Dilation theory and analytic model theory for doubly commuting sequences of $C_{.0}$-contractions

TL;DR

This paper develops dilation and analytic model theories for doubly commuting sequences of $C_{0}$-contractions, extending classical results to infinite-variable operator settings and generalizing key theorems like Beurling-Lax.

Contribution

It introduces dilation and model theories for doubly commuting sequences of $C_{0}$-contractions, expanding multivariable operator theory to infinitely many variables.

Findings

Established dilation theory for doubly commuting $C_{0}$-contraction sequences.

Developed an analytic model theory for these sequences.

Generalized Beurling-Lax theorem and Jordan blocks to infinite variables.

Abstract

Sz.-Nagy and Foias proved that each $C_{\cdot0}$-contraction has a dilation to a Hardy shift and thus established an elegant analytic functional model for contractions of class $C_{\cdot0}$. This has motivated lots of further works on model theory and generalizations to commuting tuples of $C_{\cdot0}$-contractions. In this paper, we focus on doubly commuting sequences of $C_{\cdot0}$-contractions, and establish the dilation theory and the analytic model theory for these sequences of operators. These results are applied to generalize the Beurling-Lax theorem and Jordan blocks in the multivariable operator theory to the operator theory in countably infinitely many variables.

Equations

$$T^n=P_{\mathcal{H}}{V}^n{|}_{\mathcal{H}},$$

$$D_T=(I-T^{\ast }T{)}^{\frac{1}{2}},$$

$${\mathfrak{D}}_T=\overline{D_T\mathcal{H}},$$

$$\mathcal{Q}={\mathcal{J}}_1\otimes \cdots \otimes {\mathcal{J}}_n,$$

$${\mathbb{D}}_2^{\infty }=\{\zeta =({\zeta }_1,{\zeta }_2,\cdots )\in l^2:|{\zeta }_n|<1\mathrm{for}\mathrm{all}n\ge 1\}.$$

$$H^2({\mathbb{D}}_2^{\infty })=\{F=\sum _{\alpha \in {\mathbb{Z}}_{+}^{(\infty )}}{c}_{\alpha }{\zeta }^{\alpha }:\Vert F{\Vert }^2=\sum _{\alpha \in {\mathbb{Z}}_{+}^{(\infty )}}|c_{\alpha }{|}^2<\infty \},$$

$${\zeta }^{\alpha }={\zeta }_1^{{\alpha }_1}\cdots {\zeta }_n^{{\alpha }_n}$$

$${\mathbf{K}}_{\lambda }(\zeta )=\prod _{n=1}^{\infty }\frac{1}{1-\overline{{\lambda }_n}{\zeta }_n},\lambda =({\lambda }_1,{\lambda }_2,\cdots )\in {\mathbb{D}}_2^{\infty }.$$

$$D_{\mathbf{T}}=(\mathrm{SOT})\underset{n\to \infty }{\lim }{D}_{T_1}\cdots D_{T_n}=(\mathrm{SOT})\underset{n\to \infty }{\lim }(I-T_1^{\ast }{T}_1{)}^{\frac{1}{2}}\cdots (I-T_n^{\ast }{T}_n{)}^{\frac{1}{2}}$$

$${\mathfrak{D}}_{\mathbf{T}}=\overline{D_{\mathbf{T}}\mathcal{H}}.$$

$$\Vert x{\Vert }^2=\sum _{\alpha \in {\mathbb{Z}}_{+}^n}\Vert D_{{\mathbf{T}}^{\ast }}{\mathbf{T}}^{\ast \alpha }x{\Vert }^2,x\in \mathcal{H},$$

$$\Vert x{\Vert }^2=\sum _{\alpha \in {\mathbb{Z}}_{+}^{(\infty )}}\Vert D_{{\mathbf{T}}^{\ast }}{\mathbf{T}}^{\ast \alpha }x{\Vert }^2.$$

$$[E{]}_{\mathbf{T}}=\underset{\alpha \in {\mathbb{Z}}_{+}^{(\infty )}}{\bigvee }{\mathbf{T}}^{\alpha }E,$$

$${\mathfrak{D}}_{{\mathbf{V}}^{\ast }}=\bigcap _{n=1}^{\infty }\mathrm{Ker}{V}_n^{\ast },$$

$$[{\mathfrak{D}}_{M_{\mathcal{I}}^{\ast }}{]}_{M_{\mathcal{I}}}=[\mathcal{I}\ominus z\mathcal{I}{]}_{M_{\mathcal{I}}}=\mathcal{I}.$$

$${\mathbf{M}}_{\zeta }=(M_{{\zeta }_1},M_{{\zeta }_2},\cdots )$$

$$H_{\mathcal{E}}^2({\mathbb{D}}_2^{\infty })=H^2({\mathbb{D}}_2^{\infty })\otimes \mathcal{E}$$

$${\phi }_a(z)=\frac{a-z}{1-\bar{a}z},z\in \mathbb{D}$$

$${\phi }_a(T)=(aI-T)(I-\bar{a}T{)}^{-1}(a\in \mathbb{D})$$

$${\Phi }_{\lambda }(\mathbf{T})=({\phi }_{{\lambda }_1}(T_1),{\phi }_{{\lambda }_2}(T_2),\cdots ).$$

$$S_n=U{T}_n{U}^{\ast },n\in \mathbb{N}.$$

$$\Vert x{\Vert }^2=\sum _{k=1}^{\infty }\sum _{\alpha \in {\mathbb{Z}}_{+}^{(\infty )}}\Vert D_{{\Phi }_{{\lambda }^{(k)}}(\mathbf{T}{)}^{\ast }}{\Phi }_{{\lambda }^{(k)}}^{\alpha }(\mathbf{T}{)}^{\ast }x{\Vert }^2;$$

$$\mathbf{T}\cong \underset{\gamma }{⨁}{P}_{{\mathcal{Q}}_{\gamma }}{\Phi }_{{\lambda }_{\gamma }}({\mathbf{M}}_{\zeta }){|}_{{\mathcal{Q}}_{\gamma }},$$

$$\{D_{{\Phi }_{\lambda }(\mathbf{T}{)}^{\ast }}x:\lambda \in {\mathbb{D}}^{\infty },x\in \mathcal{H}\}$$

$$ph=p(\mathbf{T})h,p\in {\mathcal{P}}_{\infty },h\in \mathcal{H}.$$

$$(M_{{\zeta }_1}{|}_{\mathcal{S}},M_{{\zeta }_2}{|}_{\mathcal{S}},\cdots )$$

$$(P_{\mathcal{Q}}{M}_{{\zeta }_1}{|}_{\mathcal{Q}},P_{\mathcal{Q}}{M}_{{\zeta }_2}{|}_{\mathcal{Q}},\cdots )$$

$$\mathcal{S}=\Psi H_{\mathcal{F}}^2({\mathbb{D}}_2^{\infty }).$$

$${\mathbf{T}}^{\alpha }=P_{\mathcal{H}}{\mathbf{V}}^{\alpha }{|}_{\mathcal{H}}.$$

$${\mathbf{T}}^{\ast \alpha }{\mathbf{T}}^{\beta }=P_{\mathcal{H}}{\mathbf{V}}^{\ast \alpha }{\mathbf{V}}^{\beta }{|}_{\mathcal{H}},$$

Full text

1. Introduction

1.1. Background

Suppose that $T$ is a contraction on a Hilbert space $\mathcal{H}$; that is, $T$ is a bounded linear operator with $\|T\|\leq 1$. The contraction $T$ is said to be in the class $C_{\cdot 0}$ if $T^{*k}\rightarrow 0$ (as $k\rightarrow\infty$) in the strong operator topology. An operator $V$ on a larger Hilbert space $\mathcal{K}\supseteq\mathcal{H}$ is said to be a dilation of $T$ if for each $n\in\mathbb{N}$,

[TABLE]

where $\mathbb{N}=\{1,2,\cdots\}$, the set of positive integers. The dilation theory developed by B. Sz.-Nagy and C. Foias is of great significance in operator theory. They built functional models for contractions, which not only reveals the structure of these operators, but also gives a way for calculations [43]. In particular, it was shown that a contraction of class $C_{\cdot 0}$ has a dilation to a Hardy shift. To be more specific, let $\mathbb{D}$ denote the open unit disk in the complex plane, and $H^{2}(\mathbb{D})$ denote the Hardy space over $\mathbb{D}$. For a contraction $T$, put

[TABLE]

the defect operator of $T$, and

[TABLE]

the defect space of $T$. If $T$ is a $C_{\cdot 0}$-contraction, then $T$ is unitarily equivalent to the compression of the multiplication operator $M_{z}$, defined on the vector-valued Hardy space $H_{\mathfrak{D}_{T^{*}}}^{2}(\mathbb{D})=H^{2}(\mathbb{D})\otimes\mathfrak{D}_{T^{*}}$, on some invariant subspace $\mathcal{J}$ of $H_{\mathfrak{D}_{T^{*}}}^{2}(\mathbb{D})$ for the backward shift $M_{z}^{*}$.

The existence of the isometric dilation of any commuting pair of contractions is proved by T. Andô [3]. However, commuting $n$-tuples of contractions have no isometric dilations in general when $n\geq 3$ [32]. Under some additional conditions, the above graceful analytic model for single contraction of class $C_{\cdot 0}$ can be generalized to the situation of a commuting finite-tuple $(T_{1},\cdots,T_{n})$ of $C_{\cdot 0}$-contractions. That is to say, this tuple $(T_{1},\cdots,T_{n})$ has a dilation to the tuple $(M_{\zeta_{1}},\cdots,M_{\zeta_{n}})$ of coordinate multiplication operators on a $\mathcal{E}$-valued analytic function space $\mathcal{H}_{\mathcal{E}}=\mathcal{H}\otimes\mathcal{E}$ with $\mathcal{H}$ consisting of holomorphic functions over some domain in $\mathbb{C}^{n}$ [13, 28, 33, 2, 9]. The particular case that the tuple being doubly commuting is rather interesting since in this case the function space $\mathcal{H}$ is exactly the Hardy space $H^{2}(\mathbb{D}^{n})$ over the $n$-polydisc, and the underlying space $\mathcal{E}$ is the defect space of the tuple $(T_{1}^{*},\cdots,T_{n}^{*})$. Moreover, we can require this dilation to be minimal and regular [9]. Recall that two operators $T$, $S$ are said to be doubly commuting if $TS=ST$ and $T^{*}S=ST^{*}$, and a tuple or a sequence of operators is said to be doubly commuting if any pair of operators in it are doubly commuting. For more about developments on model theory, we refer the readers to [42, 1, 27, 45].

A natural question arises: Does a commuting sequence of $C_{\cdot 0}$-contractions, under the conditions analogous to those given in the above-mentioned papers, have a dilation to the tuple of coordinate multiplication operators on some vector-valued Hardy space over infinite-dimensional polydisc? Except for the assumption “doubly commuting”, most of these conditions do not carry over to infinitely many operators well. For this reason, we focus on doubly commuting sequences of $C_{\cdot 0}$-contractions in this paper, and establish dilation theory and analytic model theory for these sequences of operators.

Another motivation for our study of such operator sequences comes from investigations on the structure of some special submodules and quotient modules of the Hardy module over the polydisc in [40, 37], which are said to be doubly commuting.

The analytic Hilbert module theory developed by R. Douglas and V. Paulsen opens a new door for the study of joint invariant subspaces of the Hardy space $H^{2}(\mathbb{D}^{n})\ (n\in\mathbb{N})$ for the tuple $\mathbf{M}_{\zeta}$ of coordinate multiplication operators [16, 17]. Let $\mathcal{P}_{n}$ denote the polynomial ring in $n$-complex variables. It is known that the Hardy space $H^{2}(\mathbb{D}^{n})$ carries a $\mathcal{P}_{n}$-Hilbert module structure, where the module action is defined by multiplications by polynomials, and a submodule of $H^{2}(\mathbb{D}^{n})$ is just a $\mathbf{M}_{\zeta}$-joint invariant subspace. A quotient module of $H^{2}(\mathbb{D}^{n})$ is defined to be the orthocomplement of some submodule with a $\mathcal{P}_{n}$-module structure determined by the compression of $\mathbf{M}_{\zeta}$ on it. These notions are defined analogously on vector-valued Hardy spaces $H_{\mathcal{E}}^{2}(\mathbb{D}^{n})$. For more details, we refer the reader to [18, 11].

The famous theorem of A. Beurling states that every nonzero submodule of $H^{2}(\mathbb{D})$ is of form $\eta H^{2}(\mathbb{D})$ for some inner function $\eta\in H^{\infty}(\mathbb{D})$, where $H^{\infty}(\mathbb{D})$ denotes the space of bounded holomorphic functions on $\mathbb{D}$ [5]. P. Lax generalized Beurling’s theorem to vector-valued Hardy spaces $H_{\mathcal{E}}^{2}(\mathbb{D})$ [25]: a nonzero submodule of $H_{\mathcal{E}}^{2}(\mathbb{D})$ takes the form $\theta H_{\mathcal{F}}^{2}(\mathbb{D})$ with $\theta\in H_{\mathcal{B}(\mathcal{F},\mathcal{E})}^{\infty}(\mathbb{D})$ being inner, where $H_{\mathcal{B}(\mathcal{F},\mathcal{E})}^{\infty}(\mathbb{D})$ is the space of uniformly bounded holomorphic $\mathcal{B}(\mathcal{F},\mathcal{E})$-valued function on $\mathbb{D}$. For the multi-variable situation, such Beurling-Lax type theorem fails in general [35]. Some efforts was made to determine when a submodule enjoys a Beurling-Lax type representation. In [26], V. Mandrekar considered the case of the scalar-valued Hardy space over the bidisc. He obtained a necessary and sufficient condition that the restriction of the tuple $\mathbf{M}_{\zeta}$ on the submodule is doubly commuting. This was further generalized to the vector-valued Hardy space $H_{\mathcal{E}}^{2}(\mathbb{D}^{n})$ for arbitrary positive integer $n$ in [40], and to the Hardy space over the infinite-dimensional polydisc within the language of the Hilbert space of Dirichlet series with square-summable coefficients in [31].

A proper quotient module of $H^{2}(\mathbb{D})$ is called a model space, and the compression of the Hardy shift $M_{z}$ on a model space is called a Jordan block. The notion of the Jordan block plays a central role in Sz.-Nagy and Foias’ model theory for operators of class $C_{0}$: every $C_{0}$-operator is quasi-similar to the direct sum of Jordan blocks [43]. Following former works on the Hardy quotient module over the bidisc [19, 20, 23], J. Sarkar proved that if the compression of the tuple $\mathbf{M}_{\zeta}$ on a nonzero quotient module $\mathcal{Q}$ of $H^{2}(\mathbb{D}^{n})$ is doubly commuting, then

[TABLE]

where either $\mathcal{J}_{i}$ is a model space or $\mathcal{J}_{i}=H^{2}(\mathbb{D})$ ($1\leq i\leq n$) [37]. Thus, for any $i$ so that $\mathcal{J}_{i}\neq H^{2}(\mathbb{D})$, the compression $P_{\mathcal{Q}}M_{\zeta_{i}}|_{\mathcal{Q}}$ is the tensor product of a Jordan block and an identity operator. Then the compression of the tuple $\mathbf{M}_{\zeta}$ on $\mathcal{Q}$ can be considered as the Jordan block in finitely many variables. We refer the readers to [46, 47, 24, 34, 39] for related works and further discussions. Also see [14, Remark 4.7].

In this paper, we establish dilation theory and analytic model theory for doubly commuting sequences of $C_{\cdot 0}$-contractions, and then apply them to generalize the Beurling-Lax theorem for doubly commuting submodules and the Jordan block type characterization for doubly commuting quotient module in the multi-variable case to the Hardy module in infinitely many variables.

1.2. Statements of the main results

To state our main results, we need to introduce some notations and definitions. Let $\mathbb{D}^{\infty}$ denote the Cartesian product $\mathbb{D}\times\mathbb{D}\times\cdots$ of countably infinitely many unit disks. The Hilbert’s multidisk $\mathbb{D}_{2}^{\infty}$ is defined to be

[TABLE]

The Hardy space $H^{2}(\mathbb{D}_{2}^{\infty})$ in infinitely many variables is defined as follows:

[TABLE]

where $\mathbb{Z}_{+}^{(\infty)}$ denotes the set of finitely supported sequences of nonnegative integers, and $\zeta^{\alpha}$ denotes the monimial

[TABLE]

for $\alpha=(\alpha_{1},\cdots,\alpha_{n},0,0,\cdots)\in\mathbb{Z}_{+}^{(\infty)}$. The space $H^{2}(\mathbb{D}_{2}^{\infty})$ is a reproducing kernel Hilbert space over the Hilbert’s multidisk $\mathbb{D}_{2}^{\infty}$ with the kernels [29]

[TABLE]

This space has close connections with the study of Beurling’s completeness problem and Dirichlet series [6, 22, 29], and is the expected function space upon which we build analytic models. The vector-valued Hardy space $H_{\mathcal{E}}^{2}(\mathbb{D}_{2}^{\infty})$ in infinitely many variables is defined analogously as in the finite-variable situation. See Section 2 for the details.

If $\mathbf{T}=(T_{1},T_{2},\cdots)\in\mathcal{DC}$ is a doubly commuting sequence of contractions on a Hilbert space $\mathcal{H}$, then the infinite product

[TABLE]

of defect operators of $\{T_{n}\}_{n\in\mathbb{N}}$ converges [12, Proposition 43.1], and is called the defect operator of $\mathbf{T}$. Similarly, define the defect space $\mathfrak{D}_{\mathbf{T}}$ of $\mathbf{T}$ to be

[TABLE]

Let $\mathcal{DC}(\mathcal{H})$ (or simply $\mathcal{DC}$ if no confusion is caused) denote the class of doubly commuting sequences of $C_{\cdot 0}$-contractions on $\mathcal{H}$. For convenience, we identify the class of doubly commuting finite-tuples of $C_{\cdot 0}$-contractions with the class $\mathcal{DCF}$ of sequences in class $\mathcal{DC}$ with only finitely many nonzero components. Let $\mathbf{T}^{*}$ denote the sequence $(T_{1}^{*},T_{2}^{*},\cdots)$ and $\mathbf{T}^{\alpha}$ the operator $T_{1}^{\alpha_{1}}\cdots T_{n}^{\alpha_{n}}$ for $\mathbf{T}=(T_{1},T_{2},\cdots)$ and $\alpha=(\alpha_{1},\cdots,\alpha_{n},0,0,\cdots)\in\mathbb{Z}_{+}^{(\infty)}$ ($n\in\mathbb{N}$). The key ingredient in the dilation theory for doubly commuting finite-tuples of $C_{\cdot 0}$-contractions is the following norm identity [9]:

[TABLE]

where $n\in\mathbb{N}$ and $\mathbf{T}=(T_{1},\cdots,T_{n})\in\mathcal{DCF}$. It is natural to expect that such an identity holds for sequences in class $\mathcal{DC}$ in the following sense: for each $x\in\mathcal{H}$ and $\mathbf{T}=(T_{1},T_{2},\cdots)\in\mathcal{DC}$,

[TABLE]

Unfortunately, the answer is negative in general. We will see that there exists a sequence $\mathbf{T}\in\mathcal{DC}$ such that the defect operator $D_{\mathbf{T}^{*}}$ of $\mathbf{T}^{*}$ is [math], which cannot occur in the finite-tuple case.

The class of doubly commuting sequences of pure isometries on $\mathcal{H}$ is denoted by $\mathcal{DP}(\mathcal{H})$ (or $\mathcal{DP}$). Since an isometry $V$ is of class $C_{\cdot 0}$ if and only if $V$ is pure (that is to say, the unitary part in the Wold decomposition of $V$ is [math]), $\mathcal{DP}$ is a subclass of $\mathcal{DC}$. Any doubly commuting sequence of contractions has a minimal, doubly commuting, regular isometric dilation, which is unique up to unitarily equivalence [41]. Furthermore, we will show that the minimal regular isometric dilation of a sequence in class $\mathcal{DC}$ consists of doubly commuting pure isometries. This therefore provides us with an approach to the question arisen in previous subsection via the study of the sequences in class $\mathcal{DP}$. These notions concerning the dilation will be explained in Section 2.

