

To a pair $(\lambda,\imath)\in\mathbb{Y}\times\mathbb{N}^{\ell(\lambda )}_{\ast}$, we assign the $\ell(\lambda)$-dimensional complex subspace $H_{\imath}= \mathsf{span} \{e_{\imath_{1}},\ldots,e_{\imath_{\ell (\lambda)}} \}$. On the dense subspace $\bigcup H_{\imath}$ in H there is well defined the $C_{b} (U(\infty) )$-valued linear mapping

$$ \phi\colon h\mapsto\phi_{h}(u)= \bigl\langle {u}(e)\mid h \bigr\rangle , \quad u\in U(\infty). $$

(5.1)

It can be shown that ϕ may be isometrically extended onto H as an $L^{2}_{\mu}$-valued mapping, which is still defined on $E'$ as $\phi\circ A$.

Remark 5.1

Note that in the case of a Gaussian measure μ on E there exists a unique extension $\phi\colon h\mapsto \langle\cdot\mid h \rangle$ from $\mathscr{R}({J}^{*})$ to the isometric embedding ${H\looparrowright L^{2}_{\mu}}$, which is called the Paley-Wiener map (see, e.g., [14]).

By the polarization formula for symmetric tensor products, to every $e^{\odot\lambda}_{\imath}\in{e}^{\odot\mathbb{Y}}$ there uniquely corresponds the function

$$ \phi^{\lambda}_{\imath}(u):=\prod _{k=1}^{\ell(\lambda)}\phi^{\lambda _{k}}_{e_{\imath_{k}}}(u) = \bigl\langle \bigl[u(e)\bigr]^{\otimes|\lambda|} \mid e^{\odot\lambda}_{\imath}\bigr\rangle ,\qquad \phi_{e_{\imath_{k}}}(u)= \bigl\langle {u}(e)\mid e_{\imath_{k}} \bigr\rangle , $$

(5.2)

belonging to $C_{b} (U(\infty) )$ in the variable ${u\in U(\infty )}$, where $\phi_{\imath_{k}}:=\phi_{e_{\imath_{k}}}$.

We define the Hardy space $\mathcal{H}_{\mu}^{p}$ ($1\le p\le\infty$) with respect to the Wiener measure μ associated with the covariance operator ${J}\circ{J}^{*}$ (resp., its subspace $\mathcal{H}_{\mu}^{p,n}$ with a fixed ${n\in\mathbb {Z}_{+}}$) to be the $L^{p}_{\mu}$-closed complex linear span of the system

$$\phi^{\mathbb{Y}}= \bigl\{ \phi^{\lambda}_{\imath}\colon( \lambda,\imath)\in \mathbb{Y}\times\mathbb{N}^{\ell(\lambda)}_{\ast}\bigr\} \qquad \bigl(\text{resp.}, \phi^{\mathbb{Y}_{n}}= \bigl\{ \phi^{\lambda}_{\imath}\in\phi^{\mathbb{Y}}\colon (\lambda,\imath)\in\mathbb{Y}_{n}\times \mathbb{N}^{\ell(\lambda)}_{\ast}\bigr\} \bigr), $$

where $\phi^{\emptyset}_{\imath}\equiv1$. The following theorem for a different case is proved in [12], Theorem 6.

Theorem 5.1

The system $\phi^{\mathbb{Y}}$ is orthogonal in $L_{\mu}^{2}$, and

$$ \bigl\Vert \phi^{\lambda}_{\imath}\bigr\Vert _{L^{2}_{\mu}}^{2}=\frac{(\ell(\lambda )-1)! \lambda!}{(\ell(\lambda)-1+|\lambda|)!},\quad (\lambda,\imath)\in \mathbb{Y}\times\mathbb{N}^{\ell(\lambda)}_{\ast}. $$

(5.3)

Proof

The orthogonal property ${\phi^{\lambda'}_{\jmath}\perp\phi^{\lambda}_{\imath}}$ with ${|\lambda'|\neq|\lambda|}$ follows from (4.11) since

$$\int\phi^{\lambda'}_{\jmath}\bar{\phi}^{\lambda}_{\imath}\,d\mu= \frac{1}{2\pi} \int\phi^{\lambda'}_{\jmath}\bar{\phi}^{\lambda}_{\imath}\,d\mu \int_{-\pi}^{\pi}{\exp \bigl[\mathbb {i} \bigl(\bigl\vert \lambda'\bigr\vert -|\lambda| \bigr)\vartheta \bigr]} \,d \vartheta=0 $$

for any $\lambda',\lambda\in\mathbb{Y}\setminus\{\emptyset\}$. Let $|\lambda'|=|\lambda|$ and $\ell(\lambda')>\ell(\lambda)$ for definiteness. Then there exists an index k with an appropriate nonzero integer $\lambda'_{k}$ in the diagram $\lambda'= (\lambda '_{1},\ldots,\lambda'_{k},\ldots,\lambda'_{\ell(\lambda')} )\in\mathbb {Y}\setminus\{\emptyset\}$ such that $\ell(\lambda)< k\le\ell(\lambda')$. In this case, we have ${\phi^{\lambda'}_{\jmath}\perp\phi^{\lambda}_{\imath}}$ because formula (4.11) implies

$$\int\phi^{\lambda'}_{\jmath}\bar{\phi}^{\lambda}_{\imath}\,d\mu= \frac{1}{2\pi} \int\phi^{\lambda'}_{\jmath}\bar{\phi}^{\lambda}_{\imath}\,d\mu \int_{-\pi}^{\pi}\exp \bigl(\mathbb {i} \lambda'_{k}\vartheta \bigr)\,d\vartheta=0. $$

