Free Holomorphic Functions on Polydomains

TL;DR

This paper extends the theory of free holomorphic functions on noncommutative polydomains by establishing classical complex analysis results and properties in a multivariable noncommutative setting.

Contribution

It introduces analogues of classical theorems for free holomorphic functions on noncommutative polydomains, advancing the understanding of their structure and properties.

Findings

Established Abel theorem, Hadamard formula, Cauchy inequality, and Liouville theorem in the noncommutative multivariable setting.

Proved maximum principle and Schwarz type lemma for free holomorphic functions.

Showed the algebra of free holomorphic functions forms a complete metric space.

Abstract

In this paper, we continue to develop the theory of free holomorphic functions on noncommutative regular polydomains. We find analogues of several classical results from complex analysis such as Abel theorem, Hadamard formula, Cauchy inequality, and Liouville theorem for entire functions, in our multivariable setting. We also provide a maximum principle and a Schwarz type lemma. These results are used to prove analogues of Weierstrass, Montel, and Vitali theorems for the algebra of free holomorphic functions on the regular polydomain, which turns out to be a complete metric space.

Equations

$$[B(\mathcal{H}{)}^k{]}_1:=\left\{(X_1,\dots ,X_k)\in B(\mathcal{H}{)}^k:\Vert X_1{X}_1^{\ast }+\cdots +X_k{X}_k^{\ast }{\Vert }^{1/2}<1\right\},$$

$${\mathcal{D}}_{q_1}(\mathbb{C})\times \cdots \times {\mathcal{D}}_{q_k}(\mathbb{C}),$$

$${\mathbf{\Delta }}_{\mathbf{q},\mathbf{X}}:=\left(id-{\Phi }_{q_1,X_1}\right)\circ \cdots \circ \left(id-{\Phi }_{q_k,X_k}\right).$$

$${\mathbf{D}}_{\mathbf{q}}^{-}(\mathcal{H}):=\left\{\mathbf{X}\in B(\mathcal{H}{)}^{n_1}{\times }_c\cdots {\times }_{c}B(\mathcal{H}{)}^{n_k}:{\mathbf{\Delta }}_{\mathbf{q},\mathbf{X}}^{\mathbf{p}}(I)\ge 0\text{ for any }\mathbf{p}:=(p_1,\dots ,p_k)\in \{0,1{\}}^k\right\},$$

$${\mathbf{\Delta }}_{\mathbf{q},\mathbf{X}}^{\mathbf{p}}:={\left(id-{\Phi }_{q_1,X_1}\right)}^{p_1}\circ \cdots \circ {\left(id-{\Phi }_{q_k{X}_k}\right)}^{p_k},\mathbf{p}:=(p_1,\dots ,p_k)\in {\mathbb{Z}}_{+}^k,$$

$$F^2(H_{n_i}):=\underset{p\ge 0}{⨁}{H}_{n_i}^{\otimes p},$$

$$W_{i,j}{e}_{\alpha }^i:=\frac{\sqrt{b_{i,\alpha }}}{\sqrt{b_{i,g_j\alpha }}}{e}_{g_j\alpha }^i,\alpha \in {\mathbb{F}}_{n_i}^{+},$$

$$b_{i,g_0^i}:=1\text{ and }{b}_{i,\alpha }:=\sum _{p=1}^{|\alpha |}\sum _{\genfrac{}{}{0ex}{}{{\gamma }_1,\dots ,{\gamma }_p\in {\mathbb{F}}_{n_i}^{+}}{\genfrac{}{}{0ex}{}{{\gamma }_1\cdots {\gamma }_p=\alpha }{|{\gamma }_1|\ge 1,\dots ,|{\gamma }_p|\ge 1}}}{a}_{i,{\gamma }_1}\cdots a_{i,{\gamma }_p}$$

$$W_{i,{\beta }_i}{W}_{i,{\beta }_i}^{\ast }{e}_{{\alpha }_i}^i=\{\begin{array}{ll}\frac{b_{i,{\gamma }_i}}{b_{i,{\alpha }_i}}{e}_{{\alpha }_i}^i& \text{ if }{\alpha }_i={\beta }_i{\gamma }_i,{\gamma }_i\in {\mathbb{F}}_{n_i}^{+}\\ 0& \text{ otherwise. }\end{array}$$

$${\mathbf{W}}_{i,j}:=\underset{\text{i−1 times}}{\underbrace{I\otimes \cdots \otimes I}}\otimes W_{i,j}\otimes \underset{\text{k−i times}}{\underbrace{I\otimes \cdots \otimes I}}.$$

$$(id-{\Phi }_{q_1,{\mathbf{W}}_1})\circ \cdots \circ (id-{\Phi }_{q_k,{\mathbf{W}}_k})(I)={\mathbf{P}}_{\mathbb{C}},$$

$${\mathbf{K}}_{\mathbf{q},\mathbf{X}}:\mathcal{H}\to F^2(H_{n_1})\otimes \cdots \otimes F^2(H_{n_k})\otimes \overline{{\mathbf{\Delta }}_{\mathbf{q},\mathbf{X}}(I)(\mathcal{H})}$$

$${\mathbf{K}}_{\mathbf{q},\mathbf{X}}h:=\sum _{{\beta }_i\in {\mathbb{F}}_{n_i}^{+},i=1,\dots ,k}\sqrt{b_{1,{\beta }_1}}\cdots \sqrt{b_{k,{\beta }_k}}{e}_{{\beta }_1}^1\otimes \cdots \otimes e_{{\beta }_k}^k\otimes {\mathbf{\Delta }}_{\mathbf{q},\mathbf{X}}(I{)}^{1/2}{X}_{1,{\beta }_1}^{\ast }\cdots X_{k,{\beta }_k}^{\ast }h,$$

$${\mathbf{\Delta }}_{\mathbf{q},\mathbf{X}}(I):=(id-{\Phi }_{q_1,X_1})\circ \cdots \circ (id-{\Phi }_{q_k,X_k})(I),$$

$${\mathbf{K}}_{\mathbf{q},\mathbf{X}}^{\ast }{\mathbf{K}}_{\mathbf{q},\mathbf{X}}=\underset{m_k\to \infty }{\lim }\dots \underset{m_1\to \infty }{\lim }(id-{\Phi }_{q_k,X_k}^{m_k})\circ \cdots \circ (id-{\Phi }_{q_1,X_1}^{m_1})(I),$$

$${\mathbf{K}}_{\mathbf{q},\mathbf{X}}^{\ast }{\mathbf{K}}_{\mathbf{q},\mathbf{X}}=I_{\mathcal{H}}.$$

$${\mathbf{K}}_{\mathbf{q},\mathbf{X}}{X}_{i,j}^{\ast }=({\mathbf{W}}_{i,j}^{\ast }\otimes I){\mathbf{K}}_{\mathbf{q},\mathbf{X}}.$$

$${\mathbf{D}}_{\mathbf{q}}(\mathcal{H}):=\left\{\mathbf{X}\in B(\mathcal{H}{)}^{n_1}{\times }_c\cdots {\times }_{c}B(\mathcal{H}{)}^{n_k}:{\mathbf{\Delta }}_{\mathbf{q},\mathbf{X}}(I)>0\right\}.$$

$${\mathcal{B}}_{\mathbf{X}}[g]:={\mathbf{K}}_{\mathbf{q},\mathbf{X}}^{\ast }(g\otimes I_{\mathcal{H}}){\mathbf{K}}_{\mathbf{q},\mathbf{X}},g\in B({\otimes }_{i=1}^k{F}^2(H_{n_i})).$$

$$\mathcal{S}:=\overline{\text{span}}\{p({\mathbf{W}}_{i,j})q({\mathbf{W}}_{i,j}{)}^{\ast }:p({\mathbf{W}}_{i,j}),q({\mathbf{W}}_{i,j})\in \mathcal{P}(\mathbf{W})\},$$

$${\mathcal{B}}_{\mathbf{X}}[g]:=\underset{r\to 1}{\lim }{\mathbf{K}}_{\mathbf{q},r\mathbf{X}}^{\ast }(g\otimes I_{\mathcal{H}}){\mathbf{K}}_{\mathbf{q},r\mathbf{X}},g\in \mathcal{S},$$

$${\mathcal{B}}_{\mathbf{X}}({\mathbf{W}}_{\mathbf{\alpha }}{\mathbf{W}}_{\mathbf{\beta }}^{\ast })={\mathbf{X}}_{\mathbf{\alpha }}{\mathbf{X}}_{\mathbf{\beta }}^{\ast },\mathbf{\alpha },\mathbf{\beta }\in {\mathbb{F}}_{n_1}^{+}\times \cdots \times {\mathbb{F}}_{n_k}^{+},$$

$$\underset{r\to 1}{\lim }{\mathbf{B}}_{r\mathbf{X}}[g]={\mathbf{B}}_{\mathbf{X}}[g],g\in \mathcal{S}.$$

$${\mathbf{D}}_{\mathbf{q}}(\mathcal{H})=\bigcup _{\mathbf{z}\in {\overline{\mathbb{D}}}^{n_1+\cdots +n_k}}\mathbf{z}{\mathbf{D}}_{\mathbf{q}}(\mathcal{H})=\bigcup _{\mathbf{z}\in {\mathbb{D}}^{n_1+\cdots +n_k}}\mathbf{z}{\mathbf{D}}_{\mathbf{q}}(\mathcal{H}{)}^{-}=\bigcup _{\mathbf{z}\in {\mathbb{D}}^{n_1+\cdots +n_k}}\mathbf{z}{\mathbf{D}}_{\mathbf{q}}(\mathcal{H}).$$

$${\mathbf{D}}_{\mathbf{q}}(\mathcal{H})=\bigcup _{0\le r<1}r{\mathbf{D}}_{\mathbf{q}}(\mathcal{H})=\bigcup _{0\le r<1}r{\mathbf{D}}_{\mathbf{q}}(\mathcal{H}{)}^{-}.$$

$${\mathbf{D}}_{\mathbf{q}}(\mathcal{H}{)}^{-}=\bigcup _{\mathbf{z}\in {\overline{\mathbb{D}}}^{n_1+\cdots +n_k}}\mathbf{z}{\mathbf{D}}_{\mathbf{q}}(\mathcal{H}{)}^{-}=\bigcup _{0\le r\le 1}r{\mathbf{D}}_{\mathbf{q}}(\mathcal{H}{)}^{-}.$$

$${\mathbf{\Delta }}_{\mathbf{q},\mathbf{Y}}(I)=({\mathbf{\Delta }}_{\mathbf{q},\mathbf{Y}}(I)-{\mathbf{\Delta }}_{\mathbf{q},\mathbf{X}}(I))+{\mathbf{\Delta }}_{\mathbf{q},\mathbf{X}}(I)\ge (c-d)I>0,$$

$$(id-{\Phi }_{q_1,{\lambda }_1{\mathbf{W}}_1}{)}^{p_1}\circ \cdots \circ (id-{\Phi }_{q_k{\lambda }_k{\mathbf{W}}_k}{)}^{p_k}(I)\ge \prod _{i=1}^k(1-|{\lambda }_i{|}^2{)}^{p_i}I,p_i\in \{0,1\}.$$

$$(id-{\Phi }_{q_1,{\lambda }_1{\mathbf{W}}_1}{)}^{p_1}\circ \cdots \circ (id-{\Phi }_{q_k,{\lambda }_k{\mathbf{W}}_k}{)}^{p_k}(I)=\prod _{i=1}^k(I-{\Phi }_{q_i,{\lambda }_i{\mathbf{W}}_i}(I){)}^{p_i},p_i\in \{0,1\}.$$

$${\mathbf{\Delta }}_{\mathbf{q},\mathbf{X}}^{\mathbf{p}}(I)=(id-{\Phi }_{q_1,X_1}{)}^{p_1}\circ \cdots \circ (id-{\Phi }_{q_k,X_k}{)}^{p_k}(I)\ge \prod _{i=1}^k(1-r^2{)}^{p_i}I,p_i\in \{0,1\}.$$

Full text

Research supported in part by NSF grant DMS 1500922

Introduction

We developed in [6], [7], [9], [10], [11], [12], and [14] a theory of free holomorphic functions on the open unit ball

[TABLE]

where $k\in{\mathbb{N}}:=\{1,2,\ldots\}$ and $B({\mathcal{H}})$ is the algebra of all bounded linear operators on a Hilbert space ${\mathcal{H}}$. Several classical results from complex analysis have analogues in this noncommutative multivariable setting. This theory was extended to noncommutative regular polyballs ${\bf B_{n}}$ in a series of papers, starting with [8], which led to our work on the curvature invariant [18], the Euler characteristic [19], and the group of free holomorphic automorphisms on ${\bf B_{n}}$ [20]. The regular polyball ${\bf B_{n}}$, ${\bf n}=(n_{1},\ldots,n_{k})\in{\mathbb{N}}^{k}$, is a noncommutative analogue of the scalar polyball   $({\mathbb{C}}^{n_{1}})_{1}\times\cdots\times({\mathbb{C}}^{n_{k}})_{1}$ and has a universal model ${\bf S}:=\{{\bf S}_{i,j}\}$ consisting of left creation operators acting on the tensor product $F^{2}(H_{n_{1}})\otimes\cdots\otimes F^{2}(H_{n_{k}})$ of full Fock spaces.

