Doubly commuting invariant subspaces for representations of product systems of $C^*$-correspondences

TL;DR

This paper extends Wold-type decompositions to doubly commuting covariant representations of product systems of $C^*$-correspondences, providing new insights into invariant subspaces and their structure using Fock space methods.

Contribution

It introduces a Shimorin-Wold-type decomposition for doubly commuting representations, extending previous results to a broader class of product systems.

Findings

Extended Wold-type decomposition for doubly commuting covariant representations.

Characterized wandering subspaces in the context of these representations.

Established a Beurling-type theorem for invariant subspaces using Fock space techniques.

Abstract

We obtain a Shimorin-Wold-type decomposition for a doubly commuting covariant representation of a product system of $C^*$-correspondences. This extends a recent Wold-type decomposition by Jeu and Pinto for a $q$-doubly commuting isometries. Application to the wandering subspaces of doubly commuting induced representations is explored, and a version of Mandrekar's Beurling type theorem is obtained to study doubly commuting invariant subspaces using Fock space approach due to Popescu.

Equations

$$a\xi :=\varphi (a)\xi (a\in \mathcal{M},\xi \in E).$$

$$({\zeta }_{1}a)\otimes {\xi }_1={\zeta }_1\otimes \varphi (a){\xi }_1,$$

$$⟨{\zeta }_1\otimes {\xi }_1,{\zeta }_2\otimes {\xi }_2⟩=⟨{\xi }_1,\varphi (⟨{\zeta }_1,{\zeta }_2⟩){\xi }_2⟩$$

$$T(a\xi a^{\prime })=\sigma (a)T(\xi )\sigma (a^{\prime })(\xi \in E;a,a^{\prime }\in \mathcal{M}).$$

$$T(\xi {)}^{\ast }T(\zeta )=\sigma (⟨\xi ,\zeta ⟩)(\xi ,\zeta \in E).$$

$${\mathfrak{L}}_n(\mathcal{W}):=\bigvee \{T({\xi }_1)T({\xi }_2)\cdots T({\xi }_n)h:{\xi }_i\in E,h\in \mathcal{W}\},$$

$$\mathcal{H}=\underset{n\in {\mathbb{N}}_0}{\bigvee }{\mathfrak{L}}_n(\mathcal{W}),$$

$${\mathfrak{L}}_n(\mathcal{W})\perp {\mathfrak{L}}_m(\mathcal{W}),$$

$${\mathcal{H}}_1=\underset{n\ge 0}{\bigvee }{\mathfrak{L}}_n(\mathcal{W}){\mathcal{H}}_2=\bigcap _{n\ge 1}{\tilde{T}}_n(E^{\otimes n}\otimes \mathcal{H}),$$

$${\tilde{T}}^{(i)}(I_{E_i}\otimes {\tilde{T}}^{(j)})={\tilde{T}}^{(j)}(I_{E_j}\otimes {\tilde{T}}^{(i)})(t_{i,j}\otimes I_{\mathcal{H}})$$

$$t_{ij}(x\otimes y):=\left\{\begin{array}[]{ll}z_{ij}(y\otimes x),&\hbox{if}\>i\neq j,\\ x\otimes y,&\hbox{if}\>i=j.\end{array}\right.$$

$$(\sigma ,T^{(1)},\dots ,T^{(k)})⟷(T^{(1)}(1),T^{(2)}(1),\dots ,T^{(k)}(1))$$

$${\tilde{T}}_l^{(i)}({\xi }_1\otimes \cdots \otimes {\xi }_l\otimes h):=T^{(i)}({\xi }_1)\cdots T^{(i)}({\xi }_l)h$$

$${\tilde{T}}_l^{(i)}={\tilde{T}}^{(i)}(I_{E_i}\otimes {\tilde{T}}^{(i)})\cdots (I_{E_i^{\otimes l-1}}\otimes {\tilde{T}}^{(i)}).$$

$${\tilde{T}}_{\mathbf{n}}={\tilde{T}}_{n_1}^{(1)}\left(I_{E_1^{\otimes n_1}}\otimes {\tilde{T}}_{n_2}^{(2)}\right)\cdots \left(I_{E_1^{\otimes n_1}\otimes \cdots \otimes E_{k-1}^{\otimes n_{k-1}}}\otimes {\tilde{T}}_{n_k}^{(k)}\right).$$