Consider the case $|\lambda'|=|\lambda|$ and $\ell(\lambda')=\ell (\lambda)$. If ${\phi^{\lambda'}_{\jmath}\neq\phi^{\lambda}_{\imath}}$, then $\lambda'\neq \lambda$. There exists an index $0< k\le\ell(\lambda)$ such that $\lambda'_{k}\neq \lambda_{k}$. Similarly as before, ${\phi^{\lambda'}_{\jmath}\perp\phi^{\lambda}_{\imath}}$ because

$$\int\phi^{\lambda'}_{\jmath}\bar{\phi}^{\lambda}_{\imath}\,d\mu= \frac{1}{2\pi} \int\phi^{\lambda'}_{\jmath}\bar{\phi}^{\lambda}_{\imath}\,d\mu \int_{-\pi}^{\pi}\exp \bigl[\mathbb {i}\bigl( \lambda'_{k}-\lambda_{k}\bigr)\vartheta \bigr]\,d \vartheta=0. $$

Let $H_{\imath}$ with $\imath= (\imath_{1},\ldots,\imath_{\ell(\lambda )} )\in\mathbb{N}^{\ell(\lambda)}_{\ast}$ be the ${\ell(\lambda)}$-dimensional subspace in H spanned by $\{e_{\imath_{1}},\ldots,e_{\imath_{\ell(\lambda)}} \}$, and $U(\imath)$ be the unitary subgroup of $U(\infty)$ acting in $H_{\imath}$. Let $g_{\imath}= (\mathbb {1}_{\imath},w_{\imath})\in U^{2}(\imath)$. Using (4.10) with $U(\imath)$ instead of $U(j)$ recursively by ${k=1,\ldots,\ell(\lambda)}$, we get

$$\int\bigl\vert \phi^{\lambda}_{\imath}\bigr\vert ^{2}\,d\mu= \int d\mu(u) \prod_{k=1}^{\ell(\lambda)} \int_{U(\imath)}\bigl\vert \bigl\langle w_{\imath}^{-1} u(e)\mid e_{\imath_{k}} \bigr\rangle \bigr\vert ^{2} \,d\mu_{\imath}(w_{\imath}). $$

Integrals with the Haar measures $\mu_{\imath}$ are independent of $u\in U(\infty)$. Hence,

$$\int_{U(\imath)}\bigl\vert \bigl\langle w_{\imath}^{-1} u(e)\mid e_{\imath _{k}} \bigr\rangle \bigr\vert ^{2}\,d \mu_{\imath}(w_{\imath})= \frac{(\ell(\lambda)-1)! \lambda!}{(\ell(\lambda)-1+|\lambda|)!} $$

by the well-known integral formula for unitary groups [15], n. 1.4.9. It remains to note that the last formulas immediately yield (5.3) because $\int d\mu=1$. □

Theorem 5.1 directly implies that ϕ has an isometric extension onto H and that the following orthogonal expansion holds:

$$ \mathcal{H}_{\mu}^{2}=\mathbb{C}\oplus \mathcal{H}_{\mu}^{2,1}\oplus\mathcal {H}_{\mu}^{2,2} \oplus\cdots. $$

(5.4)

Remark 5.2

In the case of a Gaussian measure μ on E, decomposition (5.4) is called the Wiener-Itô chaos expansion.

6 Inverse integral formulas

The correspondence $e^{\odot\lambda}_{\imath}\mapsto\phi^{\lambda}_{\imath}$ allows us to define a conjugate-linear isomorphism $\varGamma\rightarrow\mathcal{H}^{2}_{\mu}$. As a result, the linear isometry $\varPhi\colon\mathcal{H}^{2}\rightarrow\mathcal{H}^{2}_{\mu}$ and its adjoint ${\varPhi^{*}\colon\mathcal{H}^{2}_{\mu}\rightarrow\mathcal {H}^{2}}$ can be uniquely defined by the change of orthonormal bases

$$\varPhi\colon\mathcal{H}^{2}\ni\zeta^{\lambda}_{\imath}\bigl\Vert {e}^{\odot \lambda}_{\imath}\bigr\Vert ^{-1}_{\varGamma}\mapsto\phi^{\lambda}_{\imath}\bigl\Vert \phi^{\lambda}_{\imath}\bigr\Vert ^{-1}_{L^{2}_{\mu}}\in\mathcal{H}^{2}_{\mu}, \quad \lambda\in\mathbb{Y},\ \imath\in\mathbb{N}^{\ell(\lambda)}_{\ast}. $$