The goal of the present paper is to continue the development of the theory of free holomorphic functions on regular polydomains ${\bf D_{q}}$, which was initiated in [13], [15], [16], and [17]. This theory is related, via noncommutative Berezin transforms, to the study of operator algebras generated by the universal models associated with the regular polydomains, as well as to the theory of functions in several complex variable ([3], [21], [22]). We mention that the regular polydomain ${\bf D_{q}}$, ${\bf q}:=(q_{1},\ldots,q_{k})$, is a noncommutative analogue of the scalar polydomain

[TABLE]

where ${\mathcal{D}}_{q_{i}}({\mathbb{C}})\subset{\mathbb{C}}^{n_{i}}$ is a domain generated by a positive regular polynomial $q_{i}\in{\mathbb{C}}[Z_{1},\ldots,Z_{n_{i}}]$ (see Section 1). We remark that, in general, one can view the free holomorphic functions on noncommutative polydomains as noncommutative functions in the sense of [2]. Our approach is quite different and relies on the universal models associated with the regular polydomains.

After a few preliminaries on Berezin transforms on regular polydomains, we show, in Section 2, that the regular polydomain ${\bf D_{q}}$ is a noncommutative complete Reinhardt domain. We obtain characterizations for free holomorphic functions on regular polydomains and provide analogues of several results from complex analysis such as: Abel theorem, Hadamard formula, Cauchy inequality, and Liouville theorem for entire functions.

In Section 3, we prove a maximum principle for free holomorphic functions on regular polydomains and obtain a Schwarz lemma in this setting. In Section 4, we provide analogues of the classical results of Weierstrass, Montel, and Vitali (see [1]). These results are used to show that the algebra $Hol({\bf D_{q}})$ of all free holomorphic functions with complex coefficients on the polydomain ${\bf D_{q}}$ is a complete metric space with respect to an appropriate metric.

We mention that most of the results of this paper, including Weierstrass, Montel, and Vitali type theorems, remain true for the radial part of the polydomains ${\bf D_{f}^{m}}$, studied in [17], and the proofs are essentially the same. We also remark that the results of this paper play an important role in our project, in preparation, concerning the free biholomorphic classification of polydomains and the associated polydomain algebras.

1. Preliminaries on Berezin transforms on noncommutative polydomains

A polynomial $q\in{\mathbb{C}}\left<Z_{1},\ldots,Z_{n}\right>$ in $n$ noncommuting indeterminates is called positive regular if all its coefficients are positive, the constant term is zero, and the coefficients of the linear terms $Z_{1},\ldots,Z_{n}$ are different from zero. Let ${\mathbb{F}}_{n}^{+}$ be the unital free semigroup on $n$ generators $g_{1},\ldots,g_{n}$ and the identity $g_{0}$. We denote $Z_{\alpha}:=Z_{j_{1}}\cdots Z_{j_{p}}$ if $\alpha=g_{j_{1}}\cdots g_{j_{p}}\in{\mathbb{F}}_{n}^{+}$, and $Z_{g_{0}}=1$.

If $A:=(A_{1},\ldots,A_{n})\in B({\mathcal{H}})^{n}$ and $q=\sum_{\alpha}a_{\alpha}Z_{\alpha}$, $a_{\alpha}\in{\mathbb{C}}$, we define the map $\Phi_{q,A}:B({\mathcal{H}})\to B({\mathcal{H}})$ by setting $\Phi_{q,A}(Y):=\sum_{\alpha}a_{\alpha}A_{\alpha}YA_{\alpha}^{*}$ for $Y\in B({\mathcal{H}})$. Given ${\bf n}:=(n_{1},\ldots,n_{k})$ with $n_{i}\in{\mathbb{N}}:=\{1,2,\ldots\}$, and a $k$-tuple ${\bf q}:=(q_{1},\ldots,q_{k})$ of positive regular polynomials $q_{i}\in{\mathbb{C}}\left<Z_{1},\ldots,Z_{n_{i}}\right>$, we associate with each $k$-tuple ${\bf X}:=(X_{1},\ldots,X_{k})\in B({\mathcal{H}})^{n_{1}}\times\cdots\times B({\mathcal{H}})^{n_{k}}$ the defect mapping ${\bf\Delta_{q,X}}:B({\mathcal{H}})\to B({\mathcal{H}})$ defined by

[TABLE]

We denote by $B({\mathcal{H}})^{n_{1}}\times_{c}\cdots\times_{c}B({\mathcal{H}})^{n_{k}}$ the set of all tuples ${\bf X}:=({X}_{1},\ldots,{X}_{k})\in B({\mathcal{H}})^{n_{1}}\times\cdots\times B({\mathcal{H}})^{n_{k}}$, where ${X}_{i}:=(X_{i,1},\ldots,X_{i,n_{i}})\in B({\mathcal{H}})^{n_{i}}$, $i\in\{1,\ldots,k\}$, with the property that, for any $s,t\in\{1,\ldots,k\}$, $s\neq t$, the entries of ${X}_{s}$ are commuting with the entries of ${X}_{t}$. In this case we say that ${X}_{s}$ and ${X}_{t}$ are commuting tuples of operators. Note that the operators $X_{i,1},\ldots,X_{i,n_{i}}$ are not necessarily commuting. Consider the noncommutative regular polydomain

[TABLE]

where the defect mapping ${\bf\Delta_{X}^{p}}:B({\mathcal{H}})\to B({\mathcal{H}})$ is defined by

[TABLE]

and we use the convention that $(id-\Phi_{q_{i},X_{i}})^{0}=id$, the identity mapping on $B({\mathcal{H}})$. The abstract noncommutative polydomain ${\bf D_{q}^{-}}$ is the disjoint union $\coprod_{{\mathcal{H}}}{\bf D_{q}^{-}}({\mathcal{H}})$, over all Hilbert spaces ${\mathcal{H}}$.

For each $i\in\{1,\ldots,k\}$, let $H_{n_{i}}$ be an $n_{i}$-dimensional complex Hilbert space with orthonormal basis $e_{1}^{i},\dots,e_{n_{i}}^{i}$. We consider the full Fock space of $H_{n_{i}}$ defined by

[TABLE]

where $H_{n_{i}}^{\otimes 0}:={\mathbb{C}}1$ and $H_{n_{i}}^{\otimes p}$ is the (Hilbert) tensor product of $p$ copies of $H_{n_{i}}$. Let ${\mathbb{F}}_{n_{i}}^{+}$ be the unital free semigroup on $n_{i}$ generators $g_{1}^{i},\ldots,g_{n_{i}}^{i}$ and the identity $g_{0}^{i}$. The length of $\alpha\in{\mathbb{F}}_{n_{i}}^{+}$ is defined by $|\alpha|:=0$ if $\alpha=g_{0}^{i}$ and $|\alpha|:=p$ if $\alpha=g_{j_{1}}^{i}\cdots g_{j_{p}}^{i}$, where $j_{1},\ldots,j_{p}\in\{1,\ldots,n_{i}\}$. Set $e_{\alpha}^{i}:=e^{i}_{j_{1}}\otimes\cdots\otimes e^{i}_{j_{p}}$ if $\alpha=g^{i}_{j_{1}}\cdots g^{i}_{j_{p}}\in{\mathbb{F}}_{n_{i}}^{+}$ and $e^{i}_{g^{i}_{0}}:=1\in{\mathbb{C}}$. It is clear that $\{e^{i}_{\alpha}:\alpha\in{\mathbb{F}}_{n_{i}}^{+}\}$ is an orthonormal basis of $F^{2}(H_{n_{i}})$.

For each $j\in\{1,\ldots,n_{i}\}$, we define the weighted left creation operators $W_{i,j}:F^{2}(H_{n_{i}})\to F^{2}(H_{n_{i}})$, associated with the abstract noncommutative polydomain ${\bf D}^{-}_{q_{i}}$ by setting

[TABLE]

where

[TABLE]

for all $\alpha\in{\mathbb{F}}_{n_{i}}^{+}$ with $|\alpha|\geq 1$. Due to the definition of $W_{i,j}$, for each $\beta_{i}\in{\mathbb{F}}_{n_{i}}^{+}$, we have

[TABLE]

Now, we define the operator ${\bf W}_{i,j}$ acting on the tensor Hilbert space $F^{2}(H_{n_{1}})\otimes\cdots\otimes F^{2}(H_{n_{k}})$ by setting

[TABLE]

We recall (see [17]) that , if ${\bf W}_{i}:=({\bf W}_{i,1},\ldots,{\bf W}_{i,n_{i}})$, then

[TABLE]

where ${\bf P}_{\mathbb{C}}$ is the orthogonal projection from $\otimes_{i=1}^{k}F^{2}(H_{n_{i}})$ onto ${\mathbb{C}}1\subset\otimes_{i=1}^{k}F^{2}(H_{n_{i}})$, where ${\mathbb{C}}1$ is identified with ${\mathbb{C}}1\otimes\cdots\otimes{\mathbb{C}}1$. Moreover, ${\bf W}:=({\bf W}_{1},\ldots,{\bf W}_{k})$ is a pure $k$-tuple in the noncommutative polydomain ${\bf D_{q}^{-}}(\otimes_{i=1}^{k}F^{2}(H_{n_{i}}))$ and is called the universal model associated with the abstract noncommutative polydomain ${\bf D_{q}^{-}}$. We recall the definition of the noncommutative Berezin kernel associated with any element ${\bf X}:=({X}_{1},\ldots,{X}_{k})\in{\bf D_{q}^{-}}({\mathcal{H}})$ with $X_{i}:=(X_{i,1},\ldots,X_{i,n_{i}})$ as the operator

[TABLE]

defined by

[TABLE]

for $h\in{\mathcal{H}}$, where the defect operator is defined by

[TABLE]

and the coefficients $b_{1,\beta_{1}},\ldots,b_{k,\beta_{k}}$ are given above. Here, we use the notation $X_{i,\alpha_{i}}:=X_{i,j_{1}}\cdots X_{i,j_{p}}$ if $\alpha_{i}=g_{j_{1}}^{i}\cdots g_{j_{p}}^{i}\in{\mathbb{F}}_{n_{i}}^{+}$ and $X_{i,g_{0}^{i}}:=I$. The noncommutative Berezin kernel associated with a $k$-tuple ${\bf X}:=({X}_{1},\ldots,{X}_{k})$ in the noncommutative polydomain ${\bf D_{q}^{-}}({\mathcal{H}})$ has the following properties.

1. (i)

${\bf K_{q,X}}$ is a contraction and

[TABLE]

where the limits are in the weak operator topology. 2. (ii)

If ${\bf X}$ is pure, i.e., for each $i\in\{1,\ldots,k\}$, $\lim_{m_{i}\to\infty}\Phi_{q_{i},X_{i}}^{m_{i}}(I)=0$ in the weak operator topology, then

[TABLE] 3. (iii)

For any $i\in\{1,\ldots,k\}$ and $j\in\{1,\ldots,n_{i}\}$,

[TABLE]

Now, we introduce a class of noncommutative Berezin transforms associated with regular polydomains. If $A$ is a positive invertible operator, we write $A>0$. Define the open polydomain ${\bf D_{q}}:=\coprod_{{\mathcal{H}}}{\bf D_{q}}({\mathcal{H}})$, where

[TABLE]

The Berezin transform at ${\bf X}\in{\bf D_{q}}({\mathcal{H}})$, is the map $\boldsymbol{{\mathcal{B}}_{\bf X}}:B(\otimes_{i=1}^{k}F^{2}(H_{n_{i}}))\to B({\mathcal{H}})$ defined by

[TABLE]

Let ${\mathcal{P}}({\bf W})$ be the set of all polynomials $p({\bf W}_{i,j})$ in the operators ${\bf W}_{i,j}$, $i\in\{1,\ldots,k\}$, $j\in\{1,\ldots,n_{i}\}$, and the identity. If $g$ is in the operator space

[TABLE]

where the closure is in the operator norm, we define the Berezin transform at ${\bf X}\in{\bf D_{q}^{-}}({\mathcal{H}})$, by

[TABLE]

where the limit is in the operator norm topology. In this case, the Berezin transform at ${\bf X}$ is a unital completely positive linear map such that

[TABLE]

where ${\bf W}_{\boldsymbol{\alpha}}:={\bf W}_{1,\alpha_{1}}\cdots{\bf W}_{k,\alpha_{k}}$ if $\boldsymbol{\alpha}:=(\alpha_{1},\ldots,\alpha_{k})\in{\mathbb{F}}_{n_{1}}^{+}\times\cdots\times{\mathbb{F}}_{n_{k}}^{+}$. The polydomain algebra $\boldsymbol{{\mathcal{A}}}({\bf D_{q}})$ is the norm closed algebra generated by all ${\bf W}_{i,j}$ and the identity, while the noncommutative Hardy algebra $F^{\infty}({\bf D_{q}})$ is the weakly closed algebra generated by ${\bf W}_{i,j}$ and the identity. The restriction of noncommutative Berezin transform ${\boldsymbol{{\mathcal{B}}}_{\bf X}}$ to the polydomain algebra $\boldsymbol{{\mathcal{A}}}({\bf D_{q}})$ is a completely contractive homomorphism. If, in addition, ${\bf X}$ is a pure $k$-tuple, then

[TABLE]

The Berezin transform will play an important role in this paper. More properties concerning noncommutative Berezin transforms and multivariable operator theory on noncommutative balls and polydomains can be found in [8], [9], [12], [14], [16], and [17]. For basic results on completely positive (resp. bounded) maps we refer the reader to [4] and [5].