Unlike the finite-tuple case, sequences in class $\mathcal{DP}$ require further classification. Let us start some notations and definitions. For a family $\mathcal{T}$ of bounded linear operators on $\mathcal{H}$ and a subset $E$ of $\mathcal{H}$, let $[E]_{\mathcal{T}}$ denote the joint invariant subspace for $\mathcal{T}$ generated by $E$. In particular, if $\mathbf{T}$ is a sequence of operators, then one has

[TABLE]

where the notation $\bigvee$ denotes the closed linear span of subsets of a Hilbert space. Following [21], if a closed subspace $M$ of $\mathcal{H}$ is orthogonal to $\mathbf{T}^{\alpha}M$ for any $\alpha\in\mathbb{Z}_{+}^{(\infty)}\setminus\{0\}$, then $M$ is called a wandering subspace for the sequence $\mathbf{T}$ (see [7] for an analogous definition in the finite-tuple case).

Suppose $\mathbf{V}=(V_{1},V_{2},\cdots)\in\mathcal{DP}$. It is easy to verify that the defect spce

[TABLE]

and $\mathfrak{D}_{\mathbf{V}^{*}}$ is a wandering subspace for $\mathbf{V}$. It follows from Beurling’s theorem that if $\mathcal{I}$ is an invariant subspace of $H^{2}(\mathbb{D})$ for the Hardy shift $M_{z}$, then the wandering subspace for the isometry $M_{\mathcal{I}}=M_{z}|_{\mathcal{I}}$ always generates the entire $\mathcal{I}$; that is,

[TABLE]

The conclusion is indeed valid for the vector-valued Hardy space $H_{\mathcal{E}}^{2}(\mathbb{D})$ by P. Halmos’s observation in [21] or the Beurling-Lax theorem. This suggests the following definition.

Definition 1.1**.**

Suppose $\mathbf{V}\in\mathcal{DP}(\mathcal{H})$. The sequence $\mathbf{V}$ is said to be of Beurling type if $[\mathfrak{D}_{\mathbf{V}^{*}}]_{\mathbf{V}}=\mathcal{H}$.

Suppose $\mathbf{T}\in\mathcal{DC}(\mathcal{H})$. The sequence $\mathbf{T}$ is said to be of Beurling type if its minimal regular isometric dilation is of Beurling type.

It was shown in [38] (also see [10]) that every doubly commuting finite-tuple of pure isometries enjoys a “Beurling type” property. The tuple

[TABLE]

of coordinate multiplication operators on a vector-valued Hardy space

[TABLE]

is clearly of Beurling type. More interestingly, we will see that the converse also holds: if a sequence $\mathbf{V}\in\mathcal{DP}$ is of Beurling type, then $\mathbf{V}$ must be of the above form. Then the question reduces to the characterization for sequences in class $\mathcal{DC}$ which are of Beurling type.

Let $\varphi_{a}\ (a\in\mathbb{D})$ denote the holomorphic automorphism

[TABLE]

of $\mathbb{D}$. It is known that the Riesz functional calculus

[TABLE]

of a contraction $T$ on the Hilbert space $\mathcal{H}$ is also a contraction. Furthermore, if $T\in C_{\cdot 0}$, then $\varphi_{a}(T)\in C_{\cdot 0}$ by the dilation theory for single $C_{\cdot 0}$-contraction. For $\mathbf{T}=(T_{1},T_{2},\cdots)\in\mathcal{DC}$ and $\lambda=(\lambda_{1},\lambda_{2},\cdots)\in\mathbb{D}^{\infty}$, set

[TABLE]

Obviously, for any $\lambda\in\mathbb{D}^{\infty}$, $\Phi_{\lambda}$ defines a bijection from $\mathcal{DC}$ onto itself, and $\Phi_{0}=\Phi_{\lambda}\circ\Phi_{\lambda}=id_{\mathcal{DC}}$. Therefore, for a sequence $\mathbf{T}$ of operators, $\mathbf{T}\in\mathcal{DC}$ if and only if $\Phi_{\lambda}(\mathbf{T})\in\mathcal{DC}$. It is also easy to see that $\Phi_{\lambda}$ maps $\mathcal{DP}$ onto itself. We also write $\Phi_{\lambda}^{\alpha}(\mathbf{T})$ for the operator $\varphi_{\lambda_{1}}^{\alpha_{1}}(T_{1})\cdots\varphi_{\lambda_{n}}^{\alpha_{n}}(T_{n})$, where $\alpha=(\alpha_{1},\cdots,\alpha_{n},0,0,\cdots)\in\mathbb{Z}_{+}^{(\infty)}$ ($n\in\mathbb{N}$).

There are various examples of sequences $\mathbf{V}$ in class $\mathcal{DP}$ satisfying that the wandering space $\mathfrak{D}_{\mathbf{V}^{*}}=\{0\}$, while the defect space $\mathfrak{D}_{\Phi_{\lambda}(\mathbf{V})^{*}}$ of $\Phi_{\lambda}(\mathbf{V})^{*}$ is nonzero. For instance, let $\{\eta_{n}\}_{n\in\mathbb{N}}$ be a sequence of nonconstant inner functions in $H^{\infty}(\mathbb{D})$ and consider the sequence $\mathbf{V}=(M_{\widetilde{\eta_{1}}},M_{\widetilde{\eta_{2}}},\cdots)$ of multiplication operators on the Hardy space $H^{2}(\mathbb{D}_{2}^{\infty})$, $\widetilde{\eta_{n}}(\zeta)=\eta_{n}(\zeta_{n})\ (n\in\mathbb{N},\zeta\in\mathbb{D}^{\infty}_{2})$. We will prove that for “almost all” choices of the sequence $\{\eta_{n}\}_{n\in\mathbb{N}}$ of inner functions, the wandering space $\mathfrak{D}_{\mathbf{V}^{*}}$ for the $\mathcal{DP}$-sequence $\mathbf{V}$ is $\{0\}$, and $\Phi_{\lambda}(\mathbf{V})$ is of Beurling type for some $\lambda\in\mathbb{D}^{\infty}$ (See Proposition 3.4 and Remark 3.5). Therefore, for those $\mathcal{DP}$-sequences which are not of Beurling type, the family of maps $\Phi_{\lambda}\ (\lambda\in\mathbb{D}^{\infty})$ could be a powerful tool in building analytic models. We thus could obtain important information of such $\mathcal{DP}$-sequences $\mathbf{V}$ from the Beurling type sequences $\Phi_{\lambda}(\mathbf{V})$ for some $\lambda\in\mathbb{D}^{\infty}$.

Inspired by this, we consider the following sequences in class $\mathcal{DP}$.

Definition 1.2**.**

Suppose $\mathbf{V}\in\mathcal{DP}$. The sequence $\mathbf{V}$ is said to be of quasi-Beurling type if $\Phi_{\lambda}(\mathbf{V})$ is of Beurling type for some $\lambda\in\mathbb{D}^{\infty}$.

Suppose $\mathbf{T}\in\mathcal{DC}$. The sequence $\mathbf{T}$ is said to be of quasi-Beurling type if its minimal regular isometric dilation is of quasi-Beurling type.

Sequences of quasi-Beurling type is relatively tractable in class $\mathcal{DP}$. In Section 3, we will show that a sequence $\mathbf{V}\in\mathcal{DP}$ is of quasi-Beurling type if and only if $\mathbf{V}$ is jointly unitarily equivalent to the sequence $\Phi_{\lambda}(\mathbf{M}_{\zeta})$ on a vector-valued Hardy space $H^{2}_{\mathcal{E}}(\mathbb{D}_{2}^{\infty})$ for some $\lambda\in\mathbb{D}^{\infty}$. Recall that two sequence $\mathbf{T}=(T_{1},T_{2},\cdots)$ and $\mathbf{S}=(S_{1},S_{2},\cdots)$ of operators, defined on $\mathcal{H}$ and $\mathcal{K}$ respectively, are said to be jointly unitarily equivalent if there exists a unitary operator $U:\mathcal{H}\rightarrow\mathcal{K}$ such that

[TABLE]

The first main result in this paper is to give a complete characterization of sequences in class $\mathcal{DC}$ that can be decomposed into direct sums of sequences of quasi-Beurling type. Note that for a commuting sequence of operators on $\mathcal{H}$, by using a standard argument involving Zorn’s Lemma, one can decompose $\mathcal{H}$ into orthogonal direct sums of separable $\mathbf{T}$-joint reducing subspaces. It suffices to restrict our study to the case of separable Hilbert spaces. From now on, we only consider separable Hilbert spaces.

A sequence $\mathbf{T}\in\mathcal{DC}(\mathcal{H})$ is said to have a decomposition of quasi-Beurling type if there exists an orthogonal decomposition $\mathcal{H}=\bigoplus_{\gamma}\mathcal{H}_{\gamma}$ of the Hilbert space $\mathcal{H}$, such that each $\mathcal{H}_{\gamma}$ is $\mathbf{T}$-joint reducing and each $\mathbf{T}|_{\mathcal{H}_{\gamma}}$ is of quasi-Beurling type.

Theorem 1.3**.**

Suppose $\mathbf{T}\in\mathcal{DC}(\mathcal{H})$. The followings are equivalent:

• (1)

$\mathbf{T}$* has a decomposition of quasi-Beurling type;*

• (2)

for each $x\in\mathcal{H}$, there exists a sequence $\lambda^{(1)},\lambda^{(2)},\cdots$ of points in $\mathbb{D}^{\infty}$, such that

[TABLE]

• (3)

$\bigvee_{\lambda\in\mathbb{D}^{\infty}}\mathfrak{D}_{\Phi_{\lambda}(\mathbf{T})^{*}}=\mathcal{H}$.

In fact, the sequence $\lambda^{(1)},\lambda^{(2)},\cdots$ in condition (2) can be chosen to be independent of $x\in\mathcal{H}$.

The following particular case of Theorem 1.3 completely answers the question that when a doubly commuting sequence of $C_{\cdot 0}$-contractions has a regular dilation to the tuple of coordinate multiplication operators on some vector-valued Hardy space over the infinite-dimensional polydisc.

Corollary 1.4**.**

Let Suppose $\mathbf{T}\in\mathcal{DC}(\mathcal{H})$. The followings are equivalent:

• (1)

$\mathbf{T}$* is of Beurling type;*

• (2)

the minimal regular isometric dilation of $\mathbf{T}$ is jointly unitarily equivalent to the tuple $\mathbf{M}_{\zeta}$ of coordinate multiplication operators on a vector-valued Hardy space $H_{\mathcal{E}}^{2}(\mathbb{D}_{2}^{\infty})$;

• (3)

for each $x\in\mathcal{H}$, $\|x\|^{2}=\sum_{\alpha\in\mathbb{Z}_{+}^{(\infty)}}\|D_{\mathbf{T}^{*}}\mathbf{T}^{*\alpha}x\|^{2}$;

• (4)

$\bigvee_{\lambda\in\mathbb{D}_{2}^{\infty}}\mathfrak{D}_{\Phi_{\lambda}(\mathbf{T})^{*}}=\mathcal{H}$.

Note that for $\mathbf{T}\in\mathcal{DCF}$, the condition (3) in Corollary 1.4 coincides with the identity (1.1), Corollary 1.4 actually generalizes the finite-tuple case.

Here are some remarks for Theorem 1.3. If $\mathbf{T}\in\mathcal{DC}$ with a decomposition $\mathbf{T}=\bigoplus_{\gamma}\mathbf{T}_{\gamma}$ of quasi-Beurling type, then there correspond a point $\lambda_{\gamma}\in\mathbb{D}^{\infty}$ and a Hilbert space $\mathcal{E}_{\gamma}$ to each index $\gamma$, such that $\mathbf{T}_{\gamma}$ is jointly unitarily equivalent to $P_{\mathcal{Q}_{\gamma}}\Phi_{\lambda_{\gamma}}(\mathbf{M}_{\zeta})|_{\mathcal{Q}_{\gamma}}$, where the sequence $\Phi_{\lambda_{\gamma}}(\mathbf{M}_{\zeta})$ is defined on the vector-valued Hardy space $H^{2}_{\mathcal{E}_{\gamma}}(\mathbb{D}_{2}^{\infty})$, and $\mathcal{Q}_{\gamma}\subseteq H^{2}_{\mathcal{E}_{\gamma}}(\mathbb{D}_{2}^{\infty})$ is a $\mathbf{M}_{\zeta}^{*}$-joint invariant subspace. This gives

[TABLE]

and we therefore build an analytic model for a sequence $\mathbf{T}\in\mathcal{DC}(\mathcal{H})$ under the assumption that the subset

[TABLE]

is complete in $\mathcal{H}$. Note that this assumption always holds for the finite-tuple case (see Lemma 2.7). Condition (2) in the theorem further generalizes the identity (1.1). See Section 3 for the details in these paragraphs. Also, the following result illustrates that unlike the finite-tuple case (Lemma 2.7), some extreme phenomenon would occur in the infinite-tuple case.

Theorem 1.5**.**

There exists a sequence $\mathbf{V}\in\mathcal{DP}$ such that $\mathfrak{D}_{\Phi_{\lambda}(\mathbf{V})^{*}}=\{0\}$ for each $\lambda\in\mathbb{D}^{\infty}$.

We further refine the representation (1.2) by giving a characterization of the subspaces $\mathcal{Q}_{\gamma}$ involving characterization functions of operators in $\mathbf{T}_{\gamma}$. We will give the details in Section 4. This generalizes results in [9] to the infinite-variable case.

In section 4, we will prove that every sequence in class $\mathcal{DP}$ is jointly unitarily equivalent to a sequence of multiplication operators induced by operator-valued inner functions each of which involves one different variable (Theorem 4.1). Thus we establish operator-valued analytic functional models for general sequences in class $\mathcal{DC}$, which generalize (1.2). We also have the following application of our results.

Corollary 1.6**.**

Suppose $\mathbf{T}\in\mathcal{DC}$. Then there exists a sequence $\{B_{n}\}_{n\in\mathbb{N}}$ of finite Blaschke products, such that $\{\prod_{i=1}^{n}B_{i}(T_{i})\}_{n\in\mathbb{N}}$ converges in the strong operator topology.

In the rest of this paper, we use the language of Hilbert module by Douglas and Paulsen [17] to state the generalization of the Beurling-Lax theorem and results related with Jordan blocks in the finite-variable case to the case in infinitely many variables.

Let $\mathcal{P}_{\infty}$ denote the polynomial ring in countably infinitely many complex variables, in which each polynomial only involves finitely many variables. Similar to the finite-variable case, for each commuting sequence $\mathbf{T}$ of operators on a Hilbert space $\mathcal{H}$, one defines a $\mathcal{P}_{\infty}$-module structure on $\mathcal{H}$ by

[TABLE]

Say that this $\mathcal{P}_{\infty}$-module $\mathcal{H}$ is doubly commuting if the sequence $\mathbf{T}$ is doubly commuting. Conversely, any $\mathcal{P}_{\infty}$-module structure is determined by a commuting sequence of operators in the above way.

We can also define the Hardy module structure on a vector-valued Hardy space $H_{\mathcal{E}}^{2}(\mathbb{D}_{2}^{\infty})$ in infinitely many variables via the tuple $\mathbf{M}_{\zeta}$ of coordinate multiplication operators on $H_{\mathcal{E}}^{2}(\mathbb{D}_{2}^{\infty})$. The module action is the multiplication by polynomials and submodules of $H_{\mathcal{E}}^{2}(\mathbb{D}_{2}^{\infty})$ are exactly joint invariant subspaces for $\mathbf{M}_{\zeta}$. By the definition, a submodule $\mathcal{S}$ of $H_{\mathcal{E}}^{2}(\mathbb{D}_{2}^{\infty})$ is doubly commuting if the restriction

[TABLE]

of $\mathbf{M}_{\zeta}$ on $\mathcal{S}$ is doubly commuting; a quotient module $\mathcal{Q}$ of $H_{\mathcal{E}}^{2}(\mathbb{D}_{2}^{\infty})$ is doubly commuting if the compression

[TABLE]

of $\mathbf{M}_{\zeta}$ on $\mathcal{Q}$ is doubly commuting. We will prove that such doubly commuting restriction and compression are of Beurling type. As a consequence, we have

Theorem 1.7**.**

Let $\mathcal{S}$ be a submodule of the vector-valued Hardy module $H_{\mathcal{E}}^{2}(\mathbb{D}_{2}^{\infty})$. Then $\mathcal{S}$ is doubly commuting if and only if there exist a Hilbert space $\mathcal{F}$ and an inner function $\Psi\in H_{\mathcal{B}(\mathcal{F},\mathcal{E})}^{\infty}(\mathbb{D}_{2}^{\infty})$, such that

[TABLE]

Theorem 1.8**.**

Every doubly commuting quotient module of $H^{2}(\mathbb{D}_{2}^{\infty})$ is the tensor product of some sequence of quotient modules of $H^{2}(\mathbb{D})$.

Theorem 1.7 is a infinite-variable version of the Beurling-Lax Theorem for doubly commuting Hardy submodules. Also from Theorem 1.8, the compression of the tuple of coordinate multiplication operators on a nontrivial doubly commuting quotient module of $H^{2}(\mathbb{D}_{2}^{\infty})$ is a Jordan block in infinitely many variables. See Section 5 for the details.

It is worth mentioning that our methods presented in this paper are quite different from that in [31, 37, 9] and also valid for nonseparable Hilbert spaces.

2. Some preparations

In this section, we first establish a lemma to guarantee the validity of Definition 1.1 and 1.2. Then we list some basic properties of vector-valued Hardy spaces and operator-valued functions. Lastly, the rest of this section is dedicated to preparations for the proofs of main results in subsequent sections.

2.1. The minimal regular isometric dilation

Suppose that the sequence $\mathbf{T}=(T_{1},T_{2},\cdots)$ is a doubly commuting sequence of contractions on $\mathcal{H}$. Say, a sequence $\mathbf{V}=(V_{1},V_{2},\cdots)$ of isometries, defined on a larger Hilbert space $\mathcal{K}_{\mathbf{V}}\supseteq\mathcal{H}$, is an isometric dilation of $\mathbf{T}$ if for each $\alpha\in\mathbb{Z}_{+}^{(\infty)}$,

[TABLE]

Furthermore, if for $\alpha,\beta\in\mathbb{Z}_{+}^{(\infty)}$ satisfying $\alpha\wedge\beta=(0,0,\cdots)$,

[TABLE]

then the isometric dilation $\mathbf{V}$ of $\mathbf{T}$ is said to be regular, where

[TABLE]

An isometric dilation $\mathbf{V}$ of $\mathbf{T}$ is said to be minimal if the $\mathbf{V}$-joint invariant subspace $[\mathcal{H}]_{\mathbf{V}}$ generated by $\mathcal{H}$ is $\mathcal{K}_{\mathbf{V}}$. It is clear to see that for any regular isometric dilation $\mathbf{V}$ of $\mathbf{T}$, the restriction $\mathbf{V}|_{[\mathcal{H}]_{\mathbf{V}}}$ of the sequence $\mathbf{V}$ on $[\mathcal{H}]_{\mathbf{V}}$ is a minimal regular isometric dilation of $\mathbf{T}$. The minimal regular isometric dilation is unique up to the joint unitary equivalence in the sense that if both $\mathbf{V}=(V_{1},V_{2},\cdots)$ and $\mathbf{W}=(W_{1},W_{2},\cdots)$ are minimal regular isometric dilations of $\mathbf{T}$, then there exists a unitary operator $U:\mathcal{K}_{\mathbf{V}}\rightarrow\mathcal{K}_{\mathbf{W}}$ such that $U$ fixes vectors in $\mathcal{H}$ and

[TABLE]

Applying [41, Theorem 4.2] to the semigroup $\mathbb{Z}_{+}^{(\infty)}$, we see that the minimal regular isometric dilation of $\mathbf{T}$ always exists and is also doubly commuting. Note that the existence can be also deduced from [43, pp. 36-37] by restrict the minimal regular unitary dilation $\mathbf{U}$ of $\mathbf{T}$ on $[\mathcal{H}]_{\mathbf{U}}$. For the convenience of the readers, below we prove directly for the case $\mathbf{T}\in\mathcal{DC}$ that the minimal regular isometric dilation $\mathbf{V}$ of $\mathbf{T}$ is in class $\mathcal{DP}$. Moreover, $\mathbf{V}$ is a coextension of $\mathbf{T}$, which means that $\mathcal{H}$ is invariant for $\mathbf{V}^{*}$ and $\mathbf{T}^{*}=\mathbf{V}^{*}|_{\mathcal{H}}$, the restriction of $\mathbf{V}^{*}$ on $\mathcal{H}$.