Clearly, $\varPhi^{*}\colon\phi^{\lambda}_{\imath} \Vert \phi^{\lambda}_{\imath} \Vert ^{-1}_{L^{2}_{\mu}} \mapsto\zeta^{\lambda}_{\imath} \Vert {e}^{\odot\lambda}_{\imath} \Vert ^{-1}_{\varGamma}$ since $\langle\varPhi\zeta^{\lambda}_{\imath}\mid f \rangle_{{L^{2}_{\mu}}}= \langle\zeta^{\lambda}_{\imath}\mid\varPhi^{*}f \rangle_{\mathcal {H}^{2}}$ for all ${f\in\mathcal{H}^{2}_{\mu}}$. Hence, for any ${\psi^{\star}\in\mathcal{H}^{2}}$ with the Fourier coefficients $\tilde{\psi}^{\star}{(\lambda,\imath)}$ defined in (3.2), we obtain

$$\varPhi\psi^{\star}=\sum_{(\lambda,\imath)\in\mathbb{Y}\times\mathbb {N}^{\ell(\lambda)}_{\ast}} \tilde{ \psi}^{\star}{(\lambda,\imath)} \frac{\|{e}^{\odot\lambda}_{\imath}\|^{2}_{\varGamma}}{\|\phi^{\lambda}_{\imath}\| ^{2}_{L^{2}_{\mu}}} \phi^{\lambda}_{\imath}, \quad \text{where } \frac{\|{e}^{\odot\lambda}_{\imath}\|^{2}_{\varGamma}}{\|\phi^{\lambda}_{\imath}\| ^{2}_{L^{2}_{\mu}}}= \frac{(\ell(\lambda)-1+|\lambda|)!}{(\ell(\lambda)-1)! |\lambda|!}. $$

In particular, $\phi_{{J}^{*}z}=\sum\bar{\zeta}_{j}(z)\phi_{e_{j}}$ and $\|\phi_{{J}^{*}z}\|_{L^{2}_{\mu}}^{2}=\sum|\zeta_{j}(z)|^{2}=\|z\|_{J^{*}}^{2}$ for any ${z\in E'}$. Hence, if $E'$ is endowed with the norm $\|\cdot\|_{J^{*}}$, then the embedding

$$ \phi\circ A\colon \bigl(E',\|\cdot\|_{J^{*}} \bigr)\ni z\mapsto\phi _{J^{*}z}\in L^{2}_{\mu}$$

(6.1)

is the isometric extension of (5.1), and its image coincides with the subspace $\mathcal{H}_{\mu}^{2,1}$.

We call the isometric embedding (6.1) the Paley-Wiener map corresponding to μ.

Thus, the mapping Φ is an isometric extension of the Paley-Wiener map ${\phi\circ A}$ since $\varPhi|_{E'}={\phi\circ A}$.

Lemma 6.1

The vector-valued functions with respect to the variable ${u\in U(\infty)}$, $\mathcal{Q}\ni z\mapsto(\varPhi\circ\mathcal{C})(u,z)$ and $\mathcal{Q}\ni z\mapsto(\varPhi\circ\mathcal{P})(u,z)$, take values in the space $L_{\mu}^{\infty}$ and may be written as follows:

$$ (\varPhi\circ\mathcal{C}) (u,z)=\frac{1}{1-\phi_{{J}^{*}z}(u)},\qquad (\varPhi \circ\mathcal{P}) (u,z)=\frac{1-\|z\|^{2}_{{J}^{*}}}{|1-\phi_{{J}^{*}z}(u)|^{2}}. $$

(6.2)

Proof

Let ${h=J^{*}z}$. The Fourier decomposition $h=\sum\zeta_{j}(z)e_{j}$ yields $\phi_{h}=\sum\bar{\zeta}_{j}(z)\phi_{e_{j}}$. Applying Φ to the Fourier decomposition of $\mathcal{C}(z',z)$ under the variable ${z'\in\mathcal{Q}}$, we obtain

$$ (\varPhi\circ\mathcal{C}) (u,z)=\sum_{(\lambda,\imath)} \frac{\bar{\zeta}^{\lambda}_{\imath}(z)\phi^{\lambda}_{\imath}(u)}{\|e^{\odot \lambda}_{\imath}\|^{2}_{\varGamma}} =\sum_{n\in\mathbb{Z}_{+}} \biggl(\sum _{j\in\mathbb{N}}\bar{\zeta}_{j}(z)\phi _{e_{j}}(u) \biggr)^{n} =\frac{1}{1-\phi_{h}(u)} $$

because $\|e^{\odot\lambda}_{\imath}\|_{\varGamma}^{-2}=n!/\lambda!$ coincide with multinomial coefficients. It follows that $|(\varPhi\circ\mathcal{C})(u,z)|\le(1-|\phi _{h}|)^{-1}<\infty$ for all ${z\in\mathcal{Q}}$.

Similarly, applying Φ to the Fourier decomposition of $\mathcal {P}(\cdot,z)$, we obtain

$$ (\varPhi\circ\mathcal{P}) (u,z)=\biggl\vert \sum_{(\lambda,\imath)} \frac{\bar{\zeta}^{\lambda}_{\imath}(z)\phi^{\lambda}_{\imath}(u)}{\|e^{\odot \lambda}_{\imath}\|^{2}_{\varGamma}}\biggr\vert ^{2} \biggl(\sum _{(\lambda,\imath)} \frac{|\zeta^{\lambda}_{\imath}(z)|^{2}}{\|e^{\odot\lambda}_{\imath}\| ^{2}_{\varGamma}} \biggr)^{-1} = \frac{1-\|z\|^{2}_{J^{*}}}{|1-\phi_{h}(u)|^{2}}. $$

Again using Theorem 5.1, we get

$$ (\varPhi\circ\mathcal{P}) (u,z)=\frac{1-\|z\|^{2}_{J^{*}}}{|1-\phi_{h}(u)|^{2}} \le \bigl(1-\|z \|^{2}_{J^{*}} \bigr) \biggl(\sum_{n\in\mathbb{Z}_{+}} \|z\| ^{n}_{J^{*}} \biggr)^{2} =\frac{1-\|z\|_{J^{*}}}{(1-\|z\|_{J^{*}})^{2}}=\frac{1+\|z\|_{J^{*}}}{1-\|z\|_{J^{*}}}. $$