2. Free holomorphic functions on noncommutative polydomains

In this section, we show that the regular polydomain ${\bf D_{q}}$ is a noncommutative complete Reinhardt domain. We study free holomorphic functions on regular polydomains and provide analogues of several classical results from complex analysis such as: Abel theorem, Hadamard formula, Cauchy inequality, and Liouville theorem for entire functions.

A subset $G$ of $B({\mathcal{H}})^{n_{1}}\times\cdots\times B({\mathcal{H}})^{n_{k}}$ is called complete Reinhardt set if   ${\bf zX}\in G$ for any ${\bf X}\in G$ and ${\bf z}\in\overline{{\mathbb{D}}}^{n_{1}+\cdots+n_{k}}$, where ${\bf zX}:=\{z_{i,j}X_{i,j}\}$ if ${\bf X}:=\{X_{i,j}\}$ and ${\bf z}=\{z_{i,j}\}$ for $i\in\{1,\ldots,k\}$ and $j\in\{1,\ldots,n_{i}\}$.

Proposition 2.1**.**

The following statements hold:

1. (i)

The regular polydomain ${\bf D_{q}}({\mathcal{H}})$ is relatively open in $B({\mathcal{H}})^{n_{1}}\times_{c}\cdots\times_{c}B({\mathcal{H}})^{n_{k}}$, and its closure ${\bf D_{q}}({\mathcal{H}})^{-}$, in the operator norm topology, coincides with ${\bf D_{q}^{-}}({\mathcal{H}})$. 2. (ii)

${\bf D_{q}}({\mathcal{H}})$is a complete Reinhardt domain such that

[TABLE]

and

[TABLE] 3. (iii)

${\bf D_{q}}({\mathcal{H}})^{-}$* is a complete Reinhardt set and*

[TABLE]

Proof.

Fix ${\bf X}=(X_{1},\ldots,X_{k})\in{\bf D_{q}}({\mathcal{H}})$ and let $c>0$ be such that ${\bf\Delta_{q,X}}(I)>cI$. If $d\in(0,c)$, then there is $\epsilon>0$ such that $-dI\leq{\bf\Delta_{q,Y}}(I)-{\bf\Delta_{q,X}}(I)\leq dI$ for any ${\bf Y}=(Y_{1},\ldots,Y_{k})\in B({\mathcal{H}})^{n_{1}}\times_{c}\cdots\times_{c}B({\mathcal{H}})^{n_{k}}$ with $\max_{i\in\{1,\ldots,k\}}\|X_{i}-Y_{i}\|<\epsilon$. Hence,

[TABLE]

which proves that $Y\in{\bf D_{q}}({\mathcal{H}})$. Consequently, ${\bf D_{q}}({\mathcal{H}})$ is relatively open in $B({\mathcal{H}})^{n_{1}}\times_{c}\cdots\times_{c}B({\mathcal{H}})^{n_{k}}$ with respect to the product topology. Now, we prove that ${\bf D_{q}}({\mathcal{H}})^{-}={\bf D_{q}^{-}}({\mathcal{H}})$. First, we show that if $\lambda_{i}\in\overline{{\mathbb{D}}}$, $i\in\{1,\ldots,k\}$, and ${\bf W}=({\bf W}_{1},\ldots,{\bf W}_{k})$ is the universal model for the regular polydomain ${\bf D_{q}^{-}}$, then

[TABLE]

We recall that two operators $A,B\in B({\mathcal{H}})$ are called doubly commuting if $AB=BA$ and $AB^{*}=B^{*}A$. Since the entries of ${\bf W}_{s}=({\bf W}_{s,1},\ldots,{\bf W}_{s,n_{s}})$ are doubly commuting with the entries of ${\bf W}_{t}=({\bf W}_{t,1},\ldots,{\bf W}_{t,n_{t}})$, whenever $s,t\in\{1,\ldots,k\}$, $s\neq t$, we have

[TABLE]

Since $I-\Phi_{q_{i},\lambda_{i}{\bf W}_{i}}(I)\geq(1-|\lambda_{i}|^{2})I$, the inequality (2.1) follows. Taking into account that ${\bf D_{q}}({\mathcal{H}})$ is open in the operator norm topology, it is clear that if ${\bf X}\in{\bf D_{q}}({\mathcal{H}})$, then there is $r\in[0,1)$ such that $\frac{1}{r}{\bf X}\in{\bf D_{q}}({\mathcal{H}})$. Applying the Berezin transform at $\frac{1}{r}{\bf X}$ to the inequality of (2.1), when $\lambda_{i}=r$ for any $i\in\{1,\ldots,k\}$, we deduce that

[TABLE]

Consequently, if ${\bf Y}\in{\bf D_{q}}({\mathcal{H}})^{-}$, a limiting process implies that ${\bf\Delta_{q,Y}^{p}}(I)\geq 0$ for any ${\bf p}=(p_{1},\ldots,p_{k})$ with $p_{i}\in\{0,1\}$. Therefore, ${\bf D_{q}}({\mathcal{H}})^{-}\subseteq{\bf D_{q}^{-}}({\mathcal{H}})$. To prove the reverse inequality, let ${\bf Y}=(Y_{1},\ldots,Y_{k})\in{\bf D_{q}^{-}}({\mathcal{H}})$. For any $r\in[0,1)$, we have

[TABLE]

Hence, $\prod_{i=1}^{k}(I-\Phi_{q_{i},rW_{i}}(I))\geq(1-r^{2})^{k}I$. Applying the Berezin transform at ${\bf Y}$ and using the fact that ${\boldsymbol{{\mathcal{B}}}_{\bf X}}$ is a completely positive linear map, we deduce that ${\bf\Delta}_{q,r{\bf Y}}(I)\geq(1-r^{2})^{k}I$, which shows that $r{\bf Y}\in{\bf D_{q}}({\mathcal{H}})$. Since $r{\bf Y}\to{\bf Y}$, as $r\to 1$, we conclude that ${\bf D_{q}^{-}}({\mathcal{H}})\subseteq{\bf D_{q}}({\mathcal{H}})^{-}$, which completes the proof of item (i).

Now, we prove item (ii). Using the inequality $I-\Phi_{q_{i},{\bf z}_{i}{\bf W}_{i}}(I)\geq I-\Phi_{q_{i},{\bf W}_{i}}(I)\geq 0$ and the fact that $I-\Phi_{q_{i},{\bf W}_{i}}(I)$ commutes with $I-\Phi_{q_{s},{\bf W}_{s}}(I)$, one can deduce that if ${\bf z}=({\bf z}_{1},\ldots,{\bf z}_{k})$, where ${\bf z}_{i}=(z_{i,1},\ldots,z_{i,n_{i}})\in\overline{{\mathbb{D}}}^{n_{i}}$, then

[TABLE]

If ${\bf X}\in{\bf D_{q}}({\mathcal{H}})$, then applying the Berezin transform at ${\bf X}$ to the inequality above, we obtain ${\bf\Delta_{q,zX}^{p}}(I)\geq{\bf\Delta_{q,X}^{p}}(I)>0$ for any ${\bf p}=(p_{1},\ldots,p_{k})$ with $p_{i}\in\{0,1\}$. This implies

[TABLE]

which shows that ${\bf D_{q}}({\mathcal{H}})$ is a complete Reinhardt domain and ${\bf D_{q}}({\mathcal{H}})=\bigcup_{{\bf z}\in\overline{{\mathbb{D}}}^{n_{1}+\cdots+n_{k}}}{\bf z}{\bf D_{q}}({\mathcal{H}})$.

Now, fix ${\bf X}\in{\bf D_{q}}({\mathcal{H}})^{-}$ and ${\bf z}\in{{\mathbb{D}}}^{n_{1}+\cdots+n_{k}}$. Then there is $r\in(0,1)$ such that $\frac{1}{r}{\bf z}\in{\mathbb{D}}^{n_{1}+\cdots+n_{k}}$. Applying the Berezin transform at ${\bf X}$ to the inequality (2.1) when $\lambda_{1}=\cdots=\lambda_{k}=r$, one can see that $r{\bf X}\in{\bf D_{q}}({\mathcal{H}})$. Therefore, ${\bf zX}\in\frac{1}{r}{\bf z}{\bf D_{q}}({\mathcal{H}})\in{\bf D_{q}}({\mathcal{H}})$, which shows that

[TABLE]

Since ${\bf D_{q}}({\mathcal{H}})$ is an open set , for any ${\bf X}\in{\bf D_{q}}({\mathcal{H}})$, there is $r\in(0,1)$ such that ${\bf X}\in r{\bf D_{q}}({\mathcal{H}}))$. Consequently,

[TABLE]

and

[TABLE]

Using relations (2.2) and (2.3), one can see that the first sequence of equalities in item (ii) holds. Due to relation (2.2), for each $r\in[0,1)$, we have $r{\bf D_{q}}({\mathcal{H}})^{-}\subseteq{\bf D_{q}}({\mathcal{H}})$ which together with relation (2.4) show that the second sequence of equalities in item (ii) holds. Item (iii) follows easily from item (ii). The proof is complete. ∎

We remark that if ${\bf r}:=(r_{1},\ldots,r_{k})$, $r_{i}>0$, then we also have ${\bf D_{q}}({\mathcal{H}})=\bigcup_{0\leq r_{i}<1}{\bf r}{\bf D_{q}}({\mathcal{H}})^{-}.$

For each $i\in\{1,\ldots,k\}$, let $Z_{i}:=(Z_{i,1},\ldots,Z_{i,n_{i}})$ be an $n_{i}$-tuple of noncommuting indeterminates and assume that, for any $p,q\in\{1,\ldots,k\}$, $p\neq q$, the entries in $Z_{p}$ are commuting with the entries in $Z_{q}$. We set $Z_{i,\alpha_{i}}:=Z_{i,j_{1}}\cdots Z_{i,j_{p}}$ if $\alpha_{i}\in{\mathbb{F}}_{n_{i}}^{+}$ and $\alpha_{i}=g_{j_{1}}^{i}\cdots g_{j_{p}}^{i}$, and $Z_{i,g_{0}^{i}}:=1$, where $g_{0}^{i}$ is the identity in ${\mathbb{F}}_{n^{i}}^{+}$. If $\boldsymbol{\alpha}:=(\alpha_{1},\ldots,\alpha_{k})\in{\mathbb{F}}_{n_{1}}^{+}\times\cdots\times{\mathbb{F}}_{n_{k}}^{+}$, we denote ${\bf Z}_{\boldsymbol{\alpha}}:={Z}_{1,\alpha_{1}}\cdots{Z}_{k,\alpha_{k}}$. Let ${\mathbb{Z}}$ be the set of all integers and ${\mathbb{Z}}_{+}$ be the set of all nonnegative integers.

If $T_{1},\ldots,T_{n}\in B({\mathcal{H}})$, we use the notation $[T_{1},\ldots,T_{n}]$ to denote either the $n$-tuple $(T_{1},\ldots,T_{n})\in B({\mathcal{H}})^{n}$ or the row operator $[T_{1}\,\cdots\,T_{n}]$ acting from the direct sum ${\mathcal{H}}^{(n)}:={\mathcal{H}}\oplus\cdots\oplus{\mathcal{H}}$ to ${\mathcal{H}}$. We also set $\Lambda_{\bf p}:=\{\boldsymbol{\alpha}:=(\alpha_{1},\ldots,\alpha_{k})\in{\mathbb{F}}_{n_{1}}^{+}\times\cdots\times{\mathbb{F}}_{n_{k}}^{+}:\ |\alpha_{i}|=p_{i}\}$, where ${\bf p}:=(p_{1},\ldots,p_{k})\in{\mathbb{Z}}_{+}^{k}$.