Lemma 2.1**.**

The minimal regular isometric dilation of a sequence $\mathbf{T}\in\mathcal{DC}$ is in class $\mathcal{DP}$ and coextends $\mathbf{T}$.

Proof.

It is equivalent to show that if $\mathbf{V}=(V_{1},V_{2},\cdots)$ is the minimal regular isometric dilation of $\mathbf{T}\in\mathcal{DC}(\mathcal{H})$, then for any given $n\in\mathbb{N}$, $V_{n}$ is pure and douly commutes with $V_{m}$ for any $m\neq n$, and $\mathcal{H}$ is invariant for $V_{n}^{*}$. Assume $n=1$ without loss of generality, and put $\mathcal{E}=\mathfrak{D}_{T_{1}^{*}}$ and $\mathbf{T}^{\prime}=(T_{2},T_{3},\cdots)$. Since $\mathbf{T}$ is doubly commuting, we see that $\mathcal{E}$ is $\mathbf{T}^{\prime}$-joint reducing and the restriction $\mathbf{T}^{\prime}|_{\mathcal{E}}=(T_{2}|_{\mathcal{E}},T_{3}|_{\mathcal{E}},\cdots)$ of $\mathbf{T}^{\prime}$ on $\mathcal{E}$ is also doubly commuting. Let the sequence $\mathbf{S}=(S_{1},S_{2},\cdots)$, defined on a Hilbert space $\mathcal{F}\supseteq\mathcal{E}$, be the minimal regular isometric dilation of $\mathbf{T}^{\prime}|_{\mathcal{E}}$.

Sz.-Nagy and Foias’ functional model theory for single $C_{.0}$-contraction gives the following isometric embedding [43]

[TABLE]

By identifying $\mathcal{H}$ with the $M_{z}^{*}\otimes I_{\mathcal{E}}$-invariant subspace $V\mathcal{H}$ via the isometry $V$, one obtains a minimal isometric dilation $M_{z}\otimes I_{\mathcal{E}}$ of the contraction $T_{1}$, which is also a coextension. Set

[TABLE]

and

[TABLE]

It is routine to check that the sequence $\mathbf{W}=(W_{1},W_{2},\cdots)$ of isometries is a regular isometric dilation of $\mathbf{T}$. We claim that this dilation $\mathbf{W}$ is also minimal. Since $\mathbf{S}=(S_{1},S_{2},\cdots)$ is the minimal regular isometric dilation of $\mathbf{T}^{\prime}|_{\mathcal{E}}$, we have

[TABLE]

This together with $[\mathcal{H}]_{W_{1}}=H_{\mathcal{E}}^{2}(\mathbb{D})$ proves the claim. By the uniqueness of minimal regular isometric dilation, there exists a unitary operator $U$ that interwinds $\mathbf{V}$ and $\mathbf{W}$ and fixes vectors in $\mathcal{H}$. Therefore $V_{1}$ is pure and doubly commutes with $V_{m}\ (m\geq 2)$. Moreover,

[TABLE]

The proof is complete. ∎

We also record the following useful lemma concerning the minimal regular isometric dilation.

Lemma 2.2**.**

Let $\mathbf{T}$ be a sequence in class $\mathcal{DC}(\mathcal{H})$, and $\mathbf{V}\in\mathcal{DP}(\mathcal{K})$ be an isometric coextension of $\mathbf{T}$. Put

[TABLE]

Then the following conclusions hold:

• (1)

$\mathcal{H}_{n}\ (n\in\mathbb{N})$* is $\mathbf{V}^{*}$-joint invariant and $[\mathcal{H}]_{\mathbf{V}}$ is $\mathbf{V}$-joint reducing.*

• (2)

$P_{\mathcal{H}}V_{n}^{*}P_{\mathcal{H}_{n}}=V_{n}^{*}P_{\mathcal{H}}$* for each $n\in\mathbb{N}$.*

• (3)

Furthermore, if $\mathbf{V}$ is the minimal isometric dilation of $\mathbf{T}$, then

[TABLE]

Here, for a closed subspace $M$ of $\mathcal{K}$, $P_{M}$ denotes the orthogonal projection from $\mathcal{K}$ onto $M$.

Proof.

(1) Since the sequence $\mathbf{V}$ is doubly commuting, and $\mathcal{H}$ is joint invariant for $\mathbf{V}^{*}$, one obtains that for $n,m\in\mathbb{N}$ and $\alpha\in\mathbb{Z}_{+}^{(\infty)}$ with $\alpha_{n}=0$,

[TABLE]

[TABLE]

where

[TABLE]

This gives that each $\mathcal{H}_{n}$ is $\mathbf{V}^{*}$-joint invariant, and then

[TABLE]

is $\mathbf{V}$-joint reducing. (2) Since the sequences $\mathbf{T}$ and $\mathbf{V}$ are doubly commuting, and $\mathcal{H}$ is joint invariant for $\mathbf{V}^{*}$, we see that for $n\in\mathbb{N}$ and $\alpha\in\mathbb{Z}_{+}^{(\infty)}$ with $\alpha_{n}=0$,

[TABLE]

forcing

[TABLE]

It follows that for each $n\in\mathbb{N}$,

[TABLE]

and then

[TABLE]

(3) Write $\widetilde{\mathcal{H}}=\bigcap_{n=1}^{\infty}\mathcal{H}_{n}$. It is trivial that $\mathcal{H}\subseteq\widetilde{\mathcal{H}}$. It follows from (1) and (2) that $\widetilde{\mathcal{H}}$ is joint invariant for $\mathbf{V}^{*}$, and for each $n\in\mathbb{N}$,

[TABLE]

Assume that $x$ is an element in $\widetilde{\mathcal{H}}\ominus\mathcal{H}$. Then for each $n\in\mathbb{N}$,

[TABLE]

forcing $V_{n}^{*}x\in\widetilde{\mathcal{H}}\ominus\mathcal{H}$. Therefore, $\mathbf{V}^{*\alpha}x\in\widetilde{\mathcal{H}}\ominus\mathcal{H}$ for any $\alpha\in\mathbb{Z}_{+}^{(\infty)}$; that is, $x$ is orthogonal to the $\mathbf{V}$-invariant subspace $[\mathcal{H}]_{\mathbf{V}}$ generated by $\mathcal{H}$. Since $\mathbf{V}$ is the minimal isometric dilation of the sequence $\mathbf{T}$ defined on $\mathcal{H}$, we actually have $[\mathcal{H}]_{\mathbf{V}}=\mathcal{K}$. Thus $x=0$, and this proves $\mathcal{H}=\widetilde{\mathcal{H}}$. ∎

2.2. Some basic properties of vector-valued Hardy space and operator-valued function

Here we list some basic properties of vector-valued Hardy spaces and operator-valued functions, and the notations $\mathcal{E},\mathcal{F}$ and $\mathcal{G}$ will always denote some Hilbert spaces.

The vector-valued Hardy space $H_{\mathcal{E}}^{2}(\mathbb{D}_{2}^{\infty})$, over the domain $\mathbb{D}_{2}^{\infty}$ (connected open subset) in the Hilbert space $l^{2}$, consists of all $\mathcal{E}$-valued functions of form

[TABLE]

with each $x_{\alpha}\in\mathcal{E}$ and $\|F\|^{2}=\sum_{\alpha\in\mathbb{Z}_{+}^{(\infty)}}\|x_{\alpha}\|_{\mathcal{E}}^{2}<\infty$. By the Cauchy-Schwarz inequality, the above series converges pointwisely on $\mathbb{D}_{2}^{\infty}$ in $\mathcal{E}$-norm. We will follow the definition of holomorphic mapping given in [15] for any vector-valued function $F:\mathbb{D}_{2}^{\infty}\rightarrow\mathcal{X}$, where $\mathcal{X}$ is an arbitrary Banach space. Every function $F:\mathbb{D}_{2}^{\infty}\rightarrow\mathcal{E}$ in $H_{\mathcal{E}}^{2}(\mathbb{D}_{2}^{\infty})$ is then holomorphic in this sense.

The space $H_{\mathcal{E}}^{2}(\mathbb{D}_{2}^{\infty})$ can be considered as the tensor product of the Hardy space $H^{2}(\mathbb{D}_{2}^{\infty})$ and the Hilbert space $\mathcal{E}$ by identifying the vector-valued function $F\cdot x$ with the tensor product $F\otimes x$, where $F\in H^{2}(\mathbb{D}_{2}^{\infty})$ and $x\in\mathcal{E}$. Then the tuple of coordinate multiplication operators on $H_{\mathcal{E}}^{2}(\mathbb{D}_{2}^{\infty})$ has the form

[TABLE]

For simplicity, we often write only $\mathbf{M}_{\zeta}$ for this tuple. Moreover, one can expanse functions in $H_{\mathcal{E}}^{2}(\mathbb{D}_{2}^{\infty})$ with respect to any orthonormal basis $\{e_{k}\}_{k\in\mathbb{N}}$ of $\mathcal{E}$ as $\sum_{k=1}^{\infty}F_{k}\cdot e_{k}$, where each $F_{k}\in H^{2}(\mathbb{D}_{2}^{\infty})$ and $\sum_{k=1}^{\infty}\|F_{k}\|_{H^{2}(\mathbb{D}_{2}^{\infty})}^{2}<\infty$.

Let $\mathbf{K}_{\lambda}$ denote the reproducing kernel of $H^{2}(\mathbb{D}^{\infty}_{2})$ at the point $\lambda\in\mathbb{D}^{\infty}_{2}$. Recall that a subset $E$ of a Hilbert space $\mathcal{H}$ is said to be complete in $\mathcal{H}$ if $E$ spans a dense subspace of $\mathcal{H}$; that is, the orthocomplement $E^{\perp}$ of $E$ in $\mathcal{H}$ is $\{0\}$.

Lemma 2.3**.**

Suppose $F\in H_{\mathcal{E}}^{2}(\mathbb{D}_{2}^{\infty})$. Then the following conclusions hold:

• (1)

$\langle F,\mathbf{K}_{\lambda}\cdot x\rangle=\langle F(\lambda),x\rangle$* for any $\lambda\in\mathbb{D}_{2}^{\infty}$ and any $x\in\mathcal{E}$. Consequently, the set*

[TABLE]

is complete in $H_{\mathcal{E}}^{2}(\mathbb{D}_{2}^{\infty})$.

• (2)

If $F\neq 0$ and $M_{\zeta_{n}}^{*}F=\overline{\lambda_{n}}F$ for each $n\in\mathbb{N}$ and a sequence $\lambda=(\lambda_{1},\lambda_{2},\cdots)$ of complex numbers, then $\lambda\in\mathbb{D}_{2}^{\infty}$ and $F=\mathbf{K}_{\lambda}\cdot x$ for some $x\in\mathcal{E}$.

Proof.

(1) Assume $\|x\|=1$ without loss of generality and take any orthonormal basis $\{e_{k}\}_{k\in\mathbb{N}}$ of the subspace $\{x\}^{\perp}$ of $\mathcal{E}$. Expanse the function $F$ with respect to the orthonormal basis $\{x\}\cup\{e_{k}\}_{k\in\mathbb{N}}$ of $\mathcal{E}$ as

[TABLE]

Then we have

[TABLE]

and

[TABLE]

This proves (1).

(2) Expanse the function $F$ with respect to an orthonormal basis $\{e_{k}\}_{k\in\mathbb{N}}$ of $\mathcal{E}$ as $F=\sum_{k=1}^{\infty}F_{k}\cdot e_{k}$, then

[TABLE]

It follows that $M_{\zeta_{n}}^{*}F_{k}=\overline{\lambda_{n}}F_{k}$ for any $k,n\in\mathbb{N}$. This implies that all $F_{k}$ are in the orthocomplement of the invariant subspace generated by $\zeta_{1}-\lambda_{1}$, $\zeta_{2}-\lambda_{2}$, $\cdots$. Since $F\not=0,$ it follows that this invariant subspace is proper, and its orthocomplement is $\mathbb{C}\mathbf{K}_{\lambda}.$ Therefore, we have $\lambda\in\mathbb{D}_{2}^{\infty}$ and $F_{k}=c_{k}\mathbf{K}_{\lambda}$ with $\sum_{k=1}^{\infty}|c_{k}|^{2}<\infty$, and thus

[TABLE]

The proof is complete. ∎

The space $H_{\mathcal{B}(\mathcal{F},\mathcal{E})}^{\infty}(\mathbb{D}_{2}^{\infty})$ consists of all uniformly bounded holomorphic operator-valued functions $\Psi:\mathbb{D}_{2}^{\infty}\rightarrow\mathcal{B}(\mathcal{F},\mathcal{E})$. Here by $\Psi$ is uniformly bounded we mean $\|\Psi\|_{\infty}=\sup_{\zeta\in\mathbb{D}_{2}^{\infty}}\|\Psi(\zeta)\|<\infty$, and $\mathcal{B}(\mathcal{F},\mathcal{E})$ denotes the Banach space of bounded linear operators from $\mathcal{F}$ to $\mathcal{E}$. Every function $\Psi$ in $H_{\mathcal{B}(\mathcal{F},\mathcal{E})}^{\infty}(\mathbb{D}_{2}^{\infty})$ naturally induce a multiplication operator $M_{\Psi}$ as follows:

[TABLE]

where $\Psi F(\zeta)=\Psi(\zeta)F(\zeta)\ (\zeta\in\mathbb{D}_{2}^{\infty})$. It is clear that $M_{\Psi}$ is bounded of norm $\leq$ $\|\Psi\|_{\infty}$. We now claim that

[TABLE]

for any $\lambda\in\mathbb{D}_{2}^{\infty}$ and any $x\in\mathcal{E}$. By Lemma 2.3 (1), it suffices to show that for any fixed $\mu\in\mathbb{D}_{2}^{\infty}$ and $y\in\mathcal{F}$,

[TABLE]

Again by Lemma 2.3 (1), we have

[TABLE]

On the other hand,

[TABLE]

This proves the claim.

We are ready to prove the following.

Proposition 2.4**.**

Let $\mathbf{M}_{\zeta},\mathbf{M}_{\xi}$ be the tuple of coordinate multiplication operators on $H_{\mathcal{E}}^{2}(\mathbb{D}_{2}^{\infty})$ and $H_{\mathcal{F}}^{2}(\mathbb{D}_{2}^{\infty})$, respectively. If $T:H_{\mathcal{F}}^{2}(\mathbb{D}_{2}^{\infty})\rightarrow H_{\mathcal{E}}^{2}(\mathbb{D}_{2}^{\infty})$ is a bounded linear operator satisfying

[TABLE]

then there exists an operator-valued function $\Psi\in H_{\mathcal{B}(\mathcal{F},\mathcal{E})}^{\infty}(\mathbb{D}_{2}^{\infty})$ such that $T=M_{\Psi}$.

Proof.

By (2.1), for each $n\in\mathbb{N}$, each $\lambda\in\mathbb{D}_{2}^{\infty}$ and each $x\in\mathcal{E}$,

[TABLE]

Then by Lemma 2.3 (2), to any pair $\lambda,x$ there corresponds an element $y_{\lambda,x}\in\mathcal{F}$ such that

[TABLE]

and therefore $x\mapsto y_{\lambda,x}$ defines a linear operator $S_{\lambda}\in\mathcal{B}(\mathcal{E},\mathcal{F})$ for each $\lambda\in\mathbb{D}_{2}^{\infty}$. Now put $\Psi(\lambda)=S_{\lambda}^{*}$ ($\lambda\in\mathbb{D}_{2}^{\infty}$). It is routine to check that $\Psi:\mathbb{D}_{2}^{\infty}\rightarrow\mathcal{B}(\mathcal{F},\mathcal{E})$ is uniformly bounded and holomorphic with $\|\Psi\|_{\infty}\leq\|T\|$. Since $T^{*}$ and $M_{\Psi}^{*}$ coincide on the set $\{\mathbf{K}_{\lambda}\cdot x:\lambda\in\mathbb{D}_{2}^{\infty},x\in\mathcal{E}\}$ by (2.1), it follow from Lemma 2.3 (1) that $T=M_{\Psi}$. ∎

In [4], the infinite tensor product of a sequence of Hilbert space $\{\mathcal{H}_{n}\}_{n\in\mathbb{N}}$ with stabilizing sequence $\{e^{(n)}\}_{n\in\mathbb{N}}$ is introduced, where $e^{(n)}$ is a unit vector in $\mathcal{H}_{n}$ for each $n\in\mathbb{N}$. We remark that the Hilbert space structure of $H^{2}(\mathbb{D}_{2}^{\infty})$ coincides with the infinite tensor product

[TABLE]

of countably infinitely many Hardy spaces over $\mathbb{D}$ with stabilizing sequence $\{1\}_{n\in\mathbb{N}}$. So for any closed subspace $M$ of the Hardy space

[TABLE]

the infinite tensor product

[TABLE]

is exactly the $(M_{\zeta_{n+1}},M_{\zeta_{n+2}},\cdots)$-joint invariant subspace of $H^{2}(\mathbb{D}_{2}^{\infty})$ generated by $M(=M\otimes\mathbb{C}\otimes\mathbb{C}\cdots)$. For later use, we also note that the vector-valued Hardy space $H_{\mathcal{E}}^{2}(\mathbb{D}_{2}^{\infty})$ can be written as

[TABLE]

where

[TABLE]

One simplest class of vector-valued or operator-valued functions on $\mathbb{D}_{2}^{\infty}$ is the functions induced by those of one variable. More precisely, put $\tilde{f}(\zeta)=f(\zeta_{n})$ and $\widetilde{\theta}(\zeta)=\theta(\zeta_{n})$ ($\zeta\in\mathbb{D}_{2}^{\infty}$) for any $f\in H_{\mathcal{E}}^{2}(\mathbb{D})$, any $\theta\in H_{\mathcal{B}(\mathcal{F},\mathcal{E})}^{\infty}(\mathbb{D})$ and any $n\in\mathbb{N}$. Then we have $\tilde{f}\in H_{\mathcal{E}}^{2}(\mathbb{D}_{2}^{\infty})$ and $\widetilde{\theta}\in H_{\mathcal{B}(\mathcal{F},\mathcal{E})}^{\infty}(\mathbb{D}_{2}^{\infty})$. The multiplication operator $M_{\widetilde{\theta}}$ has the form

[TABLE]

with respect to the representation (2.2). The property of $M_{\widetilde{\theta}}$ thus relies heavily on that of $\theta$. For instance, $M_{\widetilde{\theta}}$ is isometric if and only if $M_{\theta}$ is isometric, if and only if $\theta$ is inner. Recall that $\theta\in H_{\mathcal{B}(\mathcal{F},\mathcal{E})}^{\infty}(\mathbb{D})$ is said to be inner if the boundary values of $\theta$, on some subset $E$ of the unit circle $\mathbb{T}$ with full measure, are isometries, which is defined to be the radial limit

[TABLE]

(see [30, 43] for instance). On the other hand, since $r\mathbb{D}^{\infty}\nsubseteqq\mathbb{D}_{2}^{\infty}$ for any $0<r<1$, the radial limits for bounded holomorphic functions on $\mathbb{D}_{2}^{\infty}$ do not make sense in general. In [36], E. Saksman and K. Seip defined boundary values for bounded holomorphic functions on $\mathbb{D}_{2}^{\infty}$ by taking quasi-radial limits instead. However, to avoid more discussion about the boundary behavior of functions on $\mathbb{D}_{2}^{\infty}$, we introduce an alternative definition that $\Psi\in H_{\mathcal{B}(\mathcal{F},\mathcal{E})}^{\infty}(\mathbb{D}_{2}^{\infty})$ is said to be inner if $M_{\Psi}$ is an isometry. Finally, applying Lemma 2.3 (1) to the function $\tilde{f}$ and (2.1) to the function $\widetilde{\theta}$, we have

• (i)

$\langle f,K_{a}\cdot x\rangle=\langle f(a),x\rangle$ for any $a\in\mathbb{D}$ and $x\in\mathcal{E}$. Consequently, the set

[TABLE]

is complete in $H_{\mathcal{E}}^{2}(\mathbb{D})$.