As a result, $(\varPhi\circ\mathcal{C})(\cdot,z)$ and $(\varPhi\circ \mathcal{P})(\cdot,z)$ with ${z\in\mathcal{Q}}$ take values in $L_{\mu}^{\infty}$. □

Theorem 6.2

For any ${f\in\mathcal{H}^{2}_{\mu}}$, the function

$$\mathcal{C}[f](z):= \bigl\langle \bigl(\varPhi^{*} \circ f\bigr) (\cdot)\mid \mathcal {C}(\cdot,z) \bigr\rangle _{\mathcal{H}^{2}}= \bigl\langle \bigl(\varPhi^{*} \circ f\bigr) (\cdot)\mid\mathcal{P}(\cdot,z) \bigr\rangle _{\mathcal{H}^{2}},\quad z \in\mathcal{Q}, $$

belongs to the space of analytic functions $\mathcal{H}^{2}$ and has the integral representations

$$ \mathcal{C}[f](z)= \int\frac{f \,d\mu}{1-\bar{\phi}_{{J}^{*}z}}= \int\frac{1-\|z\|^{2}_{{J}^{*}}}{|1-\bar{\phi}_{{J}^{*}z}(u)|^{2}}f(u) \,d\mu(u). $$

(6.3)

The mapping $f\mapsto\mathcal{C}[f]$ generated by $\varPhi^{*}$ produces the isometry ${\mathcal{H}^{2}_{\mu}\simeq\mathcal{H}^{2}}$.

Proof

Consider the orthogonal decomposition with respect to $\phi^{\mathbb{Y}}$ and its $\varPhi^{*}$-image

$$ f=\sum_{(\lambda,\imath)\in\mathbb{Y}\times\mathbb{N}^{\ell(\lambda )}_{\ast}}\tilde{f} {(\lambda,\imath)} \phi^{\lambda}_{\imath},\qquad \varPhi^{*}f=\sum _{(\lambda,\imath)\in\mathbb{Y}\times\mathbb{N}^{\ell (\lambda)}_{\ast}}\tilde{f} {(\lambda,\imath)} \frac{\|\phi^{\lambda}_{\imath}\|^{2}_{L^{2}_{\mu}}}{\|{e}^{\odot\lambda}_{\imath}\| ^{2}_{\varGamma}} \zeta^{\lambda}_{\imath}, $$

respectively, where $\tilde{f}{(\lambda,\imath)}:=\|\phi^{\lambda }_{\imath}\|^{-2}_{L^{2}_{\mu}}\int{f} \bar{\phi}^{\lambda}_{\imath}\,d\mu$ are the Fourier coefficients. Substituting their to $\mathcal{C}[f]$ and taking into account Lemma 6.1 together with orthogonal properties, we get the first equality in (6.3)

$$\begin{aligned} \mathcal{C}[f](z)&=\sum_{(\lambda,\imath)}\frac{\tilde{f}{(\lambda ,\imath)}\zeta^{\lambda}_{\imath}(z)\|\phi^{\lambda}_{\imath}\|^{2}_{L^{2}_{\mu}} \langle\zeta^{\lambda}_{\imath}\mid\zeta^{\lambda}_{\imath}\rangle _{\mathcal{H}^{2}}}{\|{e}^{\odot\lambda}_{\imath}\|^{4}_{\varGamma}} \\ &= \int\sum_{(\lambda,\imath)} \frac{\zeta^{\lambda}_{\imath}(z)\bar{\phi}^{\lambda}_{\imath}}{\|e^{\odot\lambda }_{\imath}\|^{2}_{\varGamma}}{f} \,d\mu= \int(\varPhi\circ\mathcal{C}) (\cdot,z)f \,d\mu= \int\frac{f \,d\mu}{1-\bar{\phi}_{{J}^{*}z}}. \end{aligned}$$

To check the second equality in (6.3), we also apply Lemma 6.1. As a result,

$$\begin{aligned} \mathcal{C}[f](z)&= \bigl\langle \bigl(\varPhi^{*} \circ f\bigr) (\cdot)\mid \mathcal {P}(\cdot,z) \bigr\rangle _{\mathcal{H}^{2}} \\ &= \int(\varPhi\circ\mathcal{P}) (z,\cdot)f \,d\mu= \int\frac{1-\|z\| ^{2}_{{J}^{*}}}{|1-\bar{\phi}_{{J}^{*}z}(u)|^{2}}f(u) \,d\mu(u). \end{aligned}$$

Hence, both integral representations in (6.3) hold. Since $\mathscr{R}(\varPhi^{*})=\mathcal{H}^{2}$, Lemma 3.2 implies that the mapping $\varPhi^{*}\colon\mathcal{H}^{2}_{\mu}\ni f\mapsto\mathcal{C}[f]\in\mathcal {H}^{2}$ is surjective. □

Remark 6.1

The $L_{\mu}^{\infty}$-valued function $\mathcal{Q}\ni z\mapsto(\varPhi\circ \mathcal{P})(\cdot,z)$ is a Poisson-type kernel for the infinite-dimensional ball $\mathcal {Q}$. The second integral formula in (6.3) is a Poisson-type formula over the Hardy space $\mathcal{H}^{2}_{\mu}$.

Remark 6.2

Since $\varPhi^{*}\colon\mathcal{H}^{2}_{\mu}\ni f\mapsto\mathcal{C}[f]\in \mathcal{H}^{2}$ is isometric and surjective, the integral formulas (6.3) are inverse to the transform Φ, which is an isometric extension of the Paley-Wiener map ${\phi\circ A}$.