Lemma 2.2**.**

Let ${\bf W}:=({\bf W}_{1},\ldots,{\bf W}_{k})$ be the universal model and $\{b_{i,\alpha_{i}}\}$ be the coefficients associated with the abstract noncommutative polydomain ${\bf D_{q}^{-}}$. If $A_{(\boldsymbol{\alpha})}$, $\boldsymbol{\alpha}\in\Lambda_{\bf p}$, are bounded linear operators on a Hilbert space ${\mathcal{K}}$, then

[TABLE]

and

[TABLE]

Proof.

First note that if $E_{1},\ldots,E_{m}$ are operators on a Hilbert space and have orthogonal ranges, then $\|[E_{1},\ldots,E_{m}]\|=\max_{j\in\{1,\ldots,m\}}\|E_{j}\|$. We know from [13] that for each $i\in\{1,\ldots,k\}$ and any $\alpha_{i}\in{\mathbb{F}}_{n_{i}}^{+}$, $\|W_{i,\alpha_{i}}\|=\frac{1}{\sqrt{b_{i,\alpha_{i}}}}$. Hence, if ${\boldsymbol{\alpha}}=(\alpha_{1},\ldots,\alpha_{k})\in{\mathbb{F}}_{n_{1}}^{+}\times\cdots\times{\mathbb{F}}_{n_{k}}^{+}$, then

[TABLE]

Since the operators ${\bf W}_{\boldsymbol{\alpha}}$, $\boldsymbol{\alpha}=(\alpha_{1},\ldots,\alpha_{k})\in\Lambda_{\bf p}$, have orthogonal ranges, we deduce that

[TABLE]

which proves the first relation of the lemma. Consequently, using the fact that

[TABLE]

we deduce that, for any $h\in{\mathcal{H}}$ with $\|h\|\leq 1$,

[TABLE]

Hence, we deduce that

[TABLE]

Since ${\bf W}_{\boldsymbol{\alpha}}$, $\boldsymbol{\alpha}=(\alpha_{1},\ldots,\alpha_{k})\in\Lambda_{\bf p}$, have orthogonal ranges, we deduce that

[TABLE]

The proof is complete. ∎

The next result is an analogue of Abel theorem from complex analysis in our noncommutative multivariable setting.

Theorem 2.3**.**

Let $\varphi:=\sum\limits_{\boldsymbol{\alpha}\in{\mathbb{F}}_{n_{1}}^{+}\times\cdots\times{\mathbb{F}}_{n_{k}}^{+}}A_{(\boldsymbol{\alpha})}\otimes{\bf Z}_{\boldsymbol{\alpha}}$ be a formal power series with $A_{(\boldsymbol{\alpha})}\in B({\mathcal{K}})$ and let ${\bf r}=(r_{1},\ldots,r_{k})$ be such that $r_{i}>0$. Then the following statements hold.

1. (i)

If the set

[TABLE]

is bounded, then the series

[TABLE]

is convergent in ${\bf r}{\bf D_{q}}({\mathcal{H}})$, the regular polydomain of polyradius ${\bf r}=(r_{1},\ldots,r_{k})$, and uniformly convergent on ${\bf s}{\bf D_{q}}({\mathcal{H}})^{-}$ for any ${\bf s}=(s_{1},\ldots,s_{k})$ with $0\leq s_{i}<r_{i}$. 2. (ii)

If the set ${\mathcal{A}}$ is unbounded, then the series

[TABLE]

are divergent for some ${\bf X}\in{\bf r}{\bf D_{q}}({\mathcal{H}})^{-}$ and some Hilbert space ${\mathcal{H}}$.

Proof.

Let $s_{i}<r_{i}$ for $i\in\{1,\ldots,k\}$, and ${\bf X}\in{\bf r}{\bf D_{q}}({\mathcal{H}})$. Assume that there is $C>0$ such that

[TABLE]

Due to the noncommutative von Neumann inequality [17] (see also [8]), we have

[TABLE]

for any ${\bf X}\in s{\bf D_{q}}({\mathcal{H}})^{-}$. On the other hand, due to Proposition 2.1, we have ${\bf r}{\bf D_{q}}({\mathcal{H}})=\bigcup_{0\leq s_{i}<r_{i}}{\bf s}{\bf D_{q}}({\mathcal{H}})^{-}$. Now, since the series $\sum_{(p_{1},\ldots,p_{k})\in{\mathbb{Z}}_{+}^{k}}\left(\frac{s_{1}}{r_{1}}\right)^{p_{1}}\cdots\left(\frac{s_{k}}{r_{k}}\right)^{p_{k}}$ is convergent, one can easily complete the proof of part (i).

To prove part (ii), assume that the set ${\mathcal{A}}$ is unbounded. We already know that the tuple ${\bf rW}:=(r_{1}{\bf W}_{1},\ldots,r_{k}{\bf W}_{k})$ is in the polydomain ${\bf rD_{q}}(\otimes_{i=1}^{k}F^{2}(H_{n_{i}}))^{-}$. Due to Lemma 2.2, we have

[TABLE]

If we assume that the series

[TABLE]

is convergent in the operator norm, then $\left\{\left\|\sum\limits_{\boldsymbol{\alpha}\in\Lambda_{\bf p}}A_{(\boldsymbol{\alpha})}\otimes r_{1}^{p_{1}}\cdots r_{k}^{p_{k}}{\bf W}_{\boldsymbol{\alpha}}\right\|\right\}_{{\bf p}:=(p_{1},\ldots,p_{k})\in{\mathbb{Z}}_{+}^{k}}$ is a bounded sequence, which contradicts that ${\mathcal{A}}$ is an unbounded set. The proof is complete. ∎

Definition 2.4**.**

A formal power series $\varphi:=\sum\limits_{\boldsymbol{\alpha}\in{\mathbb{F}}_{n_{1}}^{+}\times\cdots\times{\mathbb{F}}_{n_{k}}^{+}}A_{(\boldsymbol{\alpha})}\otimes{\bf Z}_{\boldsymbol{\alpha}}$ is called free holomorphic function (with coefficients in $B({\mathcal{K}})$) on the abstract polydomain $\boldsymbol{\rho}{\bf D_{q}}:=\coprod_{{\mathcal{H}}}\boldsymbol{\rho}{\bf D_{q}}({\mathcal{H}})$, $\boldsymbol{\rho}=(\rho_{1},\ldots,\rho_{k})$, $\rho_{i}>0$, if the series

[TABLE]

is convergent in the operator norm topology for any ${\bf X}=\{X_{i,j}\}\in\boldsymbol{\rho}{\bf D_{q}}({\mathcal{H}})$ and any Hilbert space ${\mathcal{H}}$. We denote by $Hol({\boldsymbol{\rho}{\bf D_{q}}})$ the set of all free holomorphic functions on ${\bf\boldsymbol{\rho}{\bf D_{q}}}$ with scalar coefficients.

Using Theorem 2.3, one can easily deduce the following characterization for free holomorphic functions on regular polydomains.

Corollary 2.5**.**

Let ${\bf W}$ be the universal model associated with the abstract regular polydomain ${\bf D_{q}}$. A formal power series $\varphi=\sum\limits_{\boldsymbol{\alpha}\in{\mathbb{F}}_{n_{1}}^{+}\times\cdots\times{\mathbb{F}}_{n_{k}}^{+}}A_{(\boldsymbol{\alpha})}\otimes{\bf Z}_{\boldsymbol{\alpha}}$ is a free holomorphic function (with coefficients in $B({\mathcal{K}})$) on the abstract polydomain $\boldsymbol{\rho}{\bf D_{q}}$, where $\boldsymbol{\rho}=(\rho_{1},\ldots,\rho_{k})$, $\rho_{i}>0$, if and only if the series

[TABLE]

converges for any $r_{i}\in[0,\rho_{i})$ and $i\in\{1,\ldots,k\}$.

Throughout the paper, we say that the abstract polydomain ${\bf D_{q}}$ or a free holomorphic function $F$ on ${\bf D_{q}}$ has a certain property, if the property holds for any Hilbert space representation of ${\bf D_{q}}$ and $F$, respectively. We remark that the coefficients of a free holomorphic function on a polydomain ${\bf D_{q}}$ are uniquely determined by its representation on an infinite dimensional Hilbert space.

Corollary 2.6**.**

If $\varphi:=\sum\limits_{\boldsymbol{\alpha}\in{\mathbb{F}}_{n_{1}}^{+}\times\cdots\times{\mathbb{F}}_{n_{k}}^{+}}a_{(\boldsymbol{\alpha})}{\bf Z}_{\boldsymbol{\alpha}}$, $a_{(\alpha)}\in{\mathbb{C}}$, is a free holomorphic function on the abstract polydomain $\boldsymbol{\rho}{\bf D_{q}}$, $\rho=(\rho_{1},\ldots,\rho_{k})$, then its representation on ${\mathbb{C}}$, i.e.

[TABLE]

is a holomorphic function on the scalar polydomain $\boldsymbol{\rho}{\bf D_{q}}({\mathbb{C}})$.

In what follows, we obtain Cauchy type inequalities for the coefficients of free holomorphic functions on regular polydomains.

Theorem 2.7**.**

Let $F$ be a free holomorphic function on the polydomain $\boldsymbol{\rho}{\bf D_{q}}$, with representation

[TABLE]

where $A_{(\boldsymbol{\alpha})}\in B({\mathcal{K}})$. Let ${\bf r}:=(r_{1},\ldots,r_{k})$ be such that $0<r_{i}<\rho_{i}$ and define

[TABLE]

where the supremum is taken over all ${{\bf X}\in{\bf r}{\bf D_{q}}({\mathcal{H}})^{-}}$ and any Hilbert space ${\mathcal{H}}$. Then, for each $k$-tuple ${\bf p}:=(p_{1},\ldots,p_{k})\in{\mathbb{Z}}_{+}^{k}$, we have

[TABLE]

Moreover, $M({\bf r})=\|F({\bf rW})\|$, where ${\bf W}$ is the universal model of the regular polydomain ${\bf D_{q}}$.

Proof.

Using the fact that the operators ${\bf W}_{\boldsymbol{\alpha}}$, with $\boldsymbol{\alpha}=(\alpha_{1},\ldots,\alpha_{k})\in{\mathbb{F}}_{n_{1}}^{+}\times\cdots\times{\mathbb{F}}_{n_{k}}^{+}$, $|\alpha_{i}|=p_{i}$, have orthogonal ranges, and Lemma 2.2, we deduce that

[TABLE]

for any $h\in{\mathcal{K}}$. On the other hand, we have

[TABLE]

Hence, using the inequality above , we deduce that

[TABLE]

for any $h\in{\mathcal{K}}$. Now, the inequality in the theorem follows. The fact that $M({\bf r})=\|F({\bf rW})\|$ is due to the noncommutative von Neumann inequality [8]. The proof is complete. ∎

Note that due to the fact that there is $r\in(0,1)$ such that $r{\bf P_{n}}({\mathcal{H}})\subset{\bf D_{q}}({\mathcal{H}})$, we have

[TABLE]

Assume that ${\mathcal{H}}$ is an infinite dimensional separable Hilbert space. We say that $F$ is an entire function in $B({\mathcal{H}})^{n_{1}}\times_{c}\cdots\times_{c}B({\mathcal{H}})^{n_{k}}$ if $F$ is free holomorphic on every regular polydomain $\rho{\bf D_{q}}({\mathcal{H}})$, $\rho>0$.

In what follows, we obtain an analogue of Liouville’s theorem for entire functions on $B({\mathcal{H}})^{n_{1}}\times_{c}\cdots\times_{c}B({\mathcal{H}})^{n_{k}}$.

Theorem 2.8**.**

If $F:B({\mathcal{H}})^{n_{1}}\times_{c}\cdots\times_{c}B({\mathcal{H}})^{n_{k}}\to B({\mathcal{K}})\otimes_{min}B({\mathcal{H}})$ is an entire function with the property that there is a constant $C>0$ and ${\bf m}:=(m_{1},\ldots,m_{k})\in{\mathbb{Z}}_{+}^{k}$ such that

[TABLE]

for any ${\bf X}\in B({\mathcal{H}})^{n_{1}}\times_{c}\cdots\times_{c}B({\mathcal{H}})^{n_{k}}$, then $F$ is a polynomial of degree at most $m_{1}+\cdots+m_{k}$. In particular, a bounded free holomorphic function must be constant.

Proof.