$${\tilde{T}}_{\mathbf{m}}^A={\tilde{T}}_{m_{i_1}}^{(i_1)}\left(I_{E_{i_1}^{\otimes m_{i_1}}}\otimes {\tilde{T}}_{m_{i_2}}^{(i_2)}\right)\cdots \left(I_{E_{i_1}^{\otimes m_{i_1}}\otimes \cdots \otimes E_{i_{p-1}}^{\otimes m_{i_{p-1}}}}\otimes {\tilde{T}}_{m_{i_p}}^{(i_p)}\right).$$

$${\mathfrak{L}}_{\mathbf{m}}^A(\mathcal{K}):=\bigvee \{T_{m_{i_1}}^{(i_1)}({\eta }_{i_1})\cdots T_{m_{i_p}}^{(i_p)}({\eta }_{i_p})h:{\eta }_{i_j}\in E_{i_j}^{\otimes m_{i_j}},1\le j\le p,h\in \mathcal{K}\}.$$

$$\mathcal{K}\perp {\mathfrak{L}}_{\mathbf{m}}^A(\mathcal{K})\text{mbox}foreach\mathbf{m}\in {\mathbb{N}}_0^A\setminus \{0\}.$$

$$\mathcal{H}=\underset{\mathbf{m}\in {\mathbb{N}}_0^A}{\bigvee }{\mathfrak{L}}_{\mathbf{m}}^A(\mathcal{K}).$$

$${\mathcal{W}}_A:=\bigcap _{j=1}^n(\mathcal{H}\ominus {\tilde{T}}^{(i_j)}(E_j\otimes \mathcal{H})).$$

$${\mathcal{W}}_A=\bigcap _{i_j\in A}{\mathcal{W}}_{i_j}.$$

$${\tilde{T}}^{(j{)}^{\ast }}{\tilde{T}}^{(i)}=(I_{E_j}\otimes {\tilde{T}}^{(i)})(t_{i,j}\otimes I_{\mathcal{H}})(I_{E_i}\otimes {\tilde{T}}^{(j{)}^{\ast }}).$$

$$T_i^{\ast }{T}_j=\overline{z_{ij}}{T}_j{T}_i^{\ast },\text{mbox}foralli,j\text{mbox}withi\ne j.$$

$${\tilde{T}}^{(i{)}^{\ast }}{\tilde{T}}^{(i)}{\tilde{T}}^{(j{)}^{\ast }}{\tilde{T}}^{(j)}={\tilde{T}}^{(j{)}^{\ast }}{\tilde{T}}^{(j)}{\tilde{T}}^{(i{)}^{\ast }}{\tilde{T}}^{(i)}.$$

$${\mathcal{W}}_{\alpha }\ominus {\tilde{T}}^{(j)}(E_j\otimes {\mathcal{W}}_{\alpha })={\mathcal{W}}_{\alpha }\cap {\mathcal{W}}_j,\text{for}allj\notin \alpha .$$

$${\mathcal{H}}_A$$

$$\mathcal{H}$$

$$=\underset{n_1\in {\mathbb{N}}_0}{\bigvee }{\mathfrak{L}}_{n_1}^{(1)}({\mathcal{W}}_1)⨁\left(\bigcap _{n_1\in {\mathbb{N}}_0}{\mathfrak{L}}_{n_1}^{(1)}(\mathcal{H})\right).$$

$${\mathcal{W}}_1$$

$$=\underset{n_2\in {\mathbb{N}}_0}{\bigvee }{\mathfrak{L}}_{n_2}^{(2)}({\mathcal{W}}_1\cap {\mathcal{W}}_2)⨁\left(\bigcap _{n_2\in {\mathbb{N}}_0}{\mathfrak{L}}_{n_2}^{(2)}({\mathcal{W}}_1)\right),$$