7 Directional derivatives

Now we calculate the directional derivatives of an analytic function $\psi^{\star}\in\mathcal{H}^{2}$ at any point ${z\in\mathcal{Q}}$:

$$\mathfrak{d}_{a}\psi^{\star}(z):={\biggl.\lim_{t\to0} \frac{\psi^{\star}(z+ta)-\psi ^{\star}(z)}{t}=\frac{d\psi^{\star}(z+ta)}{dt} \biggr|_{t=0}},\quad a\in\mathcal{Q},\ t \in\mathbb{R}. $$

Consider the projector $S_{1}\otimes{S}_{n-1}\colon{H}^{\otimes n}\rightarrow{H}\otimes{H}^{\odot(n- 1)}$ and its restriction $S_{n/1}:=S_{n}|_{H\otimes{H}^{\odot(n- 1)}}$ defined as $\eta\odot\psi_{n-1}={S}_{n/1}(\eta\otimes\psi_{n-1})\in{H}^{\odot n}$ for all $\eta\in{H}$ and $\psi_{n-1}\in{H}^{\odot(n-1)}$. The projector ${S}_{n}$ possesses the decomposition ${S}_{n}={{S}_{n/1}\circ ({S}_{1}\otimes{S}_{n-1})}$. For any $\lambda\in\mathbb{Y}$ such that $|\lambda|=n-1$ and ${\imath\in \mathbb{N}^{\ell(\lambda)}}$,

$$\frac{1}{n}\bigl\Vert e_{m}\otimes{e}^{\odot\lambda}_{\imath}\bigr\Vert ^{2}=\frac {1}{n}\frac{(\lambda)!}{(n-1)!} = \frac{(\lambda)!}{n!}=\bigl\Vert S_{n/1} \bigl(e_{m} \otimes{e}^{\odot\lambda }_{\imath}\bigr)\bigr\Vert ^{2},\quad \text{so } \|S_{n/1}\|=\frac{1}{n}. $$

In fact, it suffices to decompose an element of ${H\otimes{H}^{\odot(n- 1)}}$ with respect to the basis elements $e_{m}\otimes{e}^{\odot\lambda}_{\imath}$.

Define the operator $\delta_{a,n}\colon{H}^{\odot(n- 1)}\rightarrow {H}^{\odot n}$ for a nonzero ${a\in\mathcal{Q}}$ as

$$\begin{aligned} \delta_{a,n}\bigl({J}^{*}z\bigr)^{\otimes(n-1)}&:=nS_{n/1} \bigl[{J}^{*}a\otimes \bigl({J}^{*}z\bigr)^{\otimes(n-1)} \bigr] \\ &={\biggl.\frac{d({J}^{*}z+t{J}^{*}a)^{\otimes n}}{dt} \biggr|_{t=0}} =n{J}^{*}a\odot\bigl({J}^{*}z \bigr)^{\otimes(n-1)}, \end{aligned}$$

where the last equality is a consequence of the well-known tensor binomial formula $(x+ty)^{\otimes n}=\sum_{m=0}^{n}\binom{m}{n}(ty)^{\otimes m}\odot x^{\otimes(n-m)}$ with any ${x,y\in H}$. Summing over $n\ge1$, we define

$$ \delta_{a} \bigl(1-{J}^{*}z \bigr)^{-\otimes1}:= {\biggl.\bigoplus _{n\ge1}\frac{d({J}^{*}z+t{J}^{*}a)^{\otimes n}}{dt} \biggr|_{t=0}} =\bigoplus _{n\ge1}n{J}^{*}a\odot\bigl({J}^{*}z\bigr)^{\otimes(n-1)}. $$

Taking into account that $\|S_{n/1}\|=n^{-1}$, we obtain

$$\begin{aligned} \bigl\Vert \delta_{a} \bigl(1-{J}^{*}z \bigr)^{-\otimes1}\bigr\Vert ^{2}_{\varGamma}& =\sum _{n\ge1} \bigl\| n{J}^{*}a\odot\bigl({J}^{*}z\bigr)^{\otimes(n-1)} \bigr\| ^{2}_{\varGamma} \\ &\le\|a\|_{{J}^{*}}^{2}\sum_{n\ge1} \|z\|^{2(n-1)}_{{J}^{*}}= \|a\|_{{J}^{*}}^{2}\bigl\Vert \bigl(1-{J}^{*}z \bigr)^{-\otimes1}\bigr\Vert ^{2}_{\varGamma}. \end{aligned}$$

(7.1)

Inequality (7.1) and the totality of ${ \{(1-{J}^{*}z)^{-\otimes1}\colon z\in\mathcal{Q} \}}$ in Γ imply that the adjoint operator $\delta_{z}^{*}$ of $\delta_{z}$ on Γ can be defined as $\delta_{z}^{*}\psi=\bigoplus_{n\ge1}\delta_{z,n}^{*}\psi_{n}$. Here $\delta_{z,n}^{*}\colon{H}^{\odot n}\ni\psi_{n}\rightarrow\delta_{z,n}^{*}\psi _{n}\in{H}^{\odot(n- 1)}$ is defined as the adjoint operator $\delta_{z,n}^{*}$ of $\delta_{z,n}$ on ${H}^{\otimes n}$ via the equality

$$\bigl\langle \delta_{z,n}\bigl({J}^{*}z\bigr)^{\otimes(n-1)}\mid \psi_{n} \bigr\rangle = \bigl\langle \bigl({J}^{*}z\bigr)^{\otimes(n-1)}\mid \delta_{z,n}^{*}\psi_{n} \bigr\rangle . $$

In fact, the image of ${J}^{*}$ contains all elements $(e_{m})$; hence, ${ \{({J}^{*}z)^{\otimes(n-1)}\colon z\in\mathcal{Q} \}}$ is total in ${H}^{\odot(n- 1)}$. So, by Riesz’s theorem there exists unique ${\delta_{z,n}^{*}\psi_{n}\in {H}^{\odot(n- 1)}}$, and $\delta_{z,n}^{*}$ is well defined.