• (ii)

$M_{\theta}^{*}(K_{a}\cdot x)=K_{a}\cdot\theta(a)^{*}x$ for any $a\in\mathbb{D}$ and any $x\in\mathcal{E}$.

Here, $K_{a}$ denotes the reproducing kernel of $H^{2}(\mathbb{D})$ at the point $a\in\mathbb{D}$. Note that by (ii), for $a,b\in\mathbb{D}$ and $x,y\in\mathcal{E}$, we have

[TABLE]

The following lemma is also needed in the sequel.

Lemma 2.5**.**

Suppose $\theta\in H_{\mathcal{B}(\mathcal{F},\mathcal{E})}^{\infty}(\mathbb{D})$ and $\vartheta\in H_{\mathcal{B}(\mathcal{G},\mathcal{E})}^{\infty}(\mathbb{D})$. Then the following conclusions hold:

• (1)

Let $\mathcal{E}_{0}$ be a closed subspace of $\mathcal{E}$. Then $H_{\mathcal{E}_{0}}^{2}(\mathbb{D})$ is reducing for $M_{\theta}M_{\theta}^{*}$ if and only if $\mathcal{E}_{0}$ is invariant for $\theta(b)\theta(a)^{*}$ for any $a,b\in\mathbb{D}$. In this case, if $\theta$ is inner we have

[TABLE]

where $\mathcal{F}_{0}=\bigvee_{a\in\mathbb{D}}\theta(a)^{*}\mathcal{E}_{0}$.

• (2)

$M_{\theta}M_{\theta}^{*}=M_{\vartheta}M_{\vartheta}^{*}$* if and only if*

[TABLE]

for any $a,b\in\mathbb{D}$.

Proof.

(1) By (i), $H_{\mathcal{E}_{0}}^{2}(\mathbb{D})$ is reducing for $M_{\theta}M_{\theta}^{*}$ if and only if

[TABLE]

for any $a,b\in\mathbb{D}$, any $x\in\mathcal{E}_{0}$ and any $y\in\mathcal{E}\ominus\mathcal{E}_{0}$. This together with (2.3) gives that $H_{\mathcal{E}_{0}}^{2}(\mathbb{D})$ is reducing for $M_{\theta}M_{\theta}^{*}$ if and only if $\theta(b)\theta(a)^{*}\mathcal{E}_{0}\subseteq\mathcal{E}_{0}$ for any $a,b\in\mathbb{D}$. For the latter conclusion, note that by (ii), $M_{\theta}^{*}H_{\mathcal{E}_{0}}^{2}(\mathbb{D})$ is a dense subspace of $H_{\mathcal{F}_{0}}^{2}(\mathbb{D})$. Now suppose in addition $\theta$ is inner. Then $M_{\theta}M_{\theta}^{*}$ is an orthogonal projection on $H_{\mathcal{E}}^{2}(\mathbb{D})$. Since $H_{\mathcal{E}_{0}}^{2}(\mathbb{D})$ is reducing for $M_{\theta}M_{\theta}^{*}$, $M_{\theta}M_{\theta}^{*}H_{\mathcal{E}_{0}}^{2}(\mathbb{D})$ is closed, forcing $M_{\theta}^{*}H_{\mathcal{E}_{0}}^{2}(\mathbb{D})$ to be also closed. This proves (1).

(2) By (i), $M_{\theta}M_{\theta}^{*}=M_{\vartheta}M_{\vartheta}^{*}$ if and only if

[TABLE]

for any $a,b\in\mathbb{D}$ and any $x,y\in\mathcal{E}$. Then (2) follows from (2.3). ∎

2.3. Some preparations for proofs

Lemma 2.6**.**

If $\mathbf{T}$ is a doubly commuting sequence of contractions on $\mathcal{H}$ and $\mathbf{V}$ is a doubly commuting isometric coextension of $\mathbf{T}$, then for each $\lambda\in\mathbb{D}^{\infty}$ and each $x\in\mathcal{H}$,

[TABLE]

Proof.

For any fixed $\lambda\in\mathbb{D}^{\infty}$, set

[TABLE]

and

[TABLE]

It is clear that $\mathbf{W}$ is an isometric coextension of $\mathbf{S}$. Since for $x\in\mathcal{H}$,

[TABLE]

and

[TABLE]

it suffices to prove that for $n\in\mathbb{N}$,

[TABLE]

We show this by induction on $n$. For $n=1$, one has

[TABLE]

Assume that (2.4) holds for $n=k$. Since $\mathbf{S}$ and $\mathbf{W}$ are doubly commuting, we have that for $x\in\mathcal{H}$,

[TABLE]

Thus, (2.4) also holds for $n=k+1$. This completes the proof. ∎

Lemma 2.7**.**

If $(T_{1},\cdots,T_{n})$ is a doubly commuting $n$-tuple of $C_{.0}$ contractions on a Hilbert space $\mathcal{H}$, then

[TABLE]

Equivalently,

[TABLE]

Proof.

The equivalence is guaranteed by the fact that $(T_{1},\cdots,T_{n})$ is doubly commuting and then the operator

[TABLE]

is self-adjoint. We prove this lemma by induction on $n$. For $n=1$, let $T$ be a contractions of class $C_{.0}$, and assume $f\in\mathcal{H}$ so that $D_{\varphi_{a}(T)^{*}}f=0$ for any $a\in\mathbb{D}$. As in the proof of Lemma 2.1, $T$ has a coextension to the Hardy shift $M_{z}$ on the vector-valued Hardy space $H_{\mathfrak{D}_{T^{*}}}^{2}(\mathbb{D})$. Since $M_{\varphi_{a}}\ (a\in\mathbb{D})$ is isometric on $H_{\mathfrak{D}_{T^{*}}}^{2}(\mathbb{D})$, we have

[TABLE]

Note that $M_{\varphi_{a}}M_{\varphi_{a}}^{*}$ is the orthogonal projection onto $\mathrm{Ran}\ M_{\varphi_{a}}$ for each $a\in\mathbb{D}$, the above identity yields $f=M_{\varphi_{a}}g_{a}=\varphi_{a}\cdot g_{a}$ for some $g_{a}\in H_{\mathfrak{D}_{T^{*}}}^{2}(\mathbb{D})$. This gives

[TABLE]

forcing $f=0$, which proves the case $n=1$. Now assume that the conclusion holds for $n=k$, and let $(T_{1},\cdots,T_{k+1})$ be a doubly commuting tuple of $C_{.0}$ contractions. Then

[TABLE]

By induction, the proof is complete. ∎

Suppose that $\mathbf{V}=(V_{1},V_{2},\cdots)$ is a sequence in class $\mathcal{DP}$. Recall that for any $\lambda=(\lambda_{1},\lambda_{2},\cdots)\in\mathbb{D}^{\infty}$, the defect space $\mathfrak{D}_{\Phi_{\lambda}(\mathbf{V})^{*}}$ of the sequence $\Phi_{\lambda}(\mathbf{V})^{*}$ is the closure of the range of the defect operator

[TABLE]

Since $\mathbf{V}$ is doubly commuting, $\{I-\varphi_{\lambda_{n}}(V_{n})\varphi_{\lambda_{n}}(V_{n})^{*}\}_{n\in\mathbb{N}}$ is a commuting sequence of orthogonal projections, and then $D_{\Phi_{\lambda}(\mathbf{V})^{*}}$ is also an orthogonal projections onto the subspace

[TABLE]

Note that $x\in\mathrm{Ker}\ \varphi_{\lambda_{n}}(V_{n})^{*}$ if and only if $V_{n}^{*}x=\overline{\lambda_{n}}x\ (n\in\mathbb{N})$, nonzero elements in defect space $\mathfrak{D}_{\Phi_{\lambda}(\mathbf{V})^{*}}$ exactly coincide with the set of joint eigenvectors of the sequence $\mathbf{V}^{*}$ corresponding to the joint eigenvalue $\overline{\lambda}=(\overline{\lambda_{1}},\overline{\lambda_{2}},\cdots)$.

Lemma 2.8**.**

Suppose $\mathbf{V}\in\mathcal{DP}$. If $x$ is a nonzero element in the defect space $\mathfrak{D}_{\Phi_{\lambda}(\mathbf{V})^{*}}$ of the sequence $\Phi_{\lambda}(\mathbf{V})^{*}$ for some $\lambda\in\mathbb{D}^{\infty}$, then $[x]_{\mathbf{V}}(=[\{x\}]_{\mathbf{V}})$ is $\mathbf{V}$-joint reducing, and $\mathbf{V}|_{[x]_{\mathbf{V}}}$ is jointly unitarily equivalent to the sequence $\Phi_{\lambda}(\mathbf{M_{\zeta}})$, where $\mathbf{M_{\zeta}}$ is the tuple of coordinate multiplication operators on the Hardy space $H^{2}(\mathbb{D}_{2}^{\infty})$.

Proof.

Assume $\lambda=\mathbf{0}=(0,0,\cdots)$ and $\|x\|=1$ without loss of generality. Since $x\in\mathfrak{D}_{\mathbf{V}^{*}}$, we have that for $n,m\in\mathbb{N}$ and $\alpha\in\mathbb{Z}_{+}^{(\infty)}$,

[TABLE]

[TABLE]

This implies that $[x]_{\mathbf{V}}$ is $\mathbf{V}$-joint reducing. The rest of proof is given by defining a linear map $U$ from $\mathcal{P}_{\infty}$ to $[x]_{\mathbf{V}}$ as follows:

[TABLE]

where $\mathcal{P}_{\infty}=\mathbb{C}[\zeta_{1},\zeta_{2},\cdots]$, the polynomial ring in countably infinitely many variables, which is dense in $H^{2}(\mathbb{D}_{2}^{\infty})$ [29]. It is routine to check that $U$ can be extend to a unitary operator from $H^{2}(\mathbb{D}_{2}^{\infty})$ onto $[x]_{\mathbf{V}}$, and the tuple $\mathbf{M_{\zeta}}$ of coordinate multiplication operators is jointly unitarily equivalent to $\mathbf{V}|_{[x]_{\mathbf{V}}}$ via this unitary operator. ∎

Lemma 2.9**.**

Let $\lambda$ be a point in $\mathbb{D}^{\infty}$ and $\mathbf{M}_{\zeta}$ the tuple of coordinate multiplication operators on $H^{2}(\mathbb{D}^{\infty}_{2})$. Then $\mathfrak{D}_{\Phi_{\lambda}(\mathbf{M}_{\zeta})^{*}}=\{0\}$ if and only if $\lambda\notin\mathbb{D}_{2}^{\infty}$.

Proof.

By comments before the previous lemma, we have

[TABLE]

and hence

[TABLE]

where $\widetilde{\varphi_{\lambda_{n}}}(\zeta)=\varphi_{\lambda_{n}}(\zeta_{n})\ (\zeta\in\mathbb{D}^{\infty}_{2})$. By [14, Proposition 4.5],

[TABLE]

if and only if $\lambda\notin\mathbb{D}_{2}^{\infty}$. This completes the proof. ∎

By an irreducible family of operators we mean that these operators has no nontrivial joint reducing subspaces.

Lemma 2.10**.**

The tuple $\mathbf{M}_{\zeta}$ of coordinate multiplication operators on $H^{2}(\mathbb{D}^{\infty}_{2})$ is irreducible.

Proof.

It suffices to prove that for any orthogonal projection $P$ that commutes with every coordinate multiplication operator, we have $P1=1$ or $Q1=1$, where $Q=I-P$. Put $Q1=\sum_{\alpha\in\mathbb{Z}_{+}^{(\infty)}}c_{\alpha}\zeta^{\alpha}$. Since

[TABLE]

we have

[TABLE]

Note that $Q1=(I-P)1=1-P1$, the proof is complete. ∎

Lemma 2.11**.**

Let $\mathcal{H}$ and $\mathcal{K}$ be two Hilbert spaces, and $\mathcal{T}$ an irreducible family of bounded linear operators on $\mathcal{H}$. Then any bounded linear operator on $\mathcal{H}\otimes\mathcal{K}$ that doubly commutes with the family $\{T\otimes I_{\mathcal{K}}:T\in\mathcal{T}\}$ is of form $I_{\mathcal{H}}\otimes S$ for some $S\in\mathcal{B}(\mathcal{K})$.

In particular, any joint reducing subspace for the family $\{T\otimes I_{\mathcal{K}}:T\in\mathcal{T}\}$ is of form $\mathcal{H}\otimes M$ for some closed subspace $M$ of $\mathcal{K}$.

Proof.

Since the family $\mathcal{T}$ of operators has no nontrivial joint reducing subspace, the von Neumann algebra generated by $\mathcal{T}$ is the entire $\mathcal{B}(\mathcal{H})$ by the double commutant theorem [12]. This gives

[TABLE]

where $\mathcal{V}^{*}(\{T\otimes I_{\mathcal{K}}:T\in\mathcal{T}\})$ denotes the von Neumann algebra generated by the family $\{T\otimes I_{\mathcal{K}}:T\in\mathcal{T}\}$. If a bounded linear operator on $\mathcal{H}\otimes\mathcal{K}$ doubly commutes with the family $\{T\otimes I_{\mathcal{K}}:T\in\mathcal{T}\}$, then it also commutes with the algebra $\mathcal{B}(\mathcal{H})\otimes\mathbb{C}I_{\mathcal{K}}$, and therefore has form $I_{\mathcal{H}}\otimes S$ for some bounded linear operator $S$ on $\mathcal{K}$ (see [44, pp. 184]). ∎

Lemma 2.12**.**

Suppose $\mathbf{V}\in\mathcal{DP}(\mathcal{H})$. Then for any $\varepsilon>0$ and $x\in\mathcal{H}$, there exists a sequence $(k_{1},k_{2},\cdots)$ of positive integers such that

[TABLE]

where $\mathbf{W}=(V_{1}^{k_{1}},V_{2}^{k_{2}},\cdots)$.

Proof.

Take an arbitrary positive number $\varepsilon$ and assume $\|x\|=1$ without loss of generality. Since each $V_{n}\ (n\in\mathbb{N})$ is pure, there exists a sequence $(k_{1},k_{2},\cdots)$ of positive integers, such that

[TABLE]

Rewrite $(W_{1},W_{2},\cdots)=(V_{1}^{k_{1}},V_{2}^{k_{2}},\cdots)$. For each $n\in\mathbb{N}$, we have

[TABLE]

and then by induction and (2.5),

[TABLE]

Setting $n\rightarrow\infty$ in the above inequality, we obtain the desired conclusion. ∎

3. Dilation theory

In this section, we will give some operator-theoretical characterization for sequences in class $\mathcal{DC}$. Recall that a sequence $\mathbf{T}\in\mathcal{DC}(\mathcal{H})$ is said to has a decomposition of quasi-Beurling type if there exists an orthogonal decomposition $\mathcal{H}=\bigoplus_{\gamma}\mathcal{H}_{\gamma}$ of the Hilbert space $\mathcal{H}$, such that each $\mathcal{H}_{\gamma}$ is $\mathbf{T}$-joint reducing and $\mathbf{T}|_{\mathcal{H}_{\gamma}}$ is of quasi-Beurling type.

The main results in this section are restated as follows. Theorem 1.3 Suppose $\mathbf{T}\in\mathcal{DC}(\mathcal{H})$. The followings are equivalent:

• (1)

$\mathbf{T}$* has a decomposition of quasi-Beurling type;*

• (2)

for each $x\in\mathcal{H}$, there exists a sequence $\lambda^{(1)},\lambda^{(2)},\cdots$ of points in $\mathbb{D}^{\infty}$, such that

[TABLE]

• (3)

$\bigvee_{\lambda\in\mathbb{D}^{\infty}}\mathfrak{D}_{\Phi_{\lambda}(\mathbf{T})^{*}}=\mathcal{H}$.

In fact, the sequence $\lambda^{(1)},\lambda^{(2)},\cdots$ in condition (2) can be chosen to be independent of $x\in\mathcal{H}$.

Corollary 1.4 Suppose $\mathbf{T}\in\mathcal{DC}(\mathcal{H})$. The followings are equivalent:

• (1)

$\mathbf{T}$* is of Beurling type;*

• (2)

the minimal regular isometric dilation of $\mathbf{T}$ is jointly unitarily equivalent to the tuple $\mathbf{M}_{\zeta}$ of coordinate multiplication operators on a vector-valued Hardy space $H_{\mathcal{E}}^{2}(\mathbb{D}_{2}^{\infty})$;

• (3)

for each $x\in\mathcal{H}$, $\|x\|^{2}=\sum_{\alpha\in\mathbb{Z}_{+}^{(\infty)}}\|D_{\mathbf{T}^{*}}\mathbf{T}^{*\alpha}x\|^{2}$;

• (4)

$\bigvee_{\lambda\in\mathbb{D}_{2}^{\infty}}\mathfrak{D}_{\Phi_{\lambda}(\mathbf{T})^{*}}=\mathcal{H}$.

Proof of Theorem 1.3. (1)$\Rightarrow$(2). Suppose that there exists an (countable) orthogonal decomposition $\mathcal{H}=\bigoplus_{\gamma}\mathcal{H}_{\gamma}$ of the Hilbert space $\mathcal{H}$, such that each $\mathcal{H}_{\gamma}$ is $\mathbf{T}$-joint reducing and $\mathbf{T}|_{\mathcal{H}_{\gamma}}$ is of quasi-Beurling type. Take an arbitrary element $x$ in $\mathcal{H}$, and let $x_{\gamma}$ be the orthogonal projection of $x$ into the subspace $\mathcal{H}_{\gamma}$. Then $\|x\|^{2}=\sum_{\gamma}\|x_{\gamma}\|^{2}$. The implication (1)$\Rightarrow$(2) is proved once we show that for each $\gamma$, there corresponds a point $\lambda_{\gamma}\in\mathbb{D}^{\infty}$ such that

[TABLE]

Since each $\mathcal{H}_{\gamma}$ is $\mathbf{T}$-joint reducing, we see $\mathbf{T}|_{\mathcal{H}_{\gamma}}\in\mathcal{DC}(\mathcal{H}_{\gamma})$. Fix an index $\gamma$ and let $\mathbf{V}\in\mathcal{DP}(\mathcal{K})$ be the minimal regular isometric dilation of $\mathbf{T}|_{\mathcal{H}_{\gamma}}$. Then by Lemma 2.1, $\mathbf{V}$ is a coextension of $\mathbf{T}|_{\mathcal{H}_{\gamma}}$. Since $\mathbf{T}|_{\mathcal{H}_{\gamma}}$ is of quasi-Beurling type, there is a point $\lambda\in\mathbb{D}^{\infty}$ such that $\mathbf{W}=(W_{1},W_{2},\cdots)=\Phi_{\lambda}(\mathbf{V})$ is of Beurling type; that is, $[\mathfrak{D}_{\mathbf{W}^{*}}]_{\mathbf{W}}=\mathcal{K}$. Rewrite $\mathcal{E}=\mathfrak{D}_{\mathbf{W}^{*}}$. Since $\mathbf{W}$ is doubly commuting and $\mathcal{E}=\bigcap_{n=1}^{\infty}\mathrm{Ker}\ W_{n}^{*}$, we have that the family $\{\mathbf{W}^{\alpha}\mathcal{E}\}_{\alpha\in\mathbb{Z}_{+}^{(\infty)}}$ of subspaces are pairwise orthogonal, and therefore

[TABLE]

It is easy to see that for each $\alpha\in\mathbb{Z}_{+}^{(\infty)}$, the operator $\mathbf{W}^{\alpha}D_{\mathbf{W}^{*}}\mathbf{W}^{*\alpha}$ is exactly the orthogonal projection onto $\mathbf{W}^{\alpha}\mathcal{E}$, and then

[TABLE]

Also note that $\mathbf{W}$ is an isometrical coextension of $\Phi_{\lambda}(\mathbf{T})$, by Lemma 2.6 we have

[TABLE]

It is clear that the point $\lambda_{\gamma}$ that appears in (3.1) only depend the subspace $\mathcal{H}_{\gamma}$, and hence not on the choice of $x$.