$$\mathcal{H}$$

Full text

*corresponding author

1. Introduction

The theorem of Wold [34], known as Wold decomposition, says that every isometry on a Hilbert space decomposes uniquely as a direct sum of a shift operator and a unitary operator. A classical application of the Wold decomposition is Beurling’s theorem [4] which states that if $\mathcal{K}$ is a closed invariant subspace for the shift $M_{z}$ of the Hardy space $H^{2}(\mathbb{D}),$ then $\mathcal{K}$ is the image of an inner function. A wandering subspace theorem due to Halmos [8] is a generalization of the Beurling’s theorem. In [26], Rudin explained that the Beurling theorem fails in general in the multivariable case, that is, for commuting tuple of isometries. Słociński in [29] proved a Wold-type decomposition for doubly commuting isometries, which is based on a characterization of the existence of a wandering subspace for commuting tuple of isometries in terms of the doubly commuting condition. Mandrekar [14] utilized the Słociński’s decomposition and gave a Beurling’s type theorem for the Hardy space over the bidisc $H^{2}(\mathbb{D}^{2})$ under the assumption that the tuple of multiplication operators by co-ordinate functions $(M_{z_{1}},M_{z_{2}})$ is doubly commuting. Sarkar, Sasane and Wick [23] developed the Mandrekar’s result, in the polydisc case $H^{2}(\mathbb{D}^{n}),$ based on a multivariable version of the Słociński’s result given in [24]. There is a recent book [15] on this theme by Mandrekar and Redett.

Pimsner in [20] extended the notion of Cuntz-Kreiger algebras using the terminology of isometric covariant representations of $C^{*}$-correspondences. The $C^{*}$-representations of tensor algebras of a $C^{*}$-correspondence are in a bijective correspondence with the isometric covariant representations of the $C^{*}$-correspondence (cf. [20]). In [17], Muhly and Solel derived an analogue of the Wold decomposition for the isometric covariant representations which is a generalization of the Wold decomposition for a row isometry by Popescu [21]. Arveson explored the notion of tensor product system of Hilbert spaces in [1] to classify $E_{0}$-semigroups. Solel in [31] introduced the notion of doubly commuting covariant representations of product systems of $C^{*}$-correspondences and explored theory of regular dilations. Skalski and Zacharias generalized Słociński’s Wold-type decomposition for the doubly commuting isometric covariant representations.

Shimorin [27] generalized the Wold decomposition for an operator close to isometry. A version of Shimorin-Wold-type decomposition for covariant representations close to isometric is proved in [7, Theorem 3.13]. We obtain an analogue of this result for the doubly commuting covariant representations in Section 3, which provides a different proof of the result of Skalski and Zacharias, based on the techniques developed in [24, 13]. In the setting of $C^{*}$-correspondences induced representations, introduced by Rieffel [32], plays the role of a shift. In this section the wandering subspace theorem for commuting shift [24, Theorem 3.3] is extended for a covariant representation of a product system. Final section is devoted to the Beurling-Lax type characterization of the doubly commuting invariant subspaces.

1.1. Preliminaries: Shimorin Wold-type decomposition of single covariant representation

In this subsection we recall few definitions and elementary properties of covariant representations of $C^{*}$-correspondences from (see [11, 16, 19, 20]).

Let $E$ be a Hilbert $C^{*}$-module over a $C^{*}$-algebra $\mathcal{M}.$ By $\mathcal{L}(E)$ we denote the $C^{*}$-algebra of all adjointable operators on $E$. We say that the module $E$ is a $C^{*}$-correspondence over $\mathcal{M}$ if there exists a left $\mathcal{M}$-module structure through a non-zero $*$-homomorphism $\phi:\mathcal{M}\to\mathcal{L}(E)$ in the following sense

[TABLE]

Each $*$-homomorphism considered in this article is essential, that means, the closed linear span of $\phi(\mathcal{M})E$ is $E.$ Every $C^{*}$-correspondence has usual operator space structure induced from viewing it as a corner in respective linking algebra. If $F$ is another $C^{*}$-correspondence over $\mathcal{M},$ then we may consider the notion of tensor product $F\otimes_{\phi}E$ (cf. [11]) which satisfy

[TABLE]

[TABLE]

for every $\zeta_{1},\zeta_{2}\in F;$ $\xi_{1},\xi_{2}\in E$ and $a\in\mathcal{M}.$