As a consequence, from (7.1) we get $\|\delta_{a}^{*}\psi\|_{\varGamma}\le\|a\|_{{J}^{*}}\|\psi\|_{\varGamma}$ for all ${a\in\mathcal{Q}}$ and $\psi\in\varGamma$, which means that ${\delta_{a}^{*}\psi\in\varGamma}$. So we have proved the following statement.

Lemma 7.1

For any function $\psi^{\star}\in\mathcal{H}^{2}$ associated with an element ${\psi\in\varGamma}$, we have that $\mathfrak{d}_{a}\psi^{\star}\in \mathcal{H}^{2}$ and $\mathfrak{d}_{a}\psi^{\star}(z)= \langle (1-{J}^{*}z )^{-\otimes 1}\mid\delta_{a}^{*}\psi \rangle$ for all $a,z\in\mathcal{Q}$.

Theorem 7.2

For any function $f\in\mathcal{H}^{2}_{\mu}$, we have $\mathfrak {d}_{a}\mathcal{C}[f]\in\mathcal{H}^{2}$, and the following formula holds:

$$ \mathfrak{d}_{a}\mathcal{C}[f](z)= \int\frac{f(u)\bar{\phi}_{{J}^{*}a}(u) \,d\mu(u)}{(1-\bar{\phi}_{{J}^{*}z}(u))^{2}},\quad a,z\in\mathcal{Q}. $$

(7.2)

Proof

First, note that $f\phi_{{J}^{*}a}\in\mathcal{H}^{2}_{\mu}$ for all ${a\in \mathcal{Q}}$ because $\phi_{{J}^{*}a}\in\mathcal{H}^{\infty}_{\mu}$. Moreover, $\mathfrak{d}_{a}\mathcal{C}[f]\in\mathcal{H}^{2}$ by Lemma 7.1. Using the first integral formula (6.3), we can write that

$$\begin{aligned} \mathfrak{d}_{a}\mathcal{C}[f](z)&={\biggl.\frac{d\mathcal{C}[f](z+ta)}{dt} \biggr|_{t=0}} \\ &=\lim_{t\to0}\frac{1}{t} \int \biggl(\frac{f(u)}{1-\bar{\phi}_{{J}^{*}(z+ta)}(u)} -\frac{f(u)}{1-\bar{\phi}_{{J}^{*}z}(u)} \biggr)\,d\mu(u) \\ &=\lim_{t\to0}\frac{1}{t} \int \biggl(\frac{f(u)}{1-\langle{J}^{*}(z+ta)\mid u(e)\rangle}- \frac{f(u)}{1-\langle{J}^{*}z\mid u(e)\rangle} \biggr)\,d\mu(u) \\ &=\lim_{t\to0}\frac{1}{t} \int\frac{ t\langle{J}^{*}a\mid u(e)\rangle f(u) \,d\mu(u)}{ (1-\langle{J}^{*}(z+ta)\mid u(e)\rangle)(1-\langle{J}^{*}z\mid u(e)\rangle )} \\ &=\lim_{t\to0} \int\frac{\bar{\phi}_{{J}^{*}a}(u) f(u) \,d\mu(u)}{ (1-\bar{\phi}_{{{J}^{*}(z+ta)}}(u))(1-\bar{\phi}_{{J}^{*}z}(u))}. \end{aligned}$$

Now we need to prove that, as $t\to0$,

$$\begin{aligned}& \int\frac{\bar{\phi}_{{J}^{*}a}(u) f(u) \,d\mu(u)}{(1-\bar{\phi} _{{J}^{*}(z+ta)}(u))(1-\bar{\phi}_{{J}^{*}z}(u))}- \int\frac{f(u)\bar{\phi}_{{J}^{*}a}(u) \,d\mu(u)}{(1-\bar{\phi} _{{J}^{*}z}(u))^{2}} \\& \quad = \int\frac{t\bar{\phi}_{{J}^{*}a}^{2}(u)f(u) \,d\mu(u)}{(1-\bar{\phi}_{{J}^{*}(z+ta)}(u))(1-\bar{\phi}_{{J}^{*}z}(u))^{2}}\to0. \end{aligned}$$

For a fixed ${z\in\mathcal{Q}}$, we put $\alpha:=\min\{|1-\bar{\phi}_{{J}^{*}z}(u)|\colon u\in U(\infty)\}$, so $|1-\bar{\phi}_{{J}^{*}z}(u)|^{2}>\alpha^{2}$,

$$\alpha\le\bigl\vert 1-\bar{\phi}_{{J}^{*}z}(u)\bigr\vert \le\bigl\vert 1-\bar{\phi}_{{J}^{*}(z+ta)}(u)\bigr\vert +\bigl\vert t\bar{ \phi}_{{J}^{*}a}(u)\bigr\vert . $$