(2)$\Rightarrow$(3). Fix $x\in\mathcal{H}\ominus\left(\bigvee_{\lambda\in\mathbb{D}^{\infty}}\mathfrak{D}_{\Phi_{\lambda}(\mathbf{T})^{*}}\right)$. We will show that for any given $\mu=(\mu_{1},\mu_{2},\cdots)\in\mathbb{D}^{\infty}$ and $\alpha=(\alpha_{1},\cdots,\alpha_{n},0,0,\cdots)\in\mathbb{Z}_{+}^{(\infty)}\ (n\in\mathbb{N})$,

[TABLE]

so that $x=0$ by the assumption in (2).

Put $\widetilde{\mathbf{T}}=(T_{n+1},T_{n+2},\cdots)$, $\tilde{\mu}=(\mu_{n+1},\mu_{n+2},\cdots)$, and let $(\lambda,\tilde{\mu})$ denote the sequence

[TABLE]

for $\lambda=(\lambda_{1},\cdots,\lambda_{n})\in\mathbb{D}^{n}$. Since $x$ is orthogonal to

[TABLE]

for each $\lambda\in\mathbb{D}^{n}$, we have

[TABLE]

Note that for each $\lambda=(\lambda_{1},\cdots,\lambda_{n})\in\mathbb{D}^{n}$,

[TABLE]

and therefore (3.2) gives $D_{\Phi_{\tilde{\mu}}(\widetilde{\mathbf{T}})^{*}}x\in\mathrm{Ker}\ D_{\varphi_{\lambda_{1}}(T_{1})^{*}}\cdots D_{\varphi_{\lambda_{n}}(T_{n})^{*}}$. This together with Lemma 2.7 implies $D_{\Phi_{\tilde{\mu}}(\widetilde{\mathbf{T}})^{*}}x=0$. Since the sequence $\mathbf{T}$ is doubly commuting, $D_{\Phi_{\tilde{\mu}}(\widetilde{\mathbf{T}})^{*}}$ commutes with $\Phi^{\alpha}_{\mu}(\mathbf{T})^{*}$ on $\mathcal{H}$, which gives

[TABLE]

(3)$\Rightarrow$(1). In order to make this part of the proof more accessible, we divide it into several steps.

Step I. We give the construction of the subspaces $\mathcal{H}_{\gamma}$ of the Hilbert space $\mathcal{H}$, and prove that each $\mathcal{H}_{\gamma}$ is $\mathbf{T}$-joint reducing.

Define a binary relation $\sim$ on $\mathbb{D}^{\infty}$ as follows: for two points $\lambda=(\lambda_{1},\lambda_{2},\cdots)$ and $\mu=(\mu_{1},\mu_{2},\cdots)$ in $\mathbb{D}^{\infty}$, set

[TABLE]

It is straightforward to see that the binary relation $\sim$ is an equivalence relation on $\mathbb{D}^{\infty}$. The set of $\sim$-equivalence classes is denoted by $\Delta$. For $\gamma\in\Delta$, put

[TABLE]

Now we will show that $\mathcal{H}_{\gamma}$ is $\mathbf{T}$-joint reducing for any given $\gamma\in\Delta$. Put $\mathbf{T}^{\prime}=(T_{2},T_{3},\cdots)$ and $\gamma^{\prime}=\{(\lambda_{2},\lambda_{3},\cdots):\lambda=(\lambda_{1},\lambda_{2},\cdots)\in\gamma\}$. Since the sequence $\mathbf{T}$ is doubly commuting, we have that the defect operators $D_{\varphi_{a}(T_{1})^{*}}$ and $D_{\Phi_{\lambda}(\mathbf{T}^{\prime})^{*}}$ commute with each other for any $a\in\mathbb{D}$ and $\lambda\in\mathbb{D}^{\infty}$. Also, it is easy to see $\gamma=\mathbb{D}\times\gamma^{\prime}$; that is,

[TABLE]

It follows from Lemma 2.7 that

[TABLE]

and therefore

[TABLE]

This implies that $\mathcal{H}_{\gamma}$ is reducing for $T_{1}$. Similarly, $\mathcal{H}_{\gamma}$ is $T_{n}$-reducing for each $n\geq 2$.

Step II. We prove $\mathcal{H}=\bigoplus_{\gamma\in\Delta}\mathcal{H}_{\gamma}$.

By the assumption in (3), we have $\mathcal{H}=\bigvee_{\gamma\in\Delta}\mathcal{H}_{\gamma}$. It remains to show that the subspaces $\mathcal{H}_{\gamma}\ (\gamma\in\Delta)$ are pairwise orthogonal.

Let $\mathbf{V}=(V_{1},V_{2},\cdots)\in\mathcal{DP}(\mathcal{K})$ be the minimal regular isometric dilation of $\mathbf{T}$. Put

[TABLE]

for $\gamma\in\Delta$. Applying the argument in step I, we see that each $\mathcal{K}_{\gamma}$ is $\mathbf{V}$-joint reducing.

For any closed subspace $M$ of the Hilbert space $\mathcal{K}$, let $P_{M}$ denote the orthogonal projection from $\mathcal{K}$ onto $M$. We first prove the following claims. (a) For each $\gamma\in\Delta$, $\mathcal{H}_{\gamma}\subseteq\overline{P_{\mathcal{H}}\mathcal{K}_{\gamma}}$. Assume that $x\in\mathcal{H}$ is orthogonal to $P_{\mathcal{H}}\mathcal{K}_{\gamma}$. Then $x$ is orthogonal to $\mathcal{K}_{\gamma}$, which implies

[TABLE]

It follows from Lemma 2.1 that $\mathbf{V}$ is a coextension of $\mathbf{T}$. Then by Lemma 2.6 we have

[TABLE]

Therefore, $x$ is orthogonal to $\mathcal{H}_{\gamma}$. This gives $\mathcal{H}\ominus\overline{P_{\mathcal{H}}\mathcal{K}_{\gamma}}\subseteq\mathcal{H}\ominus\mathcal{H}_{\gamma}$, and then claim (a) is proved.

(b) The subspaces $\mathcal{K}_{\gamma}\ (\gamma\in\Delta)$ of $\mathcal{K}$ are pairwise orthogonal. Assume that $x\in\mathfrak{D}_{\Phi_{\lambda}(\mathbf{V})^{*}}\neq\{0\}$ for some $\lambda\in\mathbb{D}^{\infty}$. It suffices to show that for any $\mu\in\mathbb{D}^{\infty}$ not equivalent to $\lambda$,

[TABLE]

We first consider the case $\lambda=\mathbf{0}$. Then we have $\mu\notin\mathbb{D}_{2}^{\infty}$ in this case. By Lemma 2.8, $[x]_{\mathbf{V}}$ is $\mathbf{V}$-joint reducing, and $\mathbf{V}|_{[x]_{\mathbf{V}}}$ is jointly unitarily equivalent to the tuple $\mathbf{M_{\zeta}}$ of coordinate multiplication operators on the Hardy space $H^{2}(\mathbb{D}_{2}^{\infty})$. Then $[x]_{\mathbf{V}}$ is reducing for $D_{\Phi_{\mu}(\mathbf{V})^{*}}$, and $D_{\Phi_{\mu}(\mathbf{V})^{*}}|_{[x]_{\mathbf{V}}}$ is jointly unitarily equivalent to $D_{\Phi_{\mu}(\mathbf{M_{\zeta}})^{*}}$. It follows from Lemma 2.9 that $D_{\Phi_{\mu}(\mathbf{M_{\zeta}})^{*}}=0$, and hence $D_{\Phi_{\mu}(\mathbf{V})^{*}}$ vanishes on $[x]_{\mathbf{V}}$. The case for general $\lambda\in\mathbb{D}^{\infty}$ is proved by replacing the sequence $\mathbf{V}$ with $\Phi_{\lambda}(\mathbf{V})$ in the previous argument. By claims (a) and (b), it suffices to show that $P_{\mathcal{H}}P_{\mathcal{K}_{\gamma}}=P_{\mathcal{K}_{\gamma}}P_{\mathcal{H}}$ for any $\gamma\in\Delta$. Put

[TABLE]

for $n\in\mathbb{N}$. We also claim that

(c) for each $n\in\mathbb{N}$ and $\gamma\in\Delta$, $P_{\mathcal{H}_{n}}P_{\mathcal{K}_{\gamma}}=P_{\mathcal{K}_{\gamma}}P_{\mathcal{H}_{n}}$.

Assuming this claim, the desired conclusion follows immediately from Lemma 2.2 (3). Below we will give its proof.

Without loss of generality, we only prove for $n=1$. Put $\mathbf{V}^{\prime}=(V_{2},V_{3},\cdots)$. By Lemma 2.2 (1), $\mathcal{H}_{1}$ is joint invariant for $\mathbf{V}^{*}$. This gives that $\mathcal{H}_{1}$ is $\mathbf{V}^{\prime}$-joint reducing, and then for each $\lambda\in\mathbb{D}^{\infty}$,

[TABLE]

Applying a similar argument in step I, one obtains

[TABLE]

where $\gamma^{\prime}=\{(\lambda_{2},\lambda_{3},\cdots):\lambda=(\lambda_{1},\lambda_{2},\cdots)\in\gamma\}$. Thus, claim (c) follows.

Step III. We prove that the restriction $\mathbf{T}|_{\mathcal{H}_{\gamma}}$ on each nonzero $\mathcal{H}_{\gamma}\ (\gamma\in\Delta)$ is of quasi-Beurling type, and thus complete the proof.

Suppose that $\mathcal{H}_{\gamma}$ is nonzero for some $\gamma\in\Delta$. By the claims in step II of the proof, we see

[TABLE]

Then the minimal regular isometric dilation of the sequence $\mathbf{T}|_{\mathcal{H}_{\gamma}}$ is $\mathbf{V}|_{\mathcal{K}_{\gamma}}$. It remains to prove that $\mathbf{V}|_{\mathcal{K}_{\gamma}}$ is of quasi-Beurling type; that is,

[TABLE]

for some $\lambda\in\gamma$.

Fix $\lambda=(\lambda_{1},\lambda_{2},\cdots)\in\gamma$ satisfying $\mathfrak{D}_{\Phi_{\lambda}(\mathbf{V})^{*}}\neq\{0\}$. Since $\mathcal{K}_{\gamma}$ is $\mathbf{V}$-joint reducing (see step II), one has $[\mathfrak{D}_{\Phi_{\lambda}(\mathbf{V})^{*}}]_{\mathbf{V}}\subseteq\mathcal{K}_{\gamma}$. For the converse inclusion, take an arbitrary nonzero element $x\in\mathcal{K}_{\gamma}$. We will show $x\in[\mathfrak{D}_{\Phi_{\lambda}(\mathbf{V})^{*}}]_{\mathbf{V}}$. Since $x\in\mathfrak{D}_{\Phi_{\mu}(\mathbf{V})^{*}}$ for some $\mu=(\mu_{1},\mu_{2},\cdots)\in\gamma$, Lemma 2.8 implies that $[x]_{\mathbf{V}}$ is $\mathbf{V}$-joint reducing, and $\mathbf{V}|_{[x]_{\mathbf{V}}}$ is jointly unitarily equivalent to the sequence $\Phi_{\mu}(\mathbf{M_{\zeta}})$, where $\mathbf{M_{\zeta}}$ is the tuple of coordinate multiplication operators on the Hardy space $H^{2}(\mathbb{D}_{2}^{\infty})$. Then $[x]_{\mathbf{V}}$ is reducing for $D_{\Phi_{\lambda}(\mathbf{V})^{*}}$, and $D_{\Phi_{\lambda}(\mathbf{V})^{*}}|_{[x]_{\mathbf{V}}}$ is jointly unitarily equivalent to

[TABLE]

Note that for each $n\in\mathbb{N}$, $\varphi_{\lambda_{n}}\circ\varphi_{\mu_{n}}=c_{n}\varphi_{\eta_{n}}$ for some unimodular constant $c_{n}$ and $\eta_{n}\in\mathbb{D}$, we have

[TABLE]

The fact $\lambda\sim\mu$ yields $\eta=(\eta_{1},\eta_{2},\cdots)\in\mathbb{D}_{2}^{\infty}$, and then Lemma 2.9 gives $D_{\Phi_{\eta}(\mathbf{M_{\zeta}})^{*}}\neq 0$. Thus, there exists a nonzero element $y\in[x]_{\mathbf{V}}\bigcap\mathfrak{D}_{\Phi_{\lambda}(\mathbf{V})^{*}}$. Again by Lemma 2.8, $[y]_{\mathbf{V}}$ is a joint reducing subspace of $[x]_{\mathbf{V}}$ for the sequence $\mathbf{V}|_{[x]_{\mathbf{V}}}$. This together with Lemma 2.10 implies

[TABLE]

and then the proof is complete. ∎

From the proof of Theorem 1.3, we actually obtain the following collection of results, which will be used later.

Corollary 3.1**.**

Let $\mathbf{T}$ be a sequence in class $\mathcal{DC}(\mathcal{H})$, and $\mathbf{V}\in\mathcal{DP}(\mathcal{K})$ the minimal regular isometric dilation of $\mathbf{T}$. We also let $\sim$ be the equivalence relation on $\mathbb{D}^{\infty}$ so that for $\lambda,\mu\in\mathbb{D}^{\infty}$, $\lambda\sim\mu$ means that

[TABLE]

• (1)

If $\mathbf{T}$ is of Beurling type, then $\mathcal{K}$ has an orthogonal decomposition

[TABLE]

and for each $x\in\mathcal{H}$,

[TABLE]

• (2)

If $\lambda,\mu\in\mathbb{D}^{\infty}$ are not $\sim$-equivalent, then the defect spaces $\mathfrak{D}_{\Phi_{\lambda}(\mathbf{T})^{*}}$ and $\mathfrak{D}_{\Phi_{\mu}(\mathbf{T})^{*}}$ are mutually orthogonal.

• (3)

For any $\sim$-equivalence class $\gamma$, put $\mathcal{H}_{\gamma}=\bigvee_{\lambda\in\gamma}\mathfrak{D}_{\Phi_{\lambda}(\mathbf{T})^{*}}$, $\mathcal{K}_{\gamma}=\bigvee_{\lambda\in\gamma}\mathfrak{D}_{\Phi_{\lambda}(\mathbf{V})^{*}}$. Then $\mathcal{K}_{\gamma}=[\mathfrak{D}_{\Phi_{\mu}(\mathbf{V})^{*}}]_{\mathbf{V}}$ for each $\mu\in\gamma$, $\mathbf{V}|_{\mathcal{K}_{\gamma}}$ is the minimal regular isometric dilation of $\mathbf{T}|_{\mathcal{H}_{\gamma}}$, and $\mathcal{H}_{\gamma},\mathcal{K}_{\gamma}$ are joint reducing for $\mathbf{T}$ and $\mathbf{V}$, respectively.

We are ready to prove Corollary 1.4. Proof of Corollary 1.4. (1)$\Rightarrow$(2) and (1)$\Rightarrow$(3). See Corollary 3.1 (1).

(2)$\Rightarrow$(1). Obvious.

(3)$\Rightarrow$(4). This implication follows from the proof of Theorem 1.3 and the fact that $\mathbb{D}_{2}^{\infty}$ coincides with the Cartesian product

[TABLE]

(4)$\Rightarrow$(1). Let $\mathbf{V}\in\mathcal{DP}(\mathcal{K})$ be the minimal regular isometric dilation of $\mathbf{T}$. Note that $\mathbb{D}_{2}^{\infty}$ is a $\sim$-equivalence class that contains $\mathbf{0}$ and $\mathcal{H}=\mathcal{H}_{\mathbb{D}_{2}^{\infty}}$. It follows from Corollary 3.1 (3) that $\mathcal{K}=\mathcal{K}_{\mathbb{D}_{2}^{\infty}}=[\mathfrak{D}_{\mathbf{V}^{*}}]_{\mathbf{V}}$. The proof is complete. ∎

It follows immediately from Corollary 1.4 that a sequence $\mathbf{T}\in\mathcal{DC}$ is of quasi-Beurling type if and only if the minimal regular isometric dilation of $\mathbf{T}$ is jointly unitarily equivalent to the sequence $\Phi_{\lambda}(\mathbf{M}_{\zeta})$ on a vector-valued Hardy space $H^{2}_{\mathcal{E}}(\mathbb{D}_{2}^{\infty})$ for some $\lambda\in\mathbb{D}^{\infty}$.

Below we give an example to illustrate that condition (3) in Theorem 1.3 is nontrivial for sequences in class $\mathcal{DC}$ (Theorem 1.5). By comparing this with Lemma 2.7, we see that the infinite-tuple case diverges considerably from the finite-tuple case.

Theorem 3.2**.**

There exists a sequence $\mathbf{V}\in\mathcal{DP}$ such that $\mathfrak{D}_{\Phi_{\lambda}(\mathbf{V})^{*}}=\{0\}$ for each $\lambda\in\mathbb{D}^{\infty}$.

Proof.

Let $\mathbf{T}^{2}$ denote $(T_{1}^{2},T_{2}^{2},\cdots)$ for a sequence $\mathbf{T}=(T_{1},T_{2},\cdots)$ of operators, and $\mathbf{M}_{\zeta}=(M_{\zeta_{1}},M_{\zeta_{2}},\cdots)$ be the tuple of coordinate multiplication operators on $H^{2}(\mathbb{D}^{\infty}_{2})$. Put

[TABLE]

It is clear that $H^{2}(\mathbb{D}^{\infty}_{2})=\bigoplus_{k=0}^{\infty}\zeta_{n}^{k}\mathcal{E}_{n}$ for each $n\in\mathbb{N}$. Define a sequence $\mathbf{V}$ of isometries on $H^{2}(\mathbb{D}^{\infty}_{2})$ by setting

[TABLE]

for $n\in\mathbb{N}$ and $F\in\mathcal{E}_{n}$. It is routine to check that $\mathbf{V}\in\mathcal{DP}$ and $\mathbf{V}^{2}=\mathbf{M}_{\zeta}^{2}$.