Definition 1.1**.**

Let $\mathcal{H}$ be a Hilbert space, and $E$ be a $C^{*}$-correspondence over a $C^{*}$-algebra $\mathcal{M}$. Let $\sigma:\mathcal{M}\to B(\mathcal{H})$ be a representation and $T:E\to B(\mathcal{H})$ be a linear map. Then the pair $(\sigma,T)$ is said to be a covariant representation of $E$ on $\mathcal{H}$ if

[TABLE]

We say that the covariant representation is completely bounded (respectively, completely contractive) if $T$ is completely bounded (respectively, completely contractive). Moreover, it is called isometric if

[TABLE]

Example 1.2**.**

Let $\mathcal{K}$ be a finite or countably infinite dimensional Hilbert space and $\sigma$ be a representation of the complex number $\mathbb{C}$ on $\mathcal{K}.$ Let $(\sigma,T)$ be a completely contractive covariant representation of $E(:=\mathcal{K})$ on a Hilbert space $\mathcal{H}.$ If $\{e_{n}\}_{n=1}^{m}(m:=\mbox{dim }\>\mathcal{K})$ is an orthonormal basis for $E,$ then $T(\xi)=\sum\langle\xi,e_{n}\rangle T_{n}$ where $T_{n}:=T(e_{n}),1\leq n\leq m.$ Moreover, since $(\sigma,T)$ is completely contractive, $\sum_{n=1}^{m}T_{n}T^{*}_{n}\leq I.$ Conversely, let $m\in\mathbb{N}\cup\{{\infty}\}$ and $\{T_{n}\}_{n=1}^{m}$ be a sequence of operators on the Hilbert space $\mathcal{H}$ such that $\sum_{n=1}^{m}T_{n}T^{*}_{n}\leq I.$ Define a completely contractive covariant representation $(\sigma,T)$ of $E$ on $\mathcal{H}$ by $T(e_{n}):=T_{n}$ and $\sigma(x):=xI_{\mathcal{H}},$ where $E$ is Hilbert space whose dimension is $m,$ with the orthonormal basis $\{e_{n}\}_{n=1}^{m}.$ Therefore, there is a one-to-one correspondence between the completely contractive covariant representation of $E$ on Hilbert space $\mathcal{H}$ and the sequence $\{T_{n}\}_{n=1}^{m}(m:=\mbox{dim }\>E)$ of operators on $\mathcal{H}$ satisfying $\sum_{n=1}^{m}T_{n}T^{*}_{n}\leq I.$ For more details and connection with the dilation theory of contractions and the Wold decomposition, see [16, 17].

Lemma 1.3**.**

([16, Lemma 3.5]) The mapping $(\sigma,T)\mapsto\widetilde{T}$ gives a bijection between the collection of all completely bounded (respectively, completely contractive), covariant representations $(\sigma,T)$ of $E$ on $\mathcal{H}$ and the collection of all bounded (respectively, contractive) linear maps \widetilde{T}:~{}\mbox{E\otimes_{\sigma}\mathcal{H}\to\mathcal{H}} which satisfies $\widetilde{T}(\phi(a)\otimes I_{\mathcal{H}})=\sigma(a)\widetilde{T}$, $a\in\mathcal{M}$. Moreover, the covariant representation $(\sigma,T)$ is isometric if and only if $\widetilde{T}$ is an isometry.

We say that the covariant representation $(\sigma,T)$ is fully co-isometric if $\widetilde{T}$ is co-isometric, that is, $\widetilde{T}\widetilde{T}^{*}=I_{\mathcal{H}}.$ Let $E$ be a $C^{*}$-correspondence over a $C^{*}$-algebra $\mathcal{M}.$ Then for each $n\in\mathbb{N},$ let us recall the following notation and definition of Wandering subspaces from [9]:

Definition 1.4**.**

Let $E$ be a $C^{*}$-correspondence over a $C^{*}$-algebra $\mathcal{M}.$ Let $(\sigma,T)$ be a completely bounded, covariant representation of $E$ on a Hilbert space $\mathcal{H}$. For a closed $\sigma(\mathcal{M})$-invariant subspace $\mathcal{W}$ of $\mathcal{H},$ we define

[TABLE]

for $n\in\mathbb{N}$ and $\mathfrak{L}_{0}(\mathcal{W}):=\mathcal{W}$. Then $\mathcal{W}$ is called wandering for the covariant representation $(\sigma,T)$, if $\mathcal