This yields $|1-\bar{\phi}_{{J}^{*}(z+ta)}(u)|\ge\alpha-|t\bar{\phi} _{{J}^{*}a}(u)|\ge\alpha/2$ for $|t\bar{\phi}_{{J}^{*}a}(u)|\le\alpha/2$. It follows that

$$\biggl\vert \int\frac{t\bar{\phi}_{{J}^{*}a}^{2}(u) f(u) \,d\mu(u)}{(1-\bar{\phi} _{{{J}^{*}(z+ta)}}(u)) (1-\bar{\phi}_{{J}^{*}z}(u))^{2}}\biggr\vert \le \frac{|t|}{\alpha/2\cdot\alpha^{2}} \int|f| \,d\mu\le\frac{|t|}{\alpha /2\cdot\alpha^{2}}\|f\|_{L_{\mu}^{2}}\to0 $$

as $t\to0$. Hence, the integral formula (7.2) holds. □

8 Radial boundary values

Set $J^{*}z=rv(e)$ with ${z\in\mathcal{Q}}$, ${0\le r<1}$, and ${v\in U(\infty)}$, where ${e\in S\cap\mathscr{R}(J^{*})}$ is a fixed element. Note that the corresponding complex-valued function

$$U(\infty)\ni u\mapsto\phi_{J^{*}z}(u)= \bigl\langle u(e)\mid rv(e) \bigr\rangle $$

satisfies the equalities ${\phi_{J^{*}z}(u)}={\phi_{rv(e)}(u)}={r\phi _{v(e)}(u)}=r\phi_{e}(v^{-1}u)$ where $v^{-1}u=u\cdot g$ is defined as the right action with $g=(\mathbb {1},v)\in U^{2}(\infty)$. In particular, $\phi_{e}(\mathbb {1})=1$.

We define the Poisson kernel as follows:

$$\mathcal{P}_{r}(v,u):=\frac{1-r^{2}}{|1-r\bar{\phi}_{e}(v^{-1}u)|^{2}},\quad {v,u\in U(\infty)},\ {0 \le r< 1}. $$

The Poisson integral is defined for any function $f\in\mathcal {H}^{p}_{\mu}$ ($1\le p\le\infty$) as

$$\mathcal{P}_{r}[f](v):= \int\mathcal{P}_{r}(v,u)f(u) \,d\mu(u), \quad {v\in U( \infty)},\ {0\le r< 1}. $$

It is easy to see that $\mathcal{P}_{r}[ \mathsf{Re}\, f]= \mathsf{Re}\,\mathcal{P}_{r}[f]$ for all $f\in\mathcal{H}^{p}_{\mu}$. The following statement is an extension of Theorem 6.2 to the Hardy space $\mathcal{H}^{p}_{\mu}$ with an arbitrary ${1\le p\le\infty}$.

Theorem 8.1

For every function $f\in\mathcal{H}^{p}_{\mu}$ ($1\le p\le\infty$), the equalities

$$ \mathcal{P}_{r}[f](v)= \int\frac{f \,d\mu}{1-\bar{\phi}_{J^{*}z}} = \int\frac{1-\|z\|^{2}_{J^{*}}}{|1-\phi_{J^{*}z}|^{2}}f \,d\mu,\quad {z=rAv(e)\in\mathcal{Q}}, $$

(8.1)

hold, where the integrals are analytic in the variable ${z\in\mathcal{Q}}$.

Proof

The space $\mathcal{H}_{\mu}^{p}$ is defined as the $L^{p}_{\mu}$-closed linear span of the orthogonal system $\phi^{\mathbb{Y}}$. On the other hand, the kernel $\mathcal{P}_{r}$ is related to the kernel $\varPhi\circ \mathcal{P}$ in (6.2) by the equalities

$$\mathcal{P}_{r}(v,\cdot)=(\varPhi\circ\mathcal{P}) (z,\cdot)= \frac{1-\|z\| ^{2}_{J^{*}}}{|1-\phi_{J^{*}z}(\cdot)|^{2}},\quad {z=rAv(e)\in\mathcal{Q}}, $$

where $\varPhi\circ\mathcal{P}$ is an $L^{\infty}_{\mu}$-valued function in the variable z via Lemma 6.1. Therefore, equalities (8.1) hold for any $f\in\mathcal{H}_{\mu}^{p}$ by orthogonality. The $L_{\mu}^{\infty}$-valued function $\mathcal{Q}\ni z\mapsto(1-\bar{\phi}_{J^{*}z})^{-1}$ is analytic. Hence, the first integral in (8.1) is a complex-valued analytic function in the variable ${z\in\mathcal{Q}}$ as the composition of this $L_{\mu}^{\infty}$-valued function and the bounded linear functional $L_{\mu}^{\infty}\ni g\mapsto\int g f \,d\mu$ with $f\in\mathcal{H}^{p}_{\mu}$ because the embedding $L^{\infty}_{\mu}\looparrowright L^{p}_{\mu}$ ($1\le p\le \infty$) is continuous. □

Lemma 8.2

For any $u,v\in U(\infty)$ and ${0\le r<1}$, the kernel $\mathcal{P}_{r}$ satisfies the conditions

$$\mathcal{P}_{r}(u,v)=\mathcal{P}_{r}(v,u)>0, \quad \int\mathcal{P}_{r}(u,v) \,d\mu(v)=1= \int\mathcal{P}_{r}(u,v) \,d\mu(u). $$