Now we will show $\mathfrak{D}_{\Phi_{\lambda}(\mathbf{V})^{*}}=\{0\}$ for each $\lambda\in\mathbb{D}^{\infty}$. Assume conversely that there exists a point $\lambda\in\mathbb{D}^{\infty}$ such that $\mathfrak{D}_{\Phi_{\lambda}(\mathbf{V})^{*}}$ contains a function $F\neq 0$. By the comments above Lemma 2.8, $F$ is exactly an eigenvector of the sequence $\mathbf{V}^{*}$ corresponding to the joint eigenvalue $\overline{\lambda}=(\overline{\lambda_{1}},\overline{\lambda_{2}},\cdots)$, and therefore

[TABLE]

where $\lambda^{2}=(\lambda_{1}^{2},\lambda_{2}^{2},\cdots)$. Let $\mathbf{K}_{\mu}$ denote the reproducing kernel of $H^{2}(\mathbb{D}^{\infty}_{2})$ at the point $\mu\in\mathbb{D}^{\infty}_{2}$. Then

[TABLE]

which gives

[TABLE]

since the set $\{\mathbf{K}_{\mu}:\mu\in\mathbb{D}^{\infty}_{2}\}$ is complete in $H^{2}(\mathbb{D}^{\infty}_{2})$. Then by Corollary 3.1 (2), $\mathfrak{D}_{\Phi_{\xi}(\mathbf{V}^{2})^{*}}=\{0\}$ for $\xi\notin\mathbb{D}_{2}^{\infty}$, forcing $\lambda^{2}\in\mathbb{D}_{2}^{\infty}$. In particular, $\lambda_{n}\rightarrow 0\ (n\rightarrow\infty)$.

Write $\mathcal{F}=\mathfrak{D}_{\mathbf{V}^{*2}}=\mathfrak{D}_{\mathbf{M}_{\zeta}^{*2}}$. Then

[TABLE]

and

[TABLE]

Where $P_{\mathcal{F}}$ is the orthogonal projection from $H^{2}(\mathbb{D}^{\infty}_{2})$ onto $\mathcal{F}$. This implies that

[TABLE]

converges to $G=P_{\mathcal{F}}F$ ($n\rightarrow\infty$) in $H^{2}(\mathbb{D}^{\infty}_{2})$-norm. Note that the reproducing kernel $\mathbf{K}_{\lambda^{2}}$ vanishes nowhere on $\mathbb{D}^{\infty}_{2}$, and $\prod_{i=1}^{n}(1-\overline{\lambda_{n}^{2}}\zeta_{n}^{2})$ converges pointwisely to the function $\frac{1}{\mathbf{K}_{\lambda^{2}}}$ on $\mathbb{D}^{\infty}_{2}$ as $n\rightarrow\infty$, $G$ must coincide with the function $\frac{F}{\mathbf{K}_{\lambda^{2}}}$ on $\mathbb{D}^{\infty}_{2}$, forcing $G\neq 0$. Since $G\in\mathcal{F}$, there exists some $\alpha=(\alpha_{1},\alpha_{2},\cdots)\in\mathbb{Z}_{+}^{(\infty)}$ with each $\alpha_{n}\leq 1$ such that $\langle G,\zeta^{\alpha}\rangle\neq 0$.

Below we prove $\langle G,\zeta^{\alpha}\rangle=0$ to reach a contradiction. Since for each $n\in\mathbb{N}$,

[TABLE]

there corresponds a decomposition $G=G_{n}+H_{n}$ of $G$ such that $G_{n}\in\mathrm{Ker}\ V_{n}^{*}$ and $H_{n}\in\mathrm{Ker}\ V_{n}^{*2}\ominus\mathrm{Ker}\ V_{n}^{*}$. Since $\mathbf{V}$ is doubly commuting, for any $n\in\mathbb{N}$,

[TABLE]

and

[TABLE]

which gives

[TABLE]

By the fact that $\lambda_{n}\rightarrow 0\ (n\rightarrow\infty)$ and $\|G\|^{2}=\|G_{n}\|^{2}+\|H_{n}\|^{2}\ (n\in\mathbb{N})$, we see $\|H_{n}\|\rightarrow 0\ (n\rightarrow\infty)$. This gives that $G_{n}\rightarrow G\ (n\rightarrow\infty)$ in norm, and then $\langle G_{n},\zeta^{\alpha}\rangle\rightarrow\langle G,\zeta^{\alpha}\rangle\ (n\rightarrow\infty)$. Since $G_{n}\in\zeta_{n}H^{2}(\mathbb{D}^{\infty}_{2})\ (n\in\mathbb{N})$, $\langle G_{n},\zeta^{\alpha}\rangle=0$ for $n$ large sufficiently, forcing $\langle G,\zeta^{\alpha}\rangle=0$. This completes the proof. ∎

We also have the following application of Theorem 1.3 and Corollary 1.4. Set $\Gamma$ to be the set of points $\lambda=(\lambda_{1},\lambda_{2},\cdots)\in\mathbb{D}^{\infty}$ satisfying

[TABLE]

One can check that for the equivalence relation $\sim$ on $\mathbb{D}^{\infty}$ given in Corollary 3.1, $\Gamma$ is the union of some $\sim$-equivalence classes, namely for a pair $\lambda,\mu$ of $\sim$-equivalent points in $\mathbb{D}^{\infty}$, $\lambda\in\Gamma$ if and only if $\mu\in\Gamma$.

Corollary 3.3**.**

Suppose $\mathbf{T}\in\mathcal{DC}(\mathcal{H})$ and put $A_{n}=T_{1}\cdots T_{n}\ (n\in\mathbb{N})$. If $\bigvee_{\lambda\in\Gamma}\mathfrak{D}_{\Phi_{\lambda}(\mathbf{T})^{*}}=\mathcal{H}$, then $\{A_{n}\}_{n\in\mathbb{N}}$ converges in the strong operator topology.

Proof.

Assume $\bigvee_{\lambda\in\Gamma}\mathfrak{D}_{\Phi_{\lambda}(\mathbf{T})^{*}}=\mathcal{H}$. By Theorem 1.3, $\mathbf{T}$ can be decomposed into the direct sum of $\mathcal{DC}$-sequences of quasi-Beurling type. Without loss of generality, we may further assume that $\mathbf{T}$ itself is of quasi-Beurling type. Take $\lambda\in\Gamma$ so that $\Phi_{\lambda}(\mathbf{T})$ is of Beurling type. Then Corollary 1.4 implies that the minimal regular isometric dilation $\mathbf{V}$ of $\mathbf{T}$ is jointly unitarily equivalent to the sequence $(M_{\widetilde{\varphi_{\lambda_{1}}}},M_{\widetilde{\varphi_{\lambda_{2}}}},\cdots)$ of multiplication operators on a vector-valued Hardy space $H_{\mathcal{E}}^{2}(\mathbb{D}_{2}^{\infty})$, where $\widetilde{\varphi_{\lambda_{n}}}(\zeta)=\varphi_{\lambda_{n}}(\zeta_{n})\ (\zeta\in\mathbb{D}^{\infty}_{2})$.

Put $F_{n}=\prod_{i=1}^{n}\widetilde{\varphi_{\lambda_{i}}}\ (n\in\mathbb{N})$. It suffices to show that $\{M_{F_{n}}\}_{n\in\mathbb{N}}$ converges in the strong operator topology. Since this sequence has uniformly bounded operator norms and

[TABLE]

is complete in $H_{\mathcal{E}}^{2}(\mathbb{D}_{2}^{\infty})$, we only need to prove for any $p\in\mathcal{P}_{\infty}$ and any $x\in\mathcal{E}$, $\{F_{n}p\cdot x\}_{n\in\mathbb{N}}$ converges in $H_{\mathcal{E}}^{2}(\mathbb{D}_{2}^{\infty})$, where $\mathcal{P}_{\infty}$ is the polynomial ring in countably infinitely many variables. This is clearly equivalent to that $\{F_{n}\}_{n\in\mathbb{N}}$ converges in $H^{2}(\mathbb{D}_{2}^{\infty})$.

For $n>m$ a simple calculation gives

[TABLE]

Since $\lim_{m\rightarrow\infty}\prod_{n=m}^{\infty}\lambda_{m}=1$, $\{F_{n}\}_{n\in\mathbb{N}}$ is a Cauchy sequence, which completes the proof. ∎

To end this section, we record an example of $\mathcal{DC}$-sequence of quasi-Beurling type.

Proposition 3.4**.**

Let $\{f_{n}\}_{n\in\mathbb{N}}$ be a sequence of bounded analytic functions on the unit disk $\mathbb{D}$. If for each $n\in\mathbb{N}$, $\|f_{n}\|_{\infty}\leq 1$ and $f_{n}$ is not a unimodular constant, then the sequence $(M_{\widetilde{f_{1}}},M_{\widetilde{f_{2}}},\cdots)$ of multiplication operators on the Hardy space $H^{2}(\mathbb{D}_{2}^{\infty})$ is a $\mathcal{DC}$-sequence of quasi-Beurling type, where $\widetilde{f_{n}}(\zeta)=f_{n}(\zeta_{n})\ (n\in\mathbb{N},\zeta\in\mathbb{D}^{\infty}_{2})$.

Proof.

It is routine to check that $(M_{\widetilde{f_{1}}},M_{\widetilde{f_{2}}},\cdots)$ is a sequence in class $\mathcal{DC}$. Put $g_{n}=\varphi_{f_{n}(0)}\circ f_{n}\ (n\in\mathbb{N})$. Then for each $n\in\mathbb{N}$, $g_{n}(0)=0$ and $M_{\widetilde{g_{n}}}=\varphi_{f_{n}(0)}(M_{\widetilde{f_{n}}})$, where $\widetilde{g_{n}}(\zeta)=g_{n}(\zeta_{n})$. We will show that the sequence $\mathbf{M}=(M_{\widetilde{g_{1}}},M_{\widetilde{g_{2}}},\cdots)$ is of Beurling type, which implies the desired conclusion.

Note that each $n\in\mathbb{N}$ and $\lambda=(\lambda_{1},\lambda_{2},\cdots)\in\mathbb{D}^{\infty}_{2}$,

[TABLE]

which gives

[TABLE]

It follows that $D_{\Phi_{\mu(\lambda)}(\mathbf{M})^{*}}\mathbf{K}_{\lambda}=\mathbf{K}_{\lambda}$ for any $\lambda\in\mathbb{D}^{\infty}_{2}$, where

[TABLE]

Also by Schwarz’ Lemma, we have that for each $\lambda\in\mathbb{D}^{\infty}_{2}$,

[TABLE]

and then

[TABLE]

This gives $\bigvee_{\mu\in\mathbb{D}_{2}^{\infty}}\mathfrak{D}_{\Phi_{\mu}(\mathbf{M})^{*}}=H^{2}(\mathbb{D}_{2}^{\infty})$. It follows from Corollary 1.4 that the sequence $\mathbf{M}=(M_{\widetilde{g_{1}}},M_{\widetilde{g_{2}}},\cdots)$ is of Beurling type. ∎

Remark 3.5**.**

From the proof of Proposition 3.4, we see that the $\sim$-equivalence class of the point $(f_{1}(0),f_{2}(0),\cdots)\in\mathbb{D}^{\infty}$ (see Corollary 3.1) is an invariant for the sequence $(M_{\widetilde{f_{1}}},M_{\widetilde{f_{2}}},\cdots)$. More precisely, let $\{f_{n}\}_{n\in\mathbb{N}}$ and $\{g_{n}\}_{n\in\mathbb{N}}$ be two sequence of functions that satisfy the conditions given in the proposition, and put $\widetilde{f_{n}}(\zeta)=f_{n}(\zeta_{n})$, $\widetilde{g_{n}}(\zeta)=g_{n}(\zeta_{n})\ (n\in\mathbb{N},\zeta\in\mathbb{D}^{\infty}_{2})$. If the multiplication operators $(M_{\widetilde{f_{1}}},M_{\widetilde{f_{2}}},\cdots)$, $(M_{\widetilde{g_{1}}},M_{\widetilde{g_{2}}},\cdots)$ are jointly unitarily equivalent, then $(f_{1}(0),f_{2}(0),\cdots)$ and $(g_{1}(0),g_{2}(0),\cdots)$ belong to the same $\sim$-equivalence class; that is,

[TABLE]

Since the number of $\sim$-equivalence classes is countless, $\mathbb{D}_{2}^{\infty}$ is a very “small” part in $\mathbb{D}^{\infty}$. Hence, for “almost all” choices of the sequence $\{f_{n}\}_{n\in\mathbb{N}}$ of functions, the defect space of $(M_{\widetilde{f_{1}}}^{*},M_{\widetilde{f_{2}}}^{*},\cdots)$ is $\{0\}$ by Proposition 3.4 and Corollary 3.1, and then $(M_{\widetilde{f_{1}}},M_{\widetilde{f_{2}}},\cdots)$ is not of Beurling type.

4. Analytic Model

In this section, we will prove that every sequence in class $\mathcal{DP}$ is jointly unitarily equivalent to a sequence of multiplication operators induced by operator-valued inner functions each of which involves one different variables. We thus establish an operator-valued analytic functional model for general $\mathcal{DC}$-sequences.

Theorem 4.1**.**

Let $\mathbf{T}$ be a sequence in class $\mathcal{DC}(\mathcal{H})$, and $\mathbf{V}\in\mathcal{DP}(\mathcal{K})$ the minimal regular isometric dilation of $\mathbf{T}$. Then there exist a Hilbert space $\mathcal{E}$, a unitary operator $U:\mathcal{K}\rightarrow H_{\mathcal{E}}^{2}(\mathbb{D}_{2}^{\infty})$ and a sequence $\Theta=(\theta_{1},\theta_{2},\cdots)$ of inner functions in $H^{\infty}_{\mathcal{B}(\mathcal{E})}(\mathbb{D})$, such that

• (1)

for each $n\in\mathbb{N}$, $UV_{n}U^{*}=M_{\widetilde{\theta_{n}}}$, where $\widetilde{\theta_{n}}(\zeta)=\theta_{n}(\zeta_{n})\ (\zeta\in\mathbb{D}^{\infty}_{2})$;

• (2)

$\mathcal{Q}=U\mathcal{H}$* is a quotient module of $H_{\mathcal{E}}^{2}(\mathbb{D}_{2}^{\infty})$.*

The tuple $(\mathcal{Q},\Theta)$ in Theorem 4.1 is said to be an analytic model for the $\mathcal{DC}$-sequence $\mathbf{T}$, and the Hilbert space $\mathcal{E}$ is called the underlying space of the analytic model $(\mathcal{Q},\Theta)$. The sequence $(M_{\widetilde{\theta_{1}}},M_{\widetilde{\theta_{2}}},\cdots)$ is denoted by $\mathbf{M}_{\Theta}$ for simplicity. Also, for the trivial case

[TABLE]

we simply write $\mathcal{Q}$ for $(\mathcal{Q},\Theta)$. It is clear that $\mathbf{T}$ is jointly unitarily equivalent to the sequence $P_{\mathcal{Q}}\mathbf{M}_{\Theta}|_{\mathcal{Q}}$, the compression of the sequence $\mathbf{M}_{\Theta}$ on $\mathcal{Q}$.

To prove Theorem 4.1, we need the following.

Proposition 4.2**.**

Suppose $\mathbf{T}\in\mathcal{DC}$. Then there exists a sequence $(k_{1},k_{2},\cdots)$ of positive integers, such that $(T_{1}^{k_{1}},T_{2}^{k_{2}},\cdots)$ is of Beurling type.

We thank Dr. Y. Wang for discussion on the proof of Proposition 4.2.

Proof.

Assume that $\mathbf{T}\in\mathcal{DC}(\mathcal{H})$ and $\mathbf{V}\in\mathcal{DP}(\mathcal{K})$ is the minimal regular isometric dilation of $\mathbf{T}$. For a sequence $\mathbf{k}=(k_{1},k_{2},\cdots)$ of positive integers, put

[TABLE]

and $M_{\mathbf{k}}=[\mathfrak{D}_{\mathbf{V}_{\mathbf{k}}^{*}}]_{\mathbf{V}_{\mathbf{k}}}$. It suffices to prove that there exists a sequence $\mathbf{k}$ of positive integers such that $\mathcal{K}=M_{\mathbf{k}}$.

Note that we have made the convention that $\mathcal{H}$ is a separable Hilbert space in Subsection 1.2. Then $\mathcal{K}=[\mathcal{H}]_{\mathbf{V}}$ is also separable. Take a sequence $\{x_{n}\}_{n\in\mathbb{N}}$ in $\mathcal{K}$ such that all its elements constitute a dense subset of $\mathcal{K}$ and each element appears in infinitely many times. It follows from Lemma 2.12 that for each $n\in\mathbb{N}$, there exists a sequence $\mathbf{k}^{(n)}=(k_{1}^{(n)},k_{2}^{(n)},\cdots)$ of positive integers, such that

[TABLE]

Set $k_{n}=\max\{k_{n}^{(1)},\cdots,k_{n}^{(n)}\}\ (n\in\mathbb{N})$ and put $\mathbf{k}=(k_{1},k_{2},\cdots)$. Note that

[TABLE]

where $\mathbf{k}^{\prime}=(k_{2},k_{3},\cdots)$. It follows that

[TABLE]

and thus by induction,

[TABLE]

for each $n\in\mathbb{N}$. By (4.1), one obtains

[TABLE]

forcing $\|P_{M_{\mathbf{k}}}x\|=\|x\|$ for any $x\in\mathcal{K}$, where $P_{M_{\mathbf{k}}}$ is the orthogonal projection from $\mathcal{K}$ onto $M_{\mathbf{k}}$. This completes the proof. ∎

Proof of Theorem 4.1. Assume that $\mathbf{T}\in\mathcal{DC}(\mathcal{H})$ and $\mathbf{V}\in\mathcal{DP}(\mathcal{K})$ is the minimal regular isometric dilation of $\mathbf{T}$. It follows from Proposition 4.2 that there exists a sequence $(k_{1},k_{2},\cdots)$ of positive integers, such that $(T_{1}^{k_{1}},T_{2}^{k_{2}},\cdots)$ is of Beurling type. Then by Corollary 1.4, the sequence $(V_{1}^{k_{1}},V_{2}^{k_{2}},\cdots)$ is jointly unitarily equivalent to the tuple $\mathbf{M}_{\zeta}=(M_{\zeta_{1}},M_{\zeta_{2}},\cdots)$ of coordinate multiplication operators on a vector-valued Hardy space $H_{\mathcal{E}}^{2}(\mathbb{D}_{2}^{\infty})$ via a unitary operator $U:\mathcal{K}\rightarrow H_{\mathcal{E}}^{2}(\mathbb{D}_{2}^{\infty})$. This implies that for each $n\in\mathbb{N}$,

[TABLE]

and hence $\widetilde{V_{n}}=UV_{n}U^{*}$ commutes with $M_{\zeta_{n}}$ and doubly commutes with $M_{\zeta_{m}}$ for any $m\neq n$.