Let $F$ have the representation

[TABLE]

Due to the hypothesis, we have

[TABLE]

for any $r_{i}>0$. Using Theorem 2.7, we obtain

[TABLE]

for any $r_{i}>0$ and $i\in\{1,\ldots,k\}$. Note that, if there is $s\in\{1,\ldots,k\}$ such that $p_{s}>m_{s}$, then taking $r_{s}\to\infty$ we obtain

[TABLE]

which implies $A_{(\boldsymbol{\alpha})}=0$ for any $\boldsymbol{\alpha}=(\alpha_{1},\ldots,\alpha_{k})$ with $\alpha_{i}\in{\mathbb{F}}_{n_{i}}^{+}$ and $|\alpha_{i}|=p_{i}$ and any $p_{i}\in{\mathbb{Z}}^{+}$, $i\neq s$. Hence, we deduce that

[TABLE]

The proof is complete. ∎

Define the set

[TABLE]

Given a formal power series $\psi:=\sum\limits_{\boldsymbol{\alpha}\in{\mathbb{F}}_{n_{1}}^{+}\times\cdots\times{\mathbb{F}}_{n_{k}}^{+}}A_{(\boldsymbol{\alpha})}\otimes{\bf Z}_{\boldsymbol{\alpha}}$ , we define

[TABLE]

We say that ${\bf C}_{\psi}$ is logarithmically convex if $\Omega$ is log-convex, i.e. the set

[TABLE]

is convex.

Proposition 2.9**.**

Let $\psi:=\sum\limits_{\boldsymbol{\alpha}\in{\mathbb{F}}_{n_{1}}^{+}\times\cdots\times{\mathbb{F}}_{n_{k}}^{+}}A_{(\boldsymbol{\alpha})}\otimes{\bf Z}_{\boldsymbol{\alpha}}$ be a formal power series. The following statements hold.

1. (i)

$\psi$* is free holomorphic function on ${\bf C}_{\psi}$ and*

[TABLE]

where the series is convergent in the operator norm. 2. (ii)

${\bf C}_{\psi}$* is a logarithmically convex complete Reinhardt domain.*

Proof.

Item (i) is due to Theorem 2.3 and the uniqueness of the representation for free holomorphic functions on polydomains. To prove part (ii), note that Proposition 2.1 implies that ${\bf C}_{\psi}$ is a complete Reinhardt domain. It remains to show that ${\bf C}_{\psi}$ is logarithmically convex. To this end, let $(r_{1},\ldots,r_{k})$ and $(s_{1},\ldots,s_{k})$ be in $\Omega$. Then there is a constant $C>0$ such that

[TABLE]

for any ${\bf p}=(p_{1},\ldots,p_{k})\in{\mathbb{Z}}_{+}^{k}$, where $\Gamma_{\bf p}:=\sum\limits_{\boldsymbol{\alpha}\in\Lambda_{\bf p}}\frac{1}{{b_{1,\alpha_{1}}\cdots b_{k,\alpha_{k}}}}A_{(\boldsymbol{\alpha})}^{*}A_{(\boldsymbol{\alpha})}$. Due to the spectral theorem for positive operators, for any $t\in[0,1]$, we have

[TABLE]

Hence, $(r_{1}^{t}s_{1}^{1-t},\ldots,r_{k}^{t}k_{1}^{1-t})\in\Omega$, which proves that ${\bf C}_{\psi}$ is logarithmically convex. The proof is complete. ∎

We remark that, due to Theorem 2.3, if $\boldsymbol{\rho}:=(\rho_{1},\ldots,\rho_{k})\notin\Omega$, then $\sum\limits_{(p_{1},\ldots,p_{k})\in{\mathbb{Z}}_{+}^{k}}\sum\limits_{\boldsymbol{\alpha}\in\Lambda_{\bf p}}A_{(\alpha)}\otimes{\bf X}_{\boldsymbol{\alpha}}$ is divergent for some ${\bf X}\in{\boldsymbol{\rho}}{\bf D_{q}}({\mathcal{H}})^{-}$ and some Hilbert space ${\mathcal{H}}$. We call the set ${\bf C}_{\psi}$ the universal domain of convergence of the power series $\psi$. In what follows, we find the largest polydomain $r{\bf D_{q}}$, $r>0$, which is included in the universal domain of convergence ${\bf C}_{\psi}$.

Theorem 2.10**.**

Let $\psi:=\sum\limits_{\boldsymbol{\alpha}\in{\mathbb{F}}_{n_{1}}^{+}\times\cdots\times{\mathbb{F}}_{n_{k}}^{+}}A_{(\boldsymbol{\alpha})}\otimes{\bf Z}_{\boldsymbol{\alpha}}$ be a formal power series and define $\gamma\in[0,\infty]$ by setting

[TABLE]

Then the following statements hold.

1. (i)

The series

[TABLE]

is convergent. Moreover, the convergence is uniform on $r{\bf D_{q}}({\mathcal{H}})^{-}$ if $0\leq r<\gamma$. 2. (ii)

If $\gamma\in[0,\infty)$ and $s>\gamma$, then there is a Hilbert space ${\mathcal{H}}$ and ${\bf Y}\in s{\bf D_{q}}({\mathcal{H}})^{-}$ such that the series

[TABLE]

is divergent in the operator norm topology.

Proof.

First, we consider the case when $\gamma\in(0,\infty)$. Let ${\bf X}\in r{\bf D_{q}}({\mathcal{H}})^{-}$ be such that $0\leq r<\gamma$ and let $\rho\in(r,\gamma)$. Then

[TABLE]

for all but finitely many ${\bf p}:=(p_{1},\ldots,p_{k})\in{\mathbb{Z}}_{+}^{k}$. Consequently, due to the noncommutative von Neumann inequality [8], we have

[TABLE]

for all but finitely many ${\bf p}:=(p_{1},\ldots,p_{k})\in{\mathbb{Z}}_{+}^{k}$. As a consequence, item (i) holds and implies that the series $\sum\limits_{{\bf p}\in{\mathbb{Z}}_{+}^{k}}^{\infty}\left\|\sum\limits_{\boldsymbol{\alpha}\in\Lambda_{\bf p}}A_{(\boldsymbol{\alpha})}\otimes{\bf X}_{\boldsymbol{\alpha}}\right\|$ is uniformly convergent on $r{\bf D_{q}}({\mathcal{H}})^{-}$. The case when $\gamma=\infty$ can be treated in a similar manner. We leave it to the reader. Now, assume that $\gamma\in[0,\infty)$ and $\gamma<\rho<s$. Let ${\bf Y}:=s{\bf W}$, where ${\bf W}$ is the universal model of ${\bf D_{q}^{-}}$. As above if ${\bf Y}\in s{\bf D_{q}}({\mathcal{H}})^{-}$ then

[TABLE]

Since $\frac{1}{\rho}<\frac{1}{\gamma}$, there are infinitely many tuples ${\bf p}:=(p_{1},\ldots,p_{k})\in{\mathbb{Z}}_{+}^{k}$ such that

[TABLE]

Hence, and using relation (2.5), we deduce that $\|\sum\limits_{\boldsymbol{\alpha}\in\Lambda_{\bf p}}A_{(\boldsymbol{\alpha})}\otimes{\bf Y}_{\boldsymbol{\alpha}}\|>\left(\frac{s}{\rho}\right)^{p_{1}+\cdots+p_{k}}$. This shows that the series

[TABLE]

is divergent and also that item (ii) holds. The proof is complete. ∎

The number $\gamma$ satisfying properties (i) and (ii) in Theorem 2.10 is unique and is called the polydomain radius of convergence for the power series $\psi$.

In what follows, we set $\Gamma_{m}:=\{\boldsymbol{\alpha}=(\alpha_{1},\ldots,\alpha_{k})\in{\mathbb{F}}_{n_{1}}^{+}\times\cdots\times{\mathbb{F}}_{n_{k}}^{+}:\ |\alpha_{1}|+\cdots+|\alpha_{k}|=m\}$.

Theorem 2.11**.**

Let $\psi:=\sum\limits_{\boldsymbol{\alpha}\in{\mathbb{F}}_{n_{1}}^{+}\times\cdots\times{\mathbb{F}}_{n_{k}}^{+}}A_{(\boldsymbol{\alpha})}\otimes{\bf Z}_{\boldsymbol{\alpha}}$ be a formal power series and let $\gamma\in[0,\infty]$ be its polydomain radius of convergence. Then the following statements hold.

1. (i)

The series

[TABLE]

is uniformly convergent on $r{\bf D_{q}}({\mathcal{H}})^{-}$ if $0\leq r<\gamma$. 2. (ii)

For any $s>\gamma$, there is ${\bf Y}\in s{\bf D_{q}}({\mathcal{H}})^{-}$ such that the series

[TABLE]

is divergent in the operator norm topology.

Proof.

Since

[TABLE]

Theorem 2.10 implies that item (i) holds. To prove item (ii) is enough to show that, under the condition $\gamma<\rho<s$,

[TABLE]

is divergent in the operator norm topology. Assume that the series above is convergent and apply it to the vector $x\otimes 1$, where $x\in{\mathcal{K}}$. Consequently, $\sum\limits_{m=0}^{\infty}\sum\limits_{\boldsymbol{\alpha}\in\Gamma_{m}}A_{(\boldsymbol{\alpha})}x\otimes s^{m}\frac{1}{\sqrt{b_{1,\alpha_{1}}\cdots b_{k,\alpha_{k}}}}e_{\boldsymbol{\alpha}}$ is in the Hilbert space ${\mathcal{K}}\otimes\bigotimes_{i=1}^{k}F^{2}(H_{n_{i}})$. Since $\{e_{\boldsymbol{\alpha}}\}_{\boldsymbol{\alpha}\in{\mathbb{F}}_{n_{1}}^{+}\times\cdots\times{\mathbb{F}}_{n_{k}}^{+}}$ is an orthonormal basis for $\bigotimes_{i=1}^{k}F^{2}(H_{n_{i}})$, we deduce that the series $\sum_{\boldsymbol{\alpha}\in{\mathbb{F}}_{n_{1}}^{+}\times\cdots\times{\mathbb{F}}_{n_{k}}^{+}}s^{2(|\alpha_{1}|+\cdots+|\alpha_{k}|)}\frac{1}{{b_{1,\alpha_{1}}\cdots b_{k,\alpha_{k}}}}A_{(\boldsymbol{\alpha})}^{*}A_{(\boldsymbol{\alpha})}$ is WOT-convergent. For each $r\in[0,1)$, Lemma 2.2 implies

[TABLE]

Since the latter series is convergent for any $r\in[0,1)$, we deduce that

[TABLE]

which implies

[TABLE]

for any ${\bf X}\in\rho{\bf D_{q}}({\mathcal{H}})^{-}$, where $\rho\in(\gamma,s)$, which contradicts Theorem 2.10. Therefore, item (ii) holds and the proof is complete. ∎

An interesting consequence of the proofs of Theorem 2.10 and Theorem 2.11 is the following result.

Corollary 2.12**.**

The polydomain radius of convergence of a power series $\varphi:=\sum\limits_{\boldsymbol{\alpha}\in{\mathbb{F}}_{n_{1}}^{+}\times\cdots\times{\mathbb{F}}_{n_{k}}^{+}}A_{(\boldsymbol{\alpha})}\otimes{\bf Z}_{\boldsymbol{\alpha}}$ satisfies the relation

[TABLE]

We also have the following characterization for free holomorphic functions on polydomains.

Corollary 2.13**.**

Let ${\bf W}$ be the universal model associated with the abstract regular polydomain ${\bf D_{q}}$. A formal power series $\varphi:=\sum\limits_{\boldsymbol{\alpha}\in{\mathbb{F}}_{n_{1}}^{+}\times\cdots\times{\mathbb{F}}_{n_{k}}^{+}}A_{(\boldsymbol{\alpha})}\otimes{\bf Z}_{\boldsymbol{\alpha}}$ is a free holomorphic function (with coefficients in $B({\mathcal{K}})$) on the abstract polydomain $\boldsymbol{\rho}{\bf D_{q}}$, where $\boldsymbol{\rho}=(\rho_{1},\ldots,\rho_{k})$, $\rho_{i}>0$, if and only if the series

[TABLE]

is convergent in the operator norm topology for any $s_{i}\in[0,\rho_{i})$. Moreover, the set $Hol({\bf\boldsymbol{\rho}D_{q}})$ of all free holomorphic functions with scalar coefficients on ${\bf\boldsymbol{\rho}D_{q}}$ is an algebra.

3. Maximum principle and Schwarz lemma on noncommutative polydomains

In this section we prove a maximum principle and Schwarz type lemma for free holomorphic functions on regular polydomains.

Let $H^{\infty}({\bf D_{q}})$ denote the set of all elements $\varphi$ in $Hol({\bf D_{q}})$ such that

[TABLE]

where the supremum is taken over all ${\bf X}\in{\bf D_{q}}({\mathcal{H}})$ and any Hilbert space ${\mathcal{H}}$. One can show that $H^{\infty}({\bf D_{q}})$ is a Banach algebra under pointwise multiplication and the norm $\|\cdot\|_{\infty}$. For each $p\in{\mathbb{N}}$, we define the norms $\|\cdot\|_{p}:M_{p\times p}\left(H^{\infty}({\bf D_{q}})\right)\to[0,\infty)$ by setting

[TABLE]

where the supremum is taken over all ${\bf X}\in{\bf D_{q}}({\mathcal{H}})$ and any Hilbert space ${\mathcal{H}}$. It is easy to see that the norms $\|\cdot\|_{p}$, $p\in{\mathbb{N}}$, determine an operator space structure on $H^{\infty}({\bf D_{q}})$, in the sense of Ruan ([4], [5]).

In [17], we identified the noncommutative algebra ${\bf F}^{\infty}({\bf D_{q}})$ with the Hardy subalgebra $H^{\infty}({\bf D_{q}})$ of bounded free holomorphic functions on ${\bf D_{q}}$ with scalar coefficients. More precisely, we proved that the map $\Phi:H^{\infty}({\bf D_{q}})\to{\bf F}^{\infty}({\bf D_{q}})$ defined by $\Phi\left(\varphi\right):=\text{\rm SOT-}\lim_{r\to 1}\varphi(r{\bf W}),$ is a completely isometric isomorphism of operator algebras, where $\varphi(r{\bf W}):=\sum_{q=0}^{\infty}\sum_{{\boldsymbol{\alpha}\in{\mathbb{F}}_{n_{1}}^{+}\times\cdots\times{\mathbb{F}}_{n_{k}}^{+}}\atop{|\alpha_{1}|+\cdots+|\alpha_{k}|=q}}r^{q}a_{(\boldsymbol{\alpha})}{\bf W}_{\boldsymbol{\alpha}}$ and the convergence of the series is in the operator norm topology. Moreover, if $\varphi$ is a free holomorphic function on the abstract polydomain ${\bf D_{q}}$, then the following statements are equivalent:

1. (i)

$\varphi\in H^{\infty}({\bf D_{q}})$; 2. (ii)

$\sup\limits_{0\leq r<1}\|\varphi(r{\bf W})\|<\infty$; 3. (iii)

there exists $\psi\in{\bf F}^{\infty}({\bf D_{q}})$ such that $\varphi({\bf X})={\boldsymbol{{\mathcal{B}}}}_{\bf X}[\psi]$ for ${\bf X}\in{\bf D_{q}}({\mathcal{H}})$, where ${\boldsymbol{{\mathcal{B}}}}_{\bf X}$ is the noncommutative Berezin transform associated with the abstract polydomain ${\bf D_{q}}$.

Moreover, $\psi$ is uniquely determined by $\varphi$, namely, $\psi=\text{\rm SOT-}\lim_{r\to 1}\varphi(r{\bf W})$ and

[TABLE]

We denote by $A({\bf D_{q}})$ the set of all elements $g$ in $Hol({\bf D_{q}})$ such that the mapping

[TABLE]

has a continuous extension to $[{\bf D_{q}}({\mathcal{H}})]^{-}$ for any Hilbert space ${\mathcal{H}}$. One can show that $A({\bf D_{q}})$ is a Banach algebra under pointwise multiplication and the norm $\|\cdot\|_{\infty}$, and it has an operator space structure under the norms $\|\cdot\|_{p}$, $p\in{\mathbb{N}}$. Moreover, we can identify the polydomain algebra $\boldsymbol{{\mathcal{A}}}({\bf D_{q}})$ with the subalgebra $A({\bf D_{q}})$. We proved in [17] that the map $\Phi:A({\bf D_{q}})\to\boldsymbol{{\mathcal{A}}}({\bf D_{q}})$ defined by $\Phi\left(g\right):=\lim_{r\to 1}g(r{\bf W})$, in the norm topology, is a completely isometric isomorphism of operator algebras. Moreover, if $g$ is a free holomorphic function on the abstract polydomain ${\bf D_{q}}$, then the following statements are equivalent:

1. (i)

$g\in A({\bf D_{q}})$; 2. (ii)

$g(r{\bf W}):=\sum_{m=0}^{\infty}\sum_{{\boldsymbol{\alpha}\in{\mathbb{F}}_{n_{1}}^{+}\times\cdots\times{\mathbb{F}}_{n_{k}}^{+}}\atop{|\alpha_{1}|+\cdots+|\alpha_{k}|=m}}r^{q}a_{(\boldsymbol{\alpha})}{\bf W}_{\boldsymbol{\alpha}}$ is convergent in the norm topology as $r\to 1$; 3. (iii)

there exists $\varphi\in\boldsymbol{{\mathcal{A}}}({\bf D_{q}})$ such that $g({\bf X})={\boldsymbol{{\mathcal{B}}}}_{\bf X}[\varphi]$ for ${\bf X}\in{\bf D_{q}}({\mathcal{H}})$, where ${\boldsymbol{{\mathcal{B}}}}_{\bf X}$ is the noncommutative Berezin transform associated with the abstract polydomain ${\bf D_{q}}$.

Moreover, $\varphi$ is uniquely determined by $g$, namely, $\varphi=\lim_{r\to 1}g(r{\bf W})$ and

[TABLE]

Proposition 3.1**.**

Let $G=\sum\limits_{\boldsymbol{\alpha}\in{\mathbb{F}}_{n_{1}}^{+}\times\cdots\times{\mathbb{F}}_{n_{k}}^{+}}c_{(\boldsymbol{\alpha})}{\bf Z}_{\boldsymbol{\alpha}}$ be a free holomorphic function on the polydomain ${\bf D_{q}}$.

1. (i)

If $0<r_{1}<r_{2}<1$, then $r_{1}{\bf D_{q}^{-}}\subset r_{2}{\bf D_{q}}\subset{\bf D_{q}}$ and

[TABLE] 2. (ii)

If $0<r<1$, then the map  $G:r{\bf D_{q}}({\mathcal{H}})^{-}\to B({\mathcal{H}})$ defined by

[TABLE]

is continuous and  $\|G({\bf X})\|\leq\|G(r{\bf W})\|$ for any ${\bf X}\in r{\bf D_{q}}({\mathcal{H}})^{-}$. Moreover, the series defining $G$ converges uniformly on $r{\bf D_{q}}({\mathcal{H}})^{-}$ in the operator norm topology.

Proof.

If $0<r_{1}<r_{2}<1$, then the inclusions $r_{1}{\bf D_{q}^{-}}\subset r_{2}{\bf D_{q}}\subset{\bf D_{q}}$ are due to Proposition 2.1. Since $\varphi({\bf W}):=\sum_{m=0}^{\infty}\sum_{{\boldsymbol{\alpha}\in{\mathbb{F}}_{n_{1}}^{+}\times\cdots\times{\mathbb{F}}_{n_{k}}^{+}}\atop{|\alpha_{1}|+\cdots+|\alpha_{k}|=m}}c_{(\boldsymbol{\alpha})}r_{2}^{|\boldsymbol{\alpha}|}{\bf W}_{\boldsymbol{\alpha}}$ is in $\boldsymbol{{\mathcal{A}}}({\bf D_{q}})$, the noncommutative von Neumann inequality implies

[TABLE]

Taking $r:=\frac{r_{1}}{r_{2}}$, we obtain $\|G(r_{1}{\bf W})\|\leq\|G(r_{2}{\bf W})\|$.

To prove part (ii), note that $G(r{\bf W})\in\boldsymbol{{\mathcal{A}}}({\bf D_{q}}))$ and $\frac{1}{r}{\bf X}\in{\bf D_{q}}({\mathcal{H}})^{-}$. Using again the noncommutative von Neumann inequality, we obtain

[TABLE]

and

[TABLE]

for any ${\bf X}\in r{\bf D_{q}}({\mathcal{H}})^{-}$. Now, one can easily complete the proof. ∎

In what follows, we prove a maximum principle for free holomorphic functions on polydomains.

Theorem 3.2**.**

Let $F$ be a free holomorphic function on ${\bf D_{q}}$ and let $r\in[0,1)$. If ${\mathcal{H}}$ is an infinite dimensional Hilbert space, then

[TABLE]

If, in addition, $F$ has a continuous extension $\widetilde{F}$ to ${\bf D_{q}^{-}}$ in the operator norm, then

[TABLE]

Proof.

Due to the noncommutative von Neumann inequality, we have

[TABLE]

Since ${\mathcal{H}}$ is an infinite dimensional Hilbert space, there is a subspace ${\mathcal{M}}\subset{\mathcal{H}}$ and a unitary operator $U:\otimes_{i=1}^{k}F^{2}(H_{n_{i}})\to{\mathcal{M}}$. Consider the operators $A_{i,j}:=\left(\begin{matrix}U{\bf W}_{i,j}U^{*}&0\\ 0&0\end{matrix}\right)$, for any $i\in\{1,\ldots,k\}$ and $j\in\{1,\ldots,n_{i}\}$, with respect to the decomposition ${\mathcal{H}}={\mathcal{M}}\oplus{\mathcal{M}}^{\perp}$. Set ${\bf A}:=(A_{1},\ldots,A_{k})$ with $A_{i}:=(A_{i,1},\ldots,A_{i,n_{i}})$ and note that ${\bf\Delta_{q,A}}(I)=\left(\begin{matrix}UP_{\mathbb{C}}U^{*}&0\\ 0&I\end{matrix}\right)$. Since ${\bf\Delta_{q,A}}(I)\geq 0$ but ${\bf\Delta_{q,A}}(I)$ is not invertible, Proposition 2.1 shows that ${\bf A}\in\partial{\bf D_{q}}({\mathcal{H}})\subset{\bf D_{q}}({\mathcal{H}})^{-}$. Consequently, we have $r{\bf A}\in\partial(r{\bf D_{q}}({\mathcal{H}}))\subset r{\bf D_{q}}({\mathcal{H}})^{-}$ and

[TABLE]

Hence, we deduce that $\|F(r{\bf A})\|=\|F(r{\bf W})\|$. Using now the inequality (3.1), we complete the proof of the first part of the theorem.

To prove the second part, assume that $F$ has a continuous extension $\widetilde{F}$ to ${\bf D_{q}}({\mathcal{K}})^{-}$ in the operator norm, for any Hilbert space ${\mathcal{K}}$. Then $\widetilde{F}({\bf A})=\lim_{r\to 1}F(r{\bf A})$ exists in the operator norm and is equal to

[TABLE]

Hence, $\|\widetilde{F}({\bf A})\|=\lim_{r\to 1}\|F(r{\bf W})\|=\|F\|_{\infty}$. Since ${\bf A}\in\partial{\bf D_{q}}({\mathcal{H}})\subset{\bf D_{q}}({\mathcal{H}})^{-}$, the proof is complete. ∎

For the rest of this section, we assume that ${\mathcal{H}}$ is a separable infinite dimensional Hilbert space. Since ${\bf D_{q}}({\mathcal{H}})$ is a complete Reinhardt domain and

[TABLE]

we can define the Minkovski functional associated with the regular polydomain ${\bf D_{q}}({\mathcal{H}})$ to be the function $m_{\bf B_{n}}:B({\mathcal{H}})^{n_{1}}\times_{c}\cdots\times_{c}B({\mathcal{H}})^{n_{k}}\to[0,\infty)$ given by

[TABLE]

The polyball ${\bf P_{n}}$ is defined by setting ${\bf P_{n}}({\mathcal{H}}):=[B({\mathcal{H}})^{n_{1}}]_{1}\times_{c}\cdots\times_{c}[B({\mathcal{H}})^{n_{k}}]_{1}$.

Proposition 3.3**.**

The Minkovski functional associated with the regular polydomain ${\bf D_{q}}({\mathcal{H}})$ has the following properties:

1. (i)

$m_{\bf D_{q}}(\lambda{\bf X})=|\lambda|m_{\bf D_{q}}({\bf X})$* for $\lambda\in{\mathbb{C}}$;* 2. (ii)

$m_{\bf D_{q}}$* is upper semicontinuous;* 3. (iii)

${\bf D_{q}}({\mathcal{H}})=\{{\bf X}\in B({\mathcal{H}})^{n_{1}}\times_{c}\cdots\times_{c}B({\mathcal{H}})^{n_{k}}:\ m_{\bf D_{q}}({\bf X})<1\}$; 4. (iv)

${\bf D_{q}}({\mathcal{H}})^{-}=\{{\bf X}\in B({\mathcal{H}})^{n_{1}}\times_{c}\cdots\times_{c}B({\mathcal{H}})^{n_{k}}:\ m_{\bf D_{q}}({\bf X})\leq 1\}$; 5. (v)

There is a polyball $r{\bf P_{n}}({\mathcal{H}})\subset{\bf D_{q}}({\mathcal{H}})$ for some $r\in(0,1)$, where $m_{\bf D_{q}}$ is continuous.

Proof.