Proof

The first equality is a consequence of the kernel $\mathcal{P}_{r}$ definition. Putting ${f\equiv1}$ in (8.1) and using the first equality, we obtain the other equalities. □

Theorem 8.3

For every $f\in L^{p}_{\mu}$ ($1\le p\le\infty$), we have $\Vert \mathcal{P}_{r}[f]\Vert _{L_{\mu}^{p}}\le\|f\|_{L_{\mu}^{p}}$ for all ${r\in[0,1)}$. If, in addition, ${1\le p<\infty}$, then

$$ \lim_{r\to1}\bigl\Vert \mathcal{P}_{r}[f]-f \bigr\Vert _{L_{\mu}^{p}}=0, \quad f\in \mathcal{H}^{p}_{\mu}. $$

(8.2)

Proof

First, note that the invariant property (4.3) yields

$$\mathcal{P}_{r}[f](v)= \int\mathcal{P}_{r}\bigl(\mathbb {1},v^{-1}u \bigr)f(u) \,d\mu(u) = \int\mathcal{P}_{r}(\mathbb {1},s)f(vs) \,d\mu(s), \quad {f\in L^{\infty}_{\mu}}. $$

So, if $p=\infty$, then $\Vert \mathcal{P}_{r}[f]\Vert _{L_{\mu}^{\infty}}\le\|f\|_{L_{\mu}^{\infty}}\int\mathcal{P}_{r}(\mathbb {1},s) \,d\mu(s) =\|f\|_{L_{\mu}^{\infty}}$ for all ${f\in L^{\infty}_{\mu}}$.

Let $1\le p<\infty$. Using the Jensen inequality and the Fubini theorem, we get

$$ \bigl\Vert \mathcal{P}_{r}[f]\bigr\Vert _{L_{\mu}^{p}}\le \int \biggl( \int\bigl\vert f(vu)\bigr\vert ^{p} \,d\mu(v) \biggr)^{1/p} \mathcal{P}_{r}(\mathbb {1},u) \,d\mu(u)\le\|f \|_{L_{\mu}^{p}} $$

for all $f\in C_{b}(U(\infty))$. Via the denseness of $C_{b}(U(\infty))$, this inequality holds for all ${f\in L^{p}_{\mu}}$.

By Lemma 8.2, $\mathcal{P}_{r}[f](v)-f(v)=\int [f(vu)-f(v) ]\mathcal {P}_{r}(\mathbb {1},u) \,d\mu(u)$. Replacing in the previous reasoning $\mathcal{P}_{r}[f]$ by $\mathcal {P}_{r}[f]-f$, we similarly get

$$\bigl\Vert \mathcal{P}_{r}[f]-f\bigr\Vert _{L_{\mu}^{p}}\le \int \biggl( \int \bigl\vert f(vu)-f(v)\bigr\vert ^{p} \,d\mu(v) \biggr)^{1/p} \mathcal{P}_{r}(\mathbb {1},u) \,d\mu(u) $$

for all ${f\in L^{p}_{\mu}}$. Under the continuity of the shift operator in $L_{\mu}^{p}$ ($1\le p<\infty$), for every $r\in[0,1)$, there exists $\delta>0$ such that $\int|f(vu)-f(v)|^{p} \,d\mu(v)\le(1-r)^{p}$ for all $u\in U(\infty)$ such that $\mathsf{Re}\,\phi_{e}(u)<\delta$. On the other hand, if $r\to1$, then for every $\delta>0$, uniformly on $u,v\in U(\infty)$ such that $\mathsf{Re}\,\phi_{e}(v^{-1}u)\ge\delta$,

$$\mathcal{P}_{r}(v,u)=\frac{1-r^{2}}{1-2r \mathsf{Re}\,\phi _{e}(v^{-1}u)+r^{2}|\phi_{e}(v^{-1}u)|^{2}} \le\frac{1-r^{2}}{1-r^{2}-2r \mathsf{Re}\,\phi_{e}(v^{-1}u)}\to0. $$

It immediately follows that

$$\int_{ \mathsf{Re}\,\phi_{e}(u)\ge\delta}\mathcal{P}_{r}(\mathbb {1},u) \,d\mu(u) \to0 \quad \text{as } {r\to1}. $$

This proves the existence of the required limit relation (8.2) for all $f\in\mathcal{H}^{p}_{\mu}$. □

Theorem 8.4

For all functions $f\in\mathcal{H}^{\infty}_{\mu}$ and ${\eta\in L^{1}_{\mu}}$,

$$ \lim_{t\to1} \int\mathcal{P}_{r}[f]\eta \,d\mu= \int f\eta \,d\mu. $$

(8.3)

Proof

Using the Fubini theorem and Theorem 8.3 in the case $p=1$, we obtain

$$\begin{aligned} \int\mathcal{P}_{r}[f]\eta \,d\mu&= \int \int\mathcal{P}_{r}(v,u)f(u) \,d\mu(u) \eta(v) \,d\mu(v) \\ &= \int \int\mathcal{P}_{r}(v,u)\eta(v) \,d\mu(v) f(u) \,d\mu(u)\to \int\eta f \,d\mu \end{aligned}$$

for any function ${\eta\in L^{1}_{\mu}}$. □

Remark 8.1

The limit relation (8.2) holds for any $f\in L^{p}_{\mu}$ ($1\le p<\infty$). As well, (8.3) holds for any $f\in L^{\infty}_{\mu}$. However, in these cases the approximating functions $\mathcal{P}_{r}[f]$ are not analytic but harmonic in a suitable meaning.