It remains to show that $\widetilde{V_{n}}\ (n\in\mathbb{N})$ is a multiplication operator induced by an operator-valued inner function $\widetilde{\theta_{n}}\in H^{\infty}_{\mathcal{B}(\mathcal{E})}(\mathbb{D}_{2}^{\infty})$, which depends only on the $n$-th variable $\zeta_{n}$. We first prove for $n=1$. Put $\mathbf{M}_{\zeta}^{\prime}=(M_{\zeta_{2}},M_{\zeta_{3}},\cdots)$ and set

[TABLE]

Then we have $H_{\mathcal{E}}^{2}(\mathbb{D}_{2}^{\infty})=H_{\mathcal{E}}^{2}(\mathbb{D})\otimes\mathcal{L}$ (see Subsection 2.2), $M_{\zeta_{1}}=M_{z}\otimes I_{\mathcal{L}}$ and $\mathbf{M}_{\zeta}^{\prime}$ has the form $(I_{H_{\mathcal{E}}^{2}(\mathbb{D})}\otimes T_{1},I_{H_{\mathcal{E}}^{2}(\mathbb{D})}\otimes T_{2},\cdots)$ for a sequence $(T_{1},T_{2},\cdots)$ of operators on $\mathcal{L}$ jointly unitarily equivalent to the tuple of coordinate multiplication operators on $H^{2}(\mathbb{D}^{\infty}_{2})$. Since $\widetilde{V_{1}}$ doubly commutes with $\mathbf{M}_{\zeta}^{\prime}$, it follows from Lemma 2.10 and Lemma 2.11 that $\widetilde{V_{1}}=S\otimes I_{\mathcal{L}}$ for some isometry $S$ on $H_{\mathcal{E}}^{2}(\mathbb{D})$. Therefore,

[TABLE]

forcing $M_{z}=S^{k_{1}}$. In particular, $S$ commutes with $M_{z}$. Then there is a $\mathcal{B}(\mathcal{E})$-valued inner function $\theta_{1}$ in single variable $z\in\mathbb{D}$, such that $S=M_{\theta_{1}}$ (see [43, pp. 200-201] for instance), which gives

[TABLE]

where $\widetilde{\theta_{1}}(\zeta)=\theta_{1}(\zeta_{1})\ (\zeta\in\mathbb{D}^{\infty}_{2})$. Similarly, for each $n\geq 2$, $\widetilde{V_{n}}=M_{\widetilde{\theta_{n}}}$, where $\widetilde{\theta_{n}}(\zeta)=\theta_{n}(\zeta_{n})\ (\zeta\in\mathbb{D}^{\infty}_{2})$ for some inner function $\theta_{n}\in H^{\infty}_{\mathcal{B}(\mathcal{E})}(\mathbb{D})$. The proof is complete. ∎

Combining Proposition 4.2 with Corollary 3.3, we have the following corollary (Corollary 1.6).

Corollary 4.3**.**

Suppose $\mathbf{T}\in\mathcal{DC}$. Then there exists a sequence $\{B_{n}\}_{n\in\mathbb{N}}$ of finite Blaschke products, such that $\{\prod_{i=1}^{n}B_{i}(T_{i})\}_{n\in\mathbb{N}}$ converges in the strong operator topology.

Now we will establish an analytic model for sequences in class $\mathcal{DC}$ with a decomposition of quasi-Beurling type.

Suppose that $\mathbf{T}\in\mathcal{DC}(\mathcal{H})$ is of Beurling type, and $\mathbf{V}\in\mathcal{DP}(\mathcal{K})$ is the minimal regular isometric dilation of $\mathbf{T}$. By Corollary 3.1 (1), $\mathbf{V}$ is jointly unitarily equivalent to the tuple $\mathbf{M}_{\zeta}$ of coordinate multiplication operators on a vector-valued Hardy space $H_{\mathfrak{D}_{\mathbf{V}^{*}}}^{2}(\mathbb{D}_{2}^{\infty})$. We claim that the map

[TABLE]

can be extended to a unitary operator from $\mathfrak{D}_{\mathbf{V}^{*}}$ onto $\mathfrak{D}_{\mathbf{T}^{*}}$. By Lemma 2.6, it remains to prove that $D_{\mathbf{V}^{*}}\mathcal{H}$ is dense in $\mathfrak{D}_{\mathbf{V}^{*}}$. Assume that $x\in\mathfrak{D}_{\mathbf{V}^{*}}$ is orthogonal to $D_{\mathbf{V}^{*}}\mathcal{H}$. Then $x$ is orthogonal to $\mathcal{H}$ and $\mathrm{Ran}\ V_{n}$ for all $n\in\mathbb{N}$. In particular, $x$ is orthogonal to $[\mathcal{H}]_{\mathbf{V}}=\mathcal{K}$, forcing $x=0$. This proves the claim. Therefore, the above unitary operator from $\mathfrak{D}_{\mathbf{V}^{*}}$ onto $\mathfrak{D}_{\mathbf{T}^{*}}$ naturally induced a unitary operator $U_{\mathbf{T}}$ from $\mathcal{K}$ onto $H_{\mathfrak{D}_{\mathbf{T}^{*}}}^{2}(\mathbb{D}_{2}^{\infty})$. It is easy to see that $\mathcal{Q}_{\mathbf{T}}=U_{\mathbf{T}}\mathcal{H}$ is an analytic model for the sequence $\mathbf{T}$. We call $\mathcal{Q}_{\mathbf{T}}$ the canonical analytic model for $\mathbf{T}$.

As a consequence, for a sequence $\mathbf{T}\in\mathcal{DC}(\mathcal{H})$ which is of quasi-Beurling type, one can find some $\lambda=(\lambda_{1},\lambda_{2},\cdots)\in\mathbb{D}^{\infty}$ so that $\Phi_{\lambda}(\mathbf{T})$ is of Beurling type, and then $(\mathcal{Q}_{\Phi_{\lambda}(\mathbf{T})},(\varphi_{\lambda_{1}}\cdot I,\varphi_{\lambda_{2}}\cdot I,\cdots))$ is an analytic model for the sequence $\mathbf{T}$.

Suppose that $\{\mathcal{E}_{\gamma}\}_{\gamma}$ is a family of Hilbert spaces, and for each index $\gamma$, $\Theta_{\gamma}$ is a sequence in $H^{\infty}_{\mathcal{B}(\mathcal{E}_{\gamma})}(\mathbb{D})$. In a natural way, we can define the direct sum $\bigoplus_{\gamma}\Theta_{\gamma}$ of $\{\Theta_{\gamma}\}_{\gamma}$, which is then a sequence in $H^{\infty}_{\mathcal{B}(\bigoplus_{\gamma}\mathcal{E}_{\gamma})}(\mathbb{D})$. Now let $\mathbf{T}=\bigoplus_{\gamma}\mathbf{T}_{\gamma}$ be direct sum of $\mathcal{DC}$-sequences of quasi-Beurling type. For each index $\gamma$, we have previously established an analytic model $(\mathcal{Q}_{\gamma},\Theta_{\gamma})$ for $\mathbf{T}_{\gamma}$. Put $\mathcal{Q}=\bigoplus_{\gamma}\mathcal{Q}_{\gamma}$ and $\Theta=\bigoplus_{\gamma}\Theta_{\gamma}$. Then $(\mathcal{Q},\Theta)$ is an analytic model for $\mathbf{T}$.

In fact, we have

Theorem 4.4**.**

Suppose $\mathbf{T}\in\mathcal{DC}$. Then $\mathbf{T}$ has a decomposition of quasi-Beurling type if and only if $\mathbf{T}$ has an analytic model $(\mathcal{Q},\Theta)$ so that all $\theta_{n}(z)\ (n\in\mathbb{N},z\in\mathbb{D})$ are simultaneously diagonalizable with respect to some orthonormal basis of the underlying space $\mathcal{E}$ of $(\mathcal{Q},\Theta)$.

Proof.

The necessity follows from the construction in previous paragraphs. Now assume that $\mathbf{T}\in\mathcal{DC}(\mathcal{H})$ has an analytic model $(\mathcal{Q},\Theta)$ so that all $\theta_{n}(z)\ (n\in\mathbb{N},z\in\mathbb{D})$ are simultaneously diagonalizable with respect to some orthonormal basis $\{e_{i}\}_{i\in\Lambda}$ of the underlying space $\mathcal{E}$ of $(\mathcal{Q},\Theta)$. We will prove that $\mathbf{T}$ has a decomposition of quasi-Beurling type. By the assumption, we have

[TABLE]

where each $\eta_{ni}$ is an $H^{\infty}(\mathbb{D})$-inner function, $e_{i}\widehat{\otimes}e_{i}$ denotes the $1$-rank projection

[TABLE]

Then for each $i\in\Lambda$, $H_{\mathbb{C}e_{i}}^{2}(\mathbb{D}^{\infty}_{2})$, as a subspace of $H_{\mathcal{E}}^{2}(\mathbb{D}_{2}^{\infty})$, is joint reducing for $\mathbf{M}_{\Theta}$, and the restriction of $\mathbf{M}_{\Theta}$ on $H_{\mathbb{C}e_{i}}^{2}(\mathbb{D}^{\infty}_{2})$ is jointly unitarily equivalent to the sequence $(M_{\widetilde{\eta_{1i}}},M_{\widetilde{\eta_{2i}}},\cdots)$ of multiplication operators on the Hardy space $H^{2}(\mathbb{D}_{2}^{\infty})$, where $\widetilde{\eta_{ni}}(\zeta)=\eta_{ni}(\zeta_{n})\ (n\in\mathbb{N},\zeta\in\mathbb{D}^{\infty}_{2})$. It follows from Proposition 3.4 that the sequence $\mathbf{M}_{\Theta}$ has a decomposition of quasi-Beurling type, and therefore so does the minimal isometric dilation $\mathbf{V}$ of $\mathbf{T}$ by the definition of the analytic model. Then Theorem 1.3 together with Corollary 3.1 (3) implies that $\mathbf{T}$ also has a decomposition of quasi-Beurling type. ∎

To conclude this section, we give a characterization of the canonical analytic model $\mathcal{Q}_{\mathbf{T}}$ for a sequence $\mathbf{T}\in\mathcal{DC}$ which is of Beurling type. One approach, inspired by the single $C_{.0}$-contraction case, is utilizing the characteristic functions of contractions.

Recall that the characteristic function $\theta_{T}$ of a contraction $T\in\mathcal{B}(\mathcal{H})$ is defined by

[TABLE]

which is a $\mathcal{B}(\mathfrak{D}_{T},\mathfrak{D}_{T^{*}})$-valued inner function (see [43]). Then the multiplication operator $M_{\theta_{T}}$ is an isometry from $H_{\mathfrak{D}_{T}}^{2}(\mathbb{D})$ to $H_{\mathfrak{D}_{T^{*}}}^{2}(\mathbb{D})$, and thus

[TABLE]

Suppose that in addition $T\in C_{.0}$ and $V\in\mathcal{B}(\mathcal{K})$ is the minimal isometric dilation of $T$. Since $V$ is pure, $\mathcal{K}$ has a decomposition as $\mathcal{K}=\bigoplus_{k=0}^{\infty}V^{k}\mathfrak{D}_{V^{*}}$, and therefore can be identified with the vector-valued Hardy space $H_{\mathfrak{D}_{V^{*}}}^{2}(\mathbb{D})$. Similar to (4.2), the map

[TABLE]

can be extended to a unitary operator from $\mathfrak{D}_{V^{*}}$ onto $\mathfrak{D}_{T^{*}}$, and then induces a unitary operator $U_{T}$ from $\mathcal{K}$ onto $H_{\mathfrak{D}_{T^{*}}}^{2}(\mathbb{D})$. It is easy to see that $U_{T}$ is actually an extension of the isometry

[TABLE]

It follows from [43, pp. 244-245] that for $a,b\in\mathbb{D}$,

[TABLE]

This gives

[TABLE]

(see [8] or [9]); that is,

[TABLE]

Let us return to the characterization of the canonical analytic model $\mathcal{Q}_{\mathbf{T}}$ for the sequence $\mathbf{T}=(T_{1},T_{2},\cdots)$. Now consider the following spaces

[TABLE]

By (4.3), for $a,b\in\mathbb{D}$,

[TABLE]

where $\mathbf{T}^{\prime}=(T_{2},T_{3},\cdots)$. Similarly, for any $n\in\mathbb{N}$ and $a,b\in\mathbb{D}$,

[TABLE]

and then by Lemma 2.5 (1), $H_{\mathfrak{D}_{\mathbf{T}^{*}}}^{2}(\mathbb{D})$ is reducing for $M_{\theta_{T_{n}}}M_{\theta_{T_{n}}}^{*}$, and

[TABLE]

where $\mathcal{F}_{n}$ is the defect space of the sequence $(T_{1}^{*},\cdots,T_{n-1}^{*},T_{n},T_{n+1}^{*},\cdots)$. In particular,

[TABLE]

is a $M_{z}$-invariant subspace of $H_{\mathfrak{D}_{\mathbf{T}^{*}}}^{2}(\mathbb{D})$.

Put $\widetilde{\theta_{T_{n}}}(\zeta)=\theta_{T_{n}}(\zeta_{n})\ (n\in\mathbb{N},\zeta\in\mathbb{D}^{\infty}_{2})$. Then $M_{\widetilde{\theta_{T_{n}}}}M_{\widetilde{\theta_{T_{n}}}}^{*}H_{\mathfrak{D}_{\mathbf{T}^{*}}}^{2}(\mathbb{D}^{\infty}_{2})$ $(n\in\mathbb{N})$ is of form

[TABLE]

forcing it to be a joint invariant subspace of $H_{\mathfrak{D}_{\mathbf{T}^{*}}}^{2}(\mathbb{D}^{\infty}_{2})$ for the tuple of coordinate multiplication operators.

Our result presented below looks somehow similar to the above single $C_{.0}$-contraction case, or the finite-tuple case considered in [9]. However, instead of following the proof in [9], we will give an original proof.

Theorem 4.5**.**

Let $\mathbf{T}=(T_{1},T_{2},\cdots)\in\mathcal{DC}$ be of Beurling type, and $\mathcal{Q}_{\mathbf{T}}$ the canonical analytic model for $\mathbf{T}$. Then

[TABLE]

where $\widetilde{\theta_{T_{n}}}(\zeta)=\theta_{T_{n}}(\zeta_{n})\ (n\in\mathbb{N},\zeta\in\mathbb{D}^{\infty}_{2})$.

Proof.

Let $\mathbf{V}\in\mathcal{DP}(\mathcal{K})$ be the minimal regular isometric dilation of $\mathbf{T}\in\mathcal{DC}(\mathcal{H})$, and put

[TABLE]

By Lemma 2.2 (3), one has $\mathcal{Q}_{\mathbf{T}}=U_{\mathbf{T}}\mathcal{H}=\bigcap_{n=1}^{\infty}U_{\mathbf{T}}\mathcal{H}_{n}$. It thus suffices to prove that for each $n\in\mathbb{N}$,

[TABLE]

equivalently,

[TABLE]

Assume $n=1$ without loss of generality, and put $\mathbf{S}=(S_{1},S_{2},\cdots)=P_{\mathcal{H}_{1}}\mathbf{V}|_{\mathcal{H}_{1}}$, the compression of the sequence $\mathbf{V}$ on the subspace $\mathcal{H}_{1}$. Rewrite $T=T_{1}$ and $S=S_{1}$ for simplicity. One can define the unitary operators $U_{\mathbf{S}}:\mathcal{K}\rightarrow H_{\mathfrak{D}_{\mathbf{S}^{*}}}^{2}(\mathbb{D}^{\infty}_{2})$ and $U_{S}:\mathcal{K}\rightarrow H_{\mathfrak{D}_{S^{*}}}^{2}(\mathbb{D})$ as done previously in this section. Since (4.4) remain valid for $S$, (4.5) and (4.6) remain valid for $\mathbf{S}$, it follows that

[TABLE]

$H_{\mathfrak{D}_{\mathbf{S}^{*}}}^{2}(\mathbb{D})$ is reducing for $M_{\theta_{S}}M_{\theta_{S}}^{*}$ and

[TABLE]

where $\mathcal{F}$ is the defect space of the sequence $(S,S_{2}^{*},S_{3}^{*},\cdots)$.

We now claim that

[TABLE]

where

[TABLE]

Since operators on both sides of (4.11) are orthogonal projections, the claim is equivalent to

[TABLE]

The left side of (4.12) is

[TABLE]

and by (4.10) the right side of (4.12) is

[TABLE]

A calculation gives that for any $\alpha=(\alpha_{1},\alpha_{2},\cdots)\in\mathbb{Z}_{+}^{(\infty)}\ \mathrm{with}\ \alpha_{1}=0$ and $f\in H_{\mathfrak{D}_{\mathbf{S}^{*}}}^{2}(\mathbb{D})$,

[TABLE]

Since $S$ doubly commutes with $S_{n}\ (n\geq 2)$, for such $\alpha$ and $g\in H_{\mathcal{F}}^{2}(\mathbb{D})$, the above identity gives

[TABLE]

Also by Lemma 2.2 (1), $\mathcal{H}_{1}$ is reducing for $V_{n}\ (n\geq 2)$. Then $\mathbf{S}^{\prime}=(S_{2},S_{3},\cdots)$ is a $\mathcal{DP}$-sequence of Beurling type, and so is $\mathbf{S}^{\prime}|_{\mathfrak{D}_{S}}$, the restriction of $\mathbf{S}^{\prime}$ on the joint reducing subspace $\mathfrak{D}_{S}$. This together with Corollary 3.1 (1) gives

[TABLE]

Therefore, we have

[TABLE]

where the second identity follows from (4.13) and the last identity follows from (4.14). This proves the claim.

The operator $U_{\mathbf{T}}U_{\mathbf{S}}^{*}$ has the form $I_{H^{2}(\mathbb{D})}\otimes\Pi\otimes I_{\mathcal{L}}$ with respect to the representation

[TABLE]

where $\Pi:\mathfrak{D}_{\mathbf{S}^{*}}\rightarrow\mathfrak{D}_{\mathbf{T}^{*}}$ is a unitary operator satisfying

[TABLE]

Let $\Pi\cdot\theta_{S}$ denotes the operator-valued function defined by $z\mapsto\Pi\theta_{S}(z)$ $(z\in\mathbb{D})$. Then

[TABLE]

since two operators coincide on the set $\{K_{a}\cdot x:a\in\mathbb{D},x\in\mathfrak{D}_{\mathbf{T}^{*}}\}$, and thus

[TABLE]

Therefore, we have

[TABLE]

where the first identity follows from (4.9), the third identity follows from (4.11), and the fifth identity follows from (4.16). On the other hand, (4.7) gives

[TABLE]

which reduces (4.8) to

[TABLE]

By Lemma 2.5 (2), it remains to prove that for any fixed $a,b\in\mathbb{D}$ and any fixed $x\in\mathcal{H}\subseteq\mathcal{H}_{1}$,

[TABLE]

Note that Lemma 2.2 (2) implies that $\mathcal{H}$ is reducing for $S$ and $T=S|_{\mathcal{H}}$. By (4.5), we have

[TABLE]

and

[TABLE]

where

[TABLE]

It follows from (4.15) that

[TABLE]

This completes the proof. ∎

Now we are ready to refine the representation (1.2) in Subsection 1.2. Note that for each index $\gamma$, $\mathbf{S}_{\gamma}=(S_{\gamma 1},S_{\gamma 2},\cdots)=\Phi_{\lambda_{\gamma}}(\mathbf{T}_{\gamma})$ is of Beurling type. Without loss of generality, we may assume that $\mathcal{E}_{\gamma}=\mathfrak{D}_{\mathbf{S}_{\gamma}^{*}}$ and $\mathcal{Q}_{\gamma}$ is the canonical analytic model for $\mathbf{S}_{\gamma}$ in (1.2). Put $\lambda_{\gamma}=(\lambda_{\gamma 1},\lambda_{\gamma 2},\cdots)$ and let $\theta_{\gamma n}$, $\vartheta_{\gamma n}$ ($n\in\mathbb{N}$) denote the characteristic functions of $T_{\gamma n}$ and $S_{\gamma n}$, respectively. Then for each $\gamma$, $\vartheta_{\gamma n}$ coincides with $\theta_{\gamma n}\circ\varphi_{\lambda_{\gamma n}}$ (see [43, pp. 246-247]), and it follows from Theorem 4.5 that

[TABLE]

where $\widetilde{\vartheta_{\gamma n}}(\zeta)=\vartheta_{\gamma n}(\zeta_{n})\ (n\in\mathbb{N},\zeta\in\mathbb{D}^{\infty}_{2})$.

5. Doubly commuting submodules and quotient modules of $H_{\mathcal{E}}^{2}(\mathbb{D}^{\infty}_{2})$

For submodules and quotient modules of the Hardy module, we are interested in the module actions on them; that is, the restrictions of the tuple of coordinate multiplication operators on submodules and the compressions of the tuple of coordinate multiplication operators on quotient modules. In this section, we mainly consider such restrictions and compressions that are doubly commuting.