Assume that ${\bf X}\in B({\mathcal{H}})^{n_{1}}\times_{c}\cdots\times_{c}B({\mathcal{H}})^{n_{k}}$ and $\lambda\in{\mathbb{C}}$ are such that ${\bf X}\neq 0$ and $\lambda\neq 0$. It is clear that $m_{\bf D_{q}}(\lambda{\bf X})=t>0$ if and only if $\lambda{\bf X}\in c{\bf D_{q}}({\mathcal{H}})$ for any $c>t$, and $\lambda{\bf X}\notin d{\bf D_{q}}({\mathcal{H}})$ if $0<d<t$. Since ${\bf D_{q}}({\mathcal{H}})=e^{i\theta}{\bf D_{q}}({\mathcal{H}})$ for any $\theta\in{\mathbb{R}}$, we deduce that the latter conditions are equivalent to ${\bf X}\in\frac{c}{|\lambda|}{\bf D_{q}}({\mathcal{H}})$ for any $c>t$ and ${\bf X}\notin\frac{d}{|\lambda|}{\bf D_{q}}({\mathcal{H}})$ if $0<d<t$. Consequently, we obtain that $m_{\bf D_{q}}({\bf X})=\frac{t}{|\lambda|}$, which shows that item (i) holds. Item (ii) follows easily from item (i).

Due to Proposition 2.1, we have ${\bf D_{q}}({\mathcal{H}})=\bigcup_{0<r<1}r{\bf D_{q}}({\mathcal{H}})$. Consequently, one can easily deduce item (iii). As we saw in the proof of the same proposition, for any $r\in(0,1)$, we have ${\bf D_{q}}({\mathcal{H}})^{-}\subseteq\frac{1}{r}{\bf D_{q}}({\mathcal{H}})$. Hence, $m_{\bf D_{q}}({\bf X})\leq 1$ for any ${\bf X}\in{\bf D_{q}}({\mathcal{H}})^{-}$. Assume that ${\bf X}\in B({\mathcal{H}})^{n_{1}}\times_{c}\cdots\times_{c}B({\mathcal{H}})^{n_{k}}$ and $m_{\bf D_{q}}({\bf X})=1$. Then there is a sequence $\{t_{m}\}$ with $t_{m}>1$ and $t_{m}\to 1$ such that ${\bf X}\in t_{m}{\bf D_{q}}({\mathcal{H}})$ for any $m\in{\mathbb{N}}$. Taking $t_{m}\to 1$, we deduce that ${\bf X}\in{\bf D_{q}}({\mathcal{H}})^{-}$. Consequently, using item (iii), one can deduce item (iv). To prove (v), note that the fact that $r{\bf P_{n}}({\mathcal{H}})\subset{\bf D_{q}}({\mathcal{H}})$ for some $r\in(0,1)$ is obvoius. The continuity of $m_{\bf D_{q}}$ on $r{\bf P_{n}}({\mathcal{H}})$ is due to the convexity of the latter polyball. The proof is complete. ∎

Now, we present an analogue of Schwarz lemma from complex analysis in the context of free holomorphic functions on polydomains.

Theorem 3.4**.**

Let $F:{\bf D_{q}}({\mathcal{H}})\to B({\mathcal{H}})^{p}$ be a bounded free holomorphic function with $\|F\|_{\infty}\leq 1$. If $F(0)=0$, then

[TABLE]

where $m_{\bf B_{n}}$ is the Minkovski functional associated with the regular polydomain ${\bf D_{q}}({\mathcal{H}})$. In particular, if $p=1$, the free holomorphic function

[TABLE]

has the property that $\|\psi({\bf X})\|\leq m_{\bf D_{q}}({\bf X})<1$ for any ${\bf X}\in{\bf D_{q}}({\mathcal{H}})$, where $f({\bf z}):=F({\bf z})$ for any ${\bf z}=\{z_{i,j}\}\in{\bf D_{q}}({\mathbb{C}})$, is the scalar representation of $F$.

Proof.

Fix ${\bf X}\in{\bf D_{q}}({\mathcal{H}})$. Due to Proposition 3.3, we have $m_{\bf D_{q}}({\bf X})<1$. Let $t\in(0,1)$ be such that $m_{\bf D_{q}}({\bf X})<t<1$. Hence, using again Proposition 3.3 we deduce that $\frac{1}{t}{\bf X}\in{\bf D_{q}}({\mathcal{H}})$ which, due to Proposition 2.1, implies $\frac{\lambda}{t}{\bf X}\in{\bf D_{q}}({\mathcal{H}})$ for any $\lambda\in{\mathbb{D}}:=\{z\in{\mathbb{C}}:\ |z|<1\}$. For each $x,y\in{\mathcal{H}}^{(p)}:={\mathcal{H}}\oplus\cdots\oplus{\mathcal{H}}$ with $\|x\|\leq 1$ and $\|y\|\leq 1$, define the function $\varphi_{x,y}:{\mathbb{D}}\to{\mathbb{C}}$ by setting

[TABLE]

Since $F$ is a free holomorphic function on ${\bf D_{q}}({\mathcal{H}})$ and $\|F\|_{\infty}\leq 1$, we deduce that $\varphi_{x,y}$ is a holomorphic function on the unit disc ${\mathbb{D}}$ and $|\varphi_{x,y}(\lambda)|\leq 1$. Since $\varphi_{x,y}(0)=0$, the classical Schwarz lemma implies $|\varphi_{x,y}(\lambda)|\leq|\lambda|$ for any $\lambda\in{\mathbb{D}}$. Setting $\lambda=m_{\bf D_{q}}({\bf X})$, we deduce that

[TABLE]

for any $t\in(0,1)$ with $m_{\bf D_{q}}({\bf X})<t<1$. Using the fact that $F$ is continuous on ${\bf D_{q}}({\mathcal{H}})$ and taking $t\to m_{\bf D_{q}}({\bf X})$, we obtain $|\left<F({\bf X})x,y\right>|\leq m_{\bf D_{q}}({\bf X})$ for any $x,y\in{\mathcal{H}}^{(p)}$ with $\|x\|\leq 1$ and $\|y\|\leq 1$. Hence,

[TABLE]

To prove the last part of the theorem, assume that $p=1$. Due to the classical Schwarz lemma, we have $|\varphi_{x,y}^{\prime}(0)|\leq 1$. On the other hand, $\varphi_{x,y}^{\prime}(0)=\left<\frac{1}{t}\psi({\bf X})x,y\right>$, which implies $\|\psi({\bf X})\|\leq t<1$. Now, taking $t\to m_{\bf D_{q}}({\bf X})$, we deduce that $\|\psi({\bf X})\|\leq m_{\bf D_{q}}({\bf X})<1$. The proof is complete. ∎

4. Weierstrass, Montel, Vitali theorems for the algebra $Hol(\bf{D_{q}})$

In this section, we obtain Weierstrass, Montel, and Vitali type theorems for the algebra $Hol({\bf D_{q}})$ of free holomorphic functions on the noncommutative domain ${\bf D_{q}}$. This enables us to introduce a metric on $Hol({\bf D_{q}})$ with respect to which it becomes a complete metric space.

The first result is an analogue of Weierstrass theorem (see [1]) for free holomorphic functions on noncommutative polydomains.

Theorem 4.1**.**

Let $\{G_{m}\}_{m=1}^{\infty}\subset Hol({\bf D_{q}})$ be a sequence of free holomorphic functions such that, for each $r\in[0,1)$, the sequence $\{G_{m}(r{\bf W}\}_{m=1}^{\infty}$ is convergent in the operator norm topology. Then there is a free holomorphic function $F\in Hol({\bf D_{q}})$ such that $G_{m}(r{\bf W})$ converges to $F(r{\bf W})$, as $m\to\infty$, for any $r\in[0,1)$.

Proof.

Let $G_{m}$ have the representation $G_{m}:=\sum\limits_{{\bf p}=(p_{1},\ldots,p_{k})\in{\mathbb{Z}}_{+}^{k}}\sum\limits_{\boldsymbol{\alpha}\in\Lambda_{\bf p}}c_{(\boldsymbol{\alpha})}^{(m)}{\bf Z}_{\boldsymbol{\alpha}}$ and fix $r\in(0,1)$. Since $G_{m}$ is a free holomorphic function on ${\bf D_{q}}$, Corollary 2.5 shows that

[TABLE]

is in the noncommutative disc algebra $\boldsymbol{{\mathcal{A}}}({\bf D_{q}})$. Due to the hypothesis, the sequence $\{G_{m}(r{\bf W})\}_{m=1}^{\infty}$ is convergent in the operator norm of $B(F^{2}(H_{n_{1}})\otimes\cdots\otimes F^{2}(H_{n_{k}}))$. Consequently, there exists and operator $G({\bf W})\in\boldsymbol{{\mathcal{A}}}({\bf D_{q}})$ such that

[TABLE]

Let $\sum\limits_{{\bf p}=(p_{1},\ldots,p_{k})\in{\mathbb{Z}}_{+}^{k}}\sum\limits_{\boldsymbol{\alpha}\in\Lambda_{\bf p}}d_{(\boldsymbol{\alpha})}(r){\bf W}_{\boldsymbol{\alpha}}$ be the Fourier representation of $G({\bf W})$, with $d_{(\boldsymbol{\alpha})}(r)\in{\mathbb{C}}$. Since the operators ${\bf W}_{\boldsymbol{\alpha}}$, $\boldsymbol{\alpha}\in\Lambda_{\bf p}$, have orthogonal ranges and

[TABLE]

we deduce that

[TABLE]

If $\lambda_{(\boldsymbol{\beta})}\in{\mathbb{C}}$ for $\boldsymbol{\beta}=(\beta_{1},\ldots,\beta_{k})\in{\mathbb{F}}_{n_{1}}^{+}\times\cdots\times{\mathbb{F}}_{n_{k}}^{+}$ with $|\beta_{i}|=p_{i}$, $i\in\{1,\ldots,k\}$, we have

[TABLE]

Consequently, due to relation (4.2) and Lemma 2.2, we have

[TABLE]

for any $\lambda_{(\boldsymbol{\beta})}\in{\mathbb{C}}$ with $\boldsymbol{\beta}=(\beta_{1},\ldots,\beta_{k})\in{\mathbb{F}}_{n_{1}}^{+}\times\cdots\times{\mathbb{F}}_{n_{k}}^{+}$ and $|\beta_{i}|=p_{i}$. Consequently,

[TABLE]

for any ${\bf p}:=(p_{1},\ldots,p_{k})\in{\mathbb{Z}}_{+}^{k}$. Now, since $\|G_{m}(r{\bf W})-G({\bf W})\|\to 0$, as $m\to\infty$, the inequality above implies $r^{p_{1}+\cdots+p_{k}}c_{(\boldsymbol{\beta})}^{(m)}\to d_{(\boldsymbol{\beta})}(r)$, as $m\to\infty$, for any $\boldsymbol{\beta}=(\beta_{1},\ldots,\beta_{k})\in{\mathbb{F}}_{n_{1}}^{+}\times\cdots\times{\mathbb{F}}_{n_{k}}^{+}$ with $|\beta_{i}|=p_{i}$ and any ${\bf p}\in{\mathbb{Z}}_{+}^{k}$. Therefore, $c_{(\boldsymbol{\beta})}:=\lim\limits_{m\to\infty}c_{\boldsymbol{\beta})}^{(m)}$ exists and $d_{(\boldsymbol{\beta})}(r)=r^{p_{1}+\cdots+p_{k}}c_{(\boldsymbol{\beta})}$. Define the formal power series $F:=\sum\limits_{\boldsymbol{\alpha}\in{\mathbb{F}}_{n_{1}}^{+}\times\cdots\times{\mathbb{F}}_{n_{k}}^{+}}c_{(\boldsymbol{\alpha})}{\bf Z}_{\boldsymbol{\alpha}}$ and let us prove that $F$ is a free holomorphic function on the noncommutative polydomain ${\bf D_{q}}$. Using the inequality above, the fact that $\left\|\sum\limits_{\boldsymbol{\beta}\in\Lambda_{\bf p}}b_{1,\beta_{1}}\cdots b_{k,\beta_{k}}{\bf W}_{\boldsymbol{\beta}}{\bf W}_{\boldsymbol{\beta}}^{*}\right\|=1$, and Lemma 2.2, we deduce that

[TABLE]

for any $r\in[0,1)$ and any ${\bf p}:=(p_{1},\ldots,p_{k})\in{\mathbb{Z}}_{+}^{k}$. This shows that

[TABLE]

uniformly with respect to ${\bf p}\in{\mathbb{Z}}_{+}^{k}$. Since the operators ${\bf W}_{\boldsymbol{\alpha}}$, $\boldsymbol{\alpha}\in\Lambda_{\bf p}$, have orthogonal ranges, the later convergence is equivalent to

[TABLE]

uniformly with respect to ${\bf p}\in{\mathbb{Z}}_{+}^{k}$. Assume now that $\gamma>1$ and

[TABLE]

Then, there are infinitely many ${\bf p}=(p_{1},\ldots,p_{k})\in{\mathbb{Z}}_{+}^{k}$ such that

[TABLE]

Let $\lambda$ be such that $1<\lambda<\gamma$ and let $\epsilon>0$ be with the property that $\epsilon<\gamma-\lambda$. Note that $\epsilon<\gamma^{p_{1}+\cdots+p_{k}}-\lambda^{p_{1}+\cdots+p_{k}}$ for any ${\bf p}=(p_{1},\ldots,p_{k})\in{\mathbb{Z}}_{+}^{k}$. The convergence in relation (4.3) implies that there exists $K_{\epsilon}\in{\mathbb{N}}$ such that

[TABLE]

for any $m>K_{\epsilon}$ and any ${\bf p}=(p_{1},\ldots,p_{k})\in{\mathbb{Z}}_{+}^{k}$. Consequently, if we fix $m>K_{\epsilon}$ and using (4.4), we deduce that

[TABLE]

for infinitely many ${\bf p}=(p_{1},\ldots,p_{k})\in{\mathbb{Z}}_{+}^{k}$. Therefore,

[TABLE]

Now, using Theorem 2.10 and Lemma 2.2, we conclude that $G_{m}$ is not a free holomorphic function on ${\bf D_{q}}$, which is a contradiction. Therefore, we must have

[TABLE]

Using again Theorem 2.10 and Lemma 2.2, we deduce that $F$ is a free holomorphic function on ${\bf D_{q}}$. Consequently,

[TABLE]

is convergent in the operator norm topology and $G({\bf W})=F(r{\bf W})$. Due to relation (4.1), for each $r\in[0,1)$, we have

[TABLE]

The proof is complete. ∎

We say that ${\mathcal{G}}\subset Hol({\bf D_{q}})$ is a *normal set * if each sequence $\{G_{m}\}_{m=1}^{\infty}$ in ${\mathcal{G}}$ has a subsequence $\{G_{m_{k}}\}_{k=1}^{\infty}$ which converges to an element $G\in Hol({\bf D_{q}})$, i.e., for any $r\in[0,1)$,

[TABLE]

The set ${\mathcal{G}}$ is called locally bounded if, for any $r\in[0,1)$, there exists $M>0$ such that $\|F(r{\bf W})\|\leq M$ for all $F\in{\mathcal{G}}$.

An important consequence of Theorem 4.1 is the following noncommutative version of Montel theorem (see [1]).

Theorem 4.2**.**

Let ${\mathcal{G}}\subset Hol({\bf D_{q}})$ be a family of free holomorphic functions. Then the following statements are equivalent:

1. (i)

${\mathcal{G}}$* is locally bounded.* 2. (ii)

${\mathcal{G}}$* is a normal set.*

Proof.

First, we prove the implication (i)$\implies$(ii). Assume that ${\mathcal{G}}$ is locally bounded., i.e., for each $r\in[0,1)$, there is $M_{r}>0$ such that

[TABLE]

Each function $G\in{\mathcal{G}}$ has a representation

[TABLE]

where $c_{(\boldsymbol{\alpha})}^{G}\in{\mathbb{C}}$. Let $\{G_{m}\}_{m=1}^{\infty}$ be a sequence of elements in ${\mathcal{G}}$. Since $|c_{(\boldsymbol{g})}^{G}|=\|G(0)\|\leq M_{0}$ for any $G\in{\mathcal{G}}$, the classical Bolzano-Weierstrass theorem for bounded sequences of complex numbers shows that there is a subsequence $\{G_{\gamma_{k}}\}_{k=1}^{\infty}$ of $\{G_{m}\}_{m=1}^{\infty}$ such that $\{c_{(\boldsymbol{g})}^{G_{\gamma_{k}}}\}$ is convergent in ${\mathbb{C}}$. On the other hand, due to Theorem 2.7, for each ${\bf p}=(p_{1},\ldots,p_{k})\in{\mathbb{Z}}_{+}^{k}$ and $r\in(0,1)$, we have

[TABLE]

for any $G\in{\mathcal{G}}$. Using this inequality, the classical Bolzano-Weierstrass theorem, and the diagonal process, an inductive argument shows that there is a subsequence $\{G_{m_{s}}\}_{s=1}^{\infty}$ of $\{G_{m}\}_{m=1}^{\infty}$ such that, for each $\boldsymbol{\alpha}=(\alpha_{1},\ldots,\alpha_{k})\in{\mathbb{F}}_{n_{1}}^{+}\times\cdots\times{\mathbb{F}}_{n_{k}}^{+}$, the sequence $\{c_{(\boldsymbol{\alpha})}^{G_{m_{s}}}\}_{s=1}^{\infty}$ is convergent in ${\mathbb{C}}$, as $m_{s}\to\infty$.

Fix $r\in(0,1)$ and let $t>1$. We prove that the sequence $\{G_{m_{s}}(\frac{r}{t}{\bf W})\}_{s=1}^{\infty}$ is convergent in the operator norm topology of $B(F^{2}(H_{n_{1}})\otimes\cdots\otimes F^{2}(H_{n_{k}}))$. For each $N,s,\ell\in{\mathbb{N}}$, set

[TABLE]

and note that relation (4.5) implies

[TABLE]

Now, it is clear that we can choose $N$ large enough so that

[TABLE]

Since, for each $\boldsymbol{\alpha}=(\alpha_{1},\ldots,\alpha_{k})\in{\mathbb{F}}_{n_{1}}^{+}\times\cdots\times{\mathbb{F}}_{n_{k}}^{+}$, the sequence $\{c_{(\boldsymbol{\alpha})}^{G_{m_{s}}}\}_{s=1}^{\infty}$ is convergent in ${\mathbb{C}}$, as $m_{s}\to\infty$, we can choose $k_{0}\in{\mathbb{N}}$ such that $\Lambda_{N}(s,\ell)<\frac{\epsilon}{2}$ for any $s,\ell\geq k_{0}$. Putting together these results we conclude that $\left\|G_{m_{s}}\left(\frac{r}{t}{\bf W}\right)-G_{m_{\ell}}\left(\frac{r}{t}{\bf W}\right)\right\|\leq\epsilon$ for any $s,\ell\geq k_{0}$, which proves that the sequence $\{G_{m_{s}}(\frac{r}{t}{\bf W})\}_{s=1}^{\infty}$ is convergent in the operator norm topology. Since $\{\frac{r}{t}:\ r\in[0,1),t>1\}=[0,1)$, we deduce that, for each $t\in[0,1)$, $\{G_{m_{s}}({t}{\bf W})\}_{s=1}^{\infty}$ is convergent in the operator norm topology, as $m_{s}\to\infty$. Applying Theorem 4.1, we conclude that ${\mathcal{G}}$ is a normal set and the implication (i)$\implies$(ii) is true.

We prove the converse by contradiction. Assume that ${\mathcal{G}}$ is a normal set and that there is $r_{0}\in(0,1)$ such that $\sup_{G\in{\mathcal{G}}}\|G(r_{0}{\bf W})\|=\infty$. Then, there is a sequence $\{G_{m}\}_{m=1}^{\infty}\subset{\mathcal{G}}$ such that $\|G_{m}(r_{0}{\bf W})\|\to\infty$, as $m\to\infty$. Since ${\mathcal{G}}$ is a normal set, there is a subsequence $\{G_{m_{k}}\}_{k=1}^{\infty}$ and $G\in Hol({\bf D_{q}})$ such that $\|G_{m_{k}}(r{\bf W})-G(r{\bf W})\|\to 0$, as $k\to\infty$, for any $r\in[0,1)$. This contradicts the fact that $\|G_{m_{k}}(r_{0}{\bf W})\|\to\infty$, as $m_{k}\to\infty$. The proof is complete. ∎

The next result is an analogue of Vitali’s theorem in our setting.

Theorem 4.3**.**

Let $\{G_{m}\}_{m=1}^{\infty}$ be a sequence of free holomorphic function (with scalar coefficients) on the polydomain ${\bf D_{q}}$ such that , for each $r\in[0,1)$,

[TABLE]

If there is $\gamma\in(0,1)$ such that $G_{m}(\gamma{\bf W})$ converges in the operator norm, as $m\to\infty$, then there is $G\in Hol({\bf D_{q}})$ such that

[TABLE]

for any $r\in[0,1)$.

Proof.

Assume that there is $r_{0}\in[0,1)$ such that $\{G_{m}(r_{0}{\bf W})\}_{m=1}^{\infty}$ is not convergent in the operator norm. Then there is $\delta>0$ and subsequences $\{G_{m_{k}}\}_{k=1}^{\infty}$ and $\{G_{s_{k}}\}_{k=1}^{\infty}$ of $\{G_{m}\}_{m=1}^{\infty}$ such that

[TABLE]

for any $k\in{\mathbb{N}}$. Due to Theorem 4.2, there is $F\in Hol({\bf D_{q}})$ and a subsequence $\{k_{p}\}_{p=1}^{\infty}$ of $\{k\}_{k=1}^{\infty}$ such that

[TABLE]

for any $r\in[0,1)$. Note that relation (4.6) implies

[TABLE]

for any $p\in{\mathbb{N}}$. Applying again Theorem 4.2, there is $H\in Hol({\bf D_{q}})$ and a subsequence $\{p_{q}\}_{q=1}^{\infty}$ of $\{p\}_{p=1}^{\infty}$ such that

[TABLE]

for any $r\in[0,1)$. Due to relation (4.8), we have

[TABLE]

for any $q\in{\mathbb{N}}$. Taking $q\to\infty$ in the latter inequality and using relations (4.7) and (4.9), we obtain

[TABLE]

On the other hand, since the sequence $\{G_{m}(\gamma{\bf W})\}_{m=1}^{\infty}$ converges in the operator norm, as $m\to\infty$, relations (4.7) and (4.9) imply $F(\gamma{\bf W})=H(\gamma{\bf W})$. Since $\gamma\in(0,1)$ and $F,H\in Hol({\bf D_{q}})$, and due to the uniqueness of the representation of free holomorphic functions on polydomains, we deduce that $F=H$, which contradicts (4.10). Therefore, for each $r\in[0,1)$, $\{G_{m}(r{\bf W})\}_{m=1}^{\infty}$ is convergent in the operator norm, as $m\to\infty$. Applying Theorem 4.1, we complete the proof. ∎

If $F,G\in Hol({\bf D_{q}})$ and $0<r<1$, we define

[TABLE]

Due to the maximum principle of Theorem 3.2, if ${\mathcal{H}}$ is an infinite dimensional Hilbert space, then

[TABLE]

Let $0<r_{m}<1$ be such that $\{r_{m}\}_{m=1}^{\infty}$ is an increasing sequence convergent to $1$. For any $F,G\in Hol({\bf D_{q}})$, we define

[TABLE]

Using standards arguments, one can show that $\rho$ is a metric on $Hol({\bf D_{q}})$.

Theorem 4.4**.**

$\left(Hol({\bf D_{q}}),\rho\right)$* is a complete metric space.*

Proof.

First, note that if $\epsilon>0$, then there exists $\delta>0$ and $m\in{\mathbb{N}}$ such that, for any $F,G\in Hol({\bf D_{q}})$,  $d_{r_{m}}(F,G)<\delta\implies\rho(F,G)<\epsilon$. Conversely, if $\delta>0$ and $m\in{\mathbb{N}}$ are fixed, then there is $\epsilon>0$ such that, for any $F,G\in Hol({\bf D_{q}})$,  $\rho(F,G)<\epsilon\implies d_{r_{m}}(F,G)<\delta.$

Now, let $\{g_{k}\}_{k=1}^{\infty}\subset Hol({\bf D_{q}})$ be a Cauchy sequence in the metric $\rho$. An immediate consequence of the observation above is that $\{g_{k}(r_{m}{\bf W})\}_{k=1}^{\infty}$ is a Cauchy sequence in $B(F^{2}(H_{n_{1}})\otimes\cdots\otimes F^{2}(H_{n_{k}}))$, for any $m\in{\mathbb{N}}$. Consequently, for each $m\in{\mathbb{N}}$, the sequence $\{g_{k}(r_{m}{\bf W})\}_{k=1}^{\infty}$ is convergent in the operator norm. According to Theorem 4.1, there is a free holomorphic function $g\in Hol({\bf D_{q}})$ such that $g_{k}(r{\bf W})$ converges to $g(r{\bf W})$ for any $r\in[0,1)$. Using again the observation made at the beginning of this proof, we deduce that $\rho(g_{k},g)\to 0$, as $k\to\infty$, which completes the proof. ∎

We remark that, Theorem 4.2 and Theorem 4.4 imply the following compactness criterion for subsets of $Hol({\bf D_{q}})$.

Corollary 4.5**.**

A subset  ${\mathcal{G}}$ of $(Hol({\bf D_{q}}),\rho)$ is compact if and only if it is closed and locally bounded.