Recall that a submodule $\mathcal{S}$ of $H_{\mathcal{E}}^{2}(\mathbb{D}_{2}^{\infty})$ is said to be doubly commuting if the restriction

[TABLE]

of $\mathbf{M}_{\zeta}$ on $\mathcal{S}$ is doubly commuting.

Theorem 5.1**.**

Let $\mathbf{M}_{\zeta}$ be the tuple of coordinate multiplication operators on the vector-valued Hardy space $H_{\mathcal{E}}^{2}(\mathbb{D}_{2}^{\infty})$. Then the restriction of $\mathbf{M}_{\zeta}$ on a doubly commuting submodule of $H_{\mathcal{E}}^{2}(\mathbb{D}_{2}^{\infty})$ is of Beurling type.

Before giving the proof, we introduce the notion of homogeneous components of functions in $H_{\mathcal{E}}^{2}(\mathbb{D}_{2}^{\infty})$. Suppose $F\in H_{\mathcal{E}}^{2}(\mathbb{D}_{2}^{\infty})$, and let $F=\sum_{\alpha\in\mathbb{Z}_{+}^{(\infty)}}\zeta^{\alpha}\cdot x_{\alpha}$ be the power series expansion of $F$. The sum $\sum_{|\alpha|=k}\zeta^{\alpha}\cdot x_{\alpha}$ ($k=0,1,2,\cdots$) is called the $k$-th homogeneous component of $F$, where $|\alpha|=\alpha_{1}+\alpha_{2}+\cdots$. It is clear that $\|F\|^{2}$ is equal to the quadratic sum of norms of all homogeneous components of $F$.

Proof.

Let $\mathcal{S}$ be a doubly commuting quotient module of $H_{\mathcal{E}}^{2}(\mathbb{D}_{2}^{\infty})$, and set $\mathbf{R}=\mathbf{M}_{\zeta}|_{\mathcal{S}}$, the restriction of $\mathbf{M}_{\zeta}$ on $\mathcal{S}$. Assume conversely that $\mathbf{R}$ is not of Beurling type to reach a contradiction. Then by Corollary 3.1 (3), there exists an $\mathbf{R}$-joint reducing subspace $\widetilde{\mathcal{S}}$ of $\mathcal{S}$ that is orthogonal to the defect space $\mathfrak{D}_{\mathbf{R}^{*}}$ of $\mathbf{R}^{*}$; equivalently, $\mathfrak{D}_{\widetilde{\mathbf{R}}^{*}}=\{0\}$, where $\widetilde{\mathbf{R}}=\mathbf{R}|_{\widetilde{\mathcal{S}}}$. So without loss of generality, we may assume $\mathfrak{D}_{\mathbf{R}^{*}}=\{0\}$.

Set $k_{0}$ to be the minimal non-negative integer among those $k^{\prime}s$ that let nonzero $k$-th homogeneous components appear in some functions belonging to $\mathcal{S}$. Now choose a function $F\in\mathcal{S}$ so that the norm of the $k_{0}$-th homogeneous component of $F$ is $1$. Since $\mathfrak{D}_{\mathbf{R}^{*}}=\{0\}$, one has

[TABLE]

and then there exist $n\in\mathbb{N}$ and $n$ functions $F_{1},\cdots,F_{n}\in\mathcal{S}$, such that

[TABLE]

This implies that the norm of the $k_{0}$-th homogeneous component of the function $F-\sum_{i=1}^{n}\zeta_{i}F_{i}$ is less than $1$. However, it is clear that the $k_{0}$-th homogeneous component of $\sum_{i=1}^{n}\zeta_{i}F_{i}$ is [math], which contradicts with the choice of $F$. The proof is complete. ∎

We also recall that a function $\Psi\in H_{\mathcal{B}(\mathcal{F},\mathcal{E})}^{\infty}(\mathbb{D}_{2}^{\infty})$ is inner if the multiplication operator $M_{\Psi}$ induced by $\Psi$ is an isometry. It is clear that for any inner function $\Psi\in H_{\mathcal{B}(\mathcal{F},\mathcal{E})}^{\infty}(\mathbb{D}_{2}^{\infty})$, $\Psi H_{\mathcal{F}}^{2}(\mathbb{D}_{2}^{\infty})$ is a doubly commuting submodule of $H_{\mathcal{E}}^{2}(\mathbb{D}_{2}^{\infty})$. The following (Theorem 1.7) is a Beurling-Lax type Theorem for the vector-valued Hardy space in infinitely many variables.

Corollary 5.2**.**

Let $\mathcal{S}$ be a submodule of the vector-valued Hardy module $H_{\mathcal{E}}^{2}(\mathbb{D}_{2}^{\infty})$. Then $\mathcal{S}$ is doubly commuting if and only if there exist a Hilbert space $\mathcal{F}$ and an inner function $\Psi\in H_{\mathcal{B}(\mathcal{F},\mathcal{E})}^{\infty}(\mathbb{D}_{2}^{\infty})$, such that

[TABLE]

Proof.

Let $\mathbf{M}_{\zeta}$ be the tuple of coordinate multiplication operators on $H_{\mathcal{E}}^{2}(\mathbb{D}_{2}^{\infty})$, and $\mathbf{R}$ be the restriction of $\mathbf{M}_{\zeta}$ on $\mathcal{S}$. It follows from Theorem 5.1 and Corollary 1.4 that $\mathbf{R}$ is jointly unitarily equivalent to the tuple $\mathbf{M}_{\xi}$ of coordinate multiplication operators on a vector-valued Hardy space $H_{\mathcal{F}}^{2}(\mathbb{D}_{2}^{\infty})$. This naturally induces an isometry $V:H_{\mathcal{F}}^{2}(\mathbb{D}_{2}^{\infty})\rightarrow H_{\mathcal{E}}^{2}(\mathbb{D}_{2}^{\infty})$ satisfying $\mathrm{Ran}\ V=\mathcal{S}$ and

[TABLE]

Then by Proposition 2.4, $V=M_{\Psi}$ for some operator-valued inner function $\Psi\in H_{\mathcal{B}(\mathcal{F},\mathcal{E})}^{\infty}(\mathbb{D}_{2}^{\infty})$, where $M_{\Psi}$ is the multiplication operator induced by $\Psi$. This completes the proof. ∎

In particular, we reprove the known result that every doubly commuting submodule of $H^{2}(\mathbb{D}_{2}^{\infty})$ is principle (see [31]).

Corollary 5.3**.**

Let $\mathcal{S}$ be a nonzero submodule of $H^{2}(\mathbb{D}_{2}^{\infty})$. Then the followings are equivalent.

• (1)

$\mathcal{S}$* is doubly commuting;*

• (2)

$\mathcal{S}$* is generated by a single inner function in $H^{\infty}(\mathbb{D}_{2}^{\infty})$;*

• (3)

$\mathcal{S}$* as a $\mathcal{P}_{\infty}$-module is unitarily equivalent to $H^{2}(\mathbb{D}_{2}^{\infty})$.*

We say that two $\mathcal{P}_{\infty}$-modules $(\mathcal{H},\mathbf{T})$, $(\mathcal{K},\mathbf{S})$ are unitarily equivalent if $\mathbf{T}$, $\mathbf{S}$ are jointly unitarily equivalent, and the unitary operator $U:\mathcal{H}\rightarrow\mathcal{K}$ interwinding $\mathbf{T}$ and $\mathbf{S}$ is called a unitary module map.

The classification of doubly commuting Hardy submodules up to unitary equivalence of $\mathcal{P}_{\infty}$-modules is trivial, since by Corollary 5.2, it is completely determined by the dimension of $\mathcal{F}$ in the representation (5.1).

Now we turn to the situation of quotient modules. Recall that a quotient module $\mathcal{Q}$ of $H_{\mathcal{E}}^{2}(\mathbb{D}_{2}^{\infty})$ is said to be doubly commuting if the compression

[TABLE]

of $\mathbf{M}_{\zeta}$ on $\mathcal{Q}$ is doubly commuting.

Theorem 5.4**.**

Let $\mathbf{M}_{\zeta}$ be the tuple of coordinate multiplication operators on the vector-valued Hardy module $H_{\mathcal{E}}^{2}(\mathbb{D}_{2}^{\infty})$. Then the compression of $\mathbf{M}_{\zeta}$ on a doubly commuting quotient module of $H_{\mathcal{E}}^{2}(\mathbb{D}_{2}^{\infty})$ is of Beurling type.

Proof.

Let $\mathcal{Q}$ be a doubly commuting quotient module of $H_{\mathcal{E}}^{2}(\mathbb{D}_{2}^{\infty})$, and set $\mathbf{C}=P_{\mathcal{Q}}\mathbf{M}_{\zeta}|_{\mathcal{Q}}$, the compression of $\mathbf{M}_{\zeta}$ on $\mathcal{Q}$. Since both two sequences $\mathbf{M}_{\zeta}$ and $\mathbf{C}$ are doubly commuting, for $\alpha,\beta\in\mathbb{Z}_{+}^{(\infty)}$ satisfying $\alpha\wedge\beta=(0,0,\cdots)$, we have

[TABLE]

where $\alpha\wedge\beta=(\min\{\alpha_{1},\beta_{1}\},\min\{\alpha_{2},\beta_{2}\},\cdots)$. That is to say, $\mathbf{M}_{\zeta}$ is a regular isometric dilation of $\mathbf{C}$. So the minimal regular isometric dilation of $\mathbf{C}$ is the restriction of $\mathbf{M}_{\zeta}$ on the subspace $[\mathcal{Q}]_{\mathbf{M}_{\zeta}}$. It follows from Lemma 2.2 (1) that $[\mathcal{Q}]_{\mathbf{M}_{\zeta}}$ is joint reducing for $\mathbf{M}_{\zeta}$, and then by Lemma 2.11, one has

[TABLE]

for some closed subspace $\mathcal{E}_{0}$ of $\mathcal{E}$. Therefore, Corollary 1.4 gives that $\mathbf{C}$ is of Beurling type. ∎

Follow the notations in the proof of Theorem 5.4. Now we can use the characterization of the canonical analytic model $\mathcal{Q}_{\mathbf{C}}$ for $\mathbf{C}$ (see Theorem 4.5) to study the structure of the doubly commuting quotient module $\mathcal{Q}$ since as $\mathcal{P}_{\infty}$-modules, $\mathcal{Q}$ and $\mathcal{Q}_{\mathbf{C}}$ are unitarily equivalent. The unitary module map is of form $I_{H^{2}(\mathbb{D}_{2}^{\infty})}\otimes U$, where $U:\mathcal{E}_{0}\rightarrow\mathfrak{D}_{\mathbf{C}^{*}}$ is given as in (4.2). Then there exists a sequence of quotient modules $\{\mathcal{J}_{n}\}_{n\in\mathbb{N}}$ of $H_{\mathcal{E}_{0}}^{2}(\mathbb{D})$, such that

[TABLE]

Note that the joint unitary equivalence is a equivalence relation on class $\mathcal{DC}$. We can establish a one-to-one corresponding between the equivalence classes of doubly commuting quotient Hardy modules and the equivalence classes of $\mathcal{DC}$-sequences of Beurling type as illustrated below.

[TABLE]

In another word, the classification of doubly commuting Hardy quotient modules is equivalent to the classification of $\mathcal{DC}$-sequences of Beurling type.

Finally, we will consider the particular case $\mathcal{E}=\mathbb{C}$. By (5.2), we have

[TABLE]

It follows that $\mathcal{Q}=\mathcal{J}_{1}\otimes\mathcal{Q}^{\prime}$ for some closed subspace $\mathcal{Q}^{\prime}$ of

[TABLE]

and thus $P_{\mathcal{Q}}M_{\zeta_{1}}|_{\mathcal{Q}}=P_{\mathcal{J}_{1}}M_{z}|_{\mathcal{J}_{1}}\otimes I_{\mathcal{Q}^{\prime}}$. Similarly, $P_{\mathcal{Q}}M_{\zeta_{n}}|_{\mathcal{Q}}$ is the tensor product of $P_{\mathcal{J}_{n}}M_{z}|_{\mathcal{J}_{n}}$ and an identity operator for each $n\in\mathbb{N}$. Recall that a model space is a of $H^{2}(\mathbb{D})$, and the compression of the Hardy shift $M_{z}$ on a model space is called a Jordan block. Then for any $n\in\mathbb{N}$ so that $\mathcal{J}_{n}\neq H^{2}(\mathbb{D})$, the compression $P_{\mathcal{Q}}M_{\zeta_{n}}|_{\mathcal{Q}}$ is the tensor product of a Jordan block and an identity operator. So the compression of the tuple $\mathbf{M}_{\zeta}$ on $\mathcal{Q}$ can be considered as a Jordan block in the infinite-variable setting.

Recall that $C_{0}$ is the class of thoes completely nonunitary contractions $T$ for which there exists a nonzero function $f\in H^{\infty}(\mathbb{D})$ such that $f(T)=0$ [43]. The following application to operator theory is motivated by [20, Proposition 4.1].

Corollary 5.5**.**

Suppose $\mathbf{T}\in\mathcal{DC}$. If there exists a cyclic vector $x$ for $\mathbf{T}$ such that $T_{n}^{*}x=\overline{\lambda_{n}}x\ (n\in\mathbb{N})$ for some $\lambda\in\mathbb{D}^{\infty}$, then for each $n\in\mathbb{N}$, $T_{n}$ is either a $C_{0}$-contraction or a pure isometry.

Proof.

Assume $\mathbf{T}\in\mathcal{DC}(\mathcal{H})$ Since for each $n\in\mathbb{N}$, $T_{n}$ is a $C_{0}$-contraction (pure isometry) if and only if $\varphi_{\lambda_{n}}(T_{n})$ is a $C_{0}$-contraction (pure isometry), we may assume $\lambda=\mathbf{0}$ without loss of generality.

Put $\widetilde{\mathcal{H}}=\bigvee_{\mu\in\mathbb{D}_{2}^{\infty}}\mathfrak{D}_{\Phi_{\mu}(\mathbf{T})^{*}}$. Then by Corollary 3.1 (3), $\widetilde{\mathcal{H}}$ is joint reducing for $\mathbf{T}$. Therefore,

[TABLE]

forcing $\widetilde{\mathcal{H}}=\mathcal{H}$. This together with Corollary 1.4 imples that $\mathbf{T}$ is of Beurling type. Since $\dim\mathfrak{D}_{\mathbf{T}^{*}}=1$ (otherwise $x$ is not cyclic for $\mathbf{T}$), the canonical analytic model $\mathcal{Q}_{\mathbf{T}}$ for $\mathbf{T}$ can be viewed as a doubly commuting quotient module $\mathcal{Q}$ of $H^{2}(\mathbb{D}_{2}^{\infty})$, and then $\mathbf{T}$ is jointly unitarily equivalent to the compression $P_{\mathcal{Q}}\mathbf{M}_{\zeta}|_{\mathcal{Q}}$ of $\mathbf{M}_{\zeta}$ on $\mathcal{Q}$, where $\mathbf{M}_{\zeta}$ is the tuple of coordinate multiplication operators on $H^{2}(\mathbb{D}_{2}^{\infty})$. This completes the proof. ∎

Also from (5.3), it seems plausible to view every doubly commuting quotient module of $H^{2}(\mathbb{D}_{2}^{\infty})$ as the tensor product of infinitely many model spaces or $H^{2}(\mathbb{D})$’s. This can be realized after giving an appropriate definition for the infinite tensor product.

Let $\{M_{n}\}_{n\in\mathbb{N}}$ be a sequence of closed subspaces of $H^{2}(\mathbb{D})$, the tensor product of $\{M_{n}\}_{n\in\mathbb{N}}$ in $H^{2}(\mathbb{D}_{2}^{\infty})$, denoted by $\bigotimes_{n=1}^{\infty}M_{n}$, is definied to be the closed subspace of $H^{2}(\mathbb{D}_{2}^{\infty})$ spanned by the functions in $H^{2}(\mathbb{D}_{2}^{\infty})$ of form $\prod_{n=1}^{\infty}f_{n}(\zeta_{n})$ (in pointwise convergence for $\zeta\in\mathbb{D}_{2}^{\infty}$) with $f_{n}\in M_{n}$ for each $n\in\mathbb{N}$. Note that for infinite tensor products of form

[TABLE]

this new definition coincides with the original one that the $(M_{\zeta_{n+1}},M_{\zeta_{n+2}},\cdots)$-joint invariant subspace generated by $M_{1}\otimes\cdots\otimes M_{n}$ (see Subsection 2.2).

Corollary 5.6**.**

Every doubly commuting quotient module of $H^{2}(\mathbb{D}_{2}^{\infty})$ is the tensor product of some sequence of quotient modules of $H^{2}(\mathbb{D})$.

Proof.

Let $\mathcal{Q}$ be a doubly commuting quotient module of $H^{2}(\mathbb{D}_{2}^{\infty})$. Then there exists a sequence $\{\mathcal{J}_{n}\}_{n\in\mathbb{N}}$ of quotient modules of $H^{2}(\mathbb{D})$, such that

[TABLE]

Now we will prove $\mathcal{Q}=\bigotimes_{n=1}^{\infty}\mathcal{J}_{n}$.

For simplicity, rewrite

[TABLE]

The inclusion $\bigotimes_{n=1}^{\infty}\mathcal{J}_{n}\subseteq\bigcap_{n=1}^{\infty}\mathcal{Q}_{n}=\mathcal{Q}$ is trivial to see. For the reverse inclusion, note that the set $\{P_{\mathcal{Q}}\zeta^{\alpha}:\alpha\in\mathbb{Z}_{+}^{(\infty)}\}$ is complete in $\mathcal{Q}$, where $P_{\mathcal{Q}}$ is the orthogonal projection from $H^{2}(\mathbb{D}_{2}^{\infty})$ onto $\mathcal{Q}$. It suffices to show that for any fixed $\alpha=(\alpha_{1},\alpha_{2},\cdots)\in\mathbb{Z}_{+}^{(\infty)}$, $P_{\mathcal{Q}}\zeta^{\alpha}$ belongs to the infinite tensor product $\bigotimes_{n=1}^{\infty}\mathcal{J}_{n}$. Let $P_{\mathcal{Q}_{n}}\ (n\in\mathbb{N})$ denote the orthogonal projection from $H^{2}(\mathbb{D}_{2}^{\infty})$ onto $\mathcal{Q}_{n}$, and $P_{\mathcal{J}_{n}}$ denote the orthogonal projection from $H^{2}(\mathbb{D})$ onto $\mathcal{J}_{n}$. Then for each $n\in\mathbb{N}$,

[TABLE]

Taking $m\in\mathbb{N}$ so that $\alpha_{m+1}=\alpha_{m+2}=\cdots=0$ and setting $f_{n}=P_{\mathcal{J}_{n}}1\ (n\in\mathbb{N})$, we further have

[TABLE]

for any $n\geq m+1$. On the other hand, since $\{P_{\mathcal{Q}_{n}}\}_{n\in\mathbb{N}}$ converges to $P_{\mathcal{Q}}$ $(n\rightarrow\infty)$ in the strong operator topology, $P_{\mathcal{Q}_{n}}\zeta^{\alpha}$ converges to $P_{\mathcal{Q}}\zeta^{\alpha}$ $(n\rightarrow\infty)$ in $H^{2}(\mathbb{D}^{\infty}_{2})$-norm. In particular, $P_{\mathcal{Q}_{n}}\zeta^{\alpha}$ converges pointwisely to $P_{\mathcal{Q}}\zeta^{\alpha}$ as $n\rightarrow\infty$, and then $P_{\mathcal{Q}}\zeta^{\alpha}$ is of form

[TABLE]

This gives $P_{\mathcal{Q}}\zeta^{\alpha}\in\bigotimes_{n=1}^{\infty}\mathcal{J}_{n}$, and the proof is complete.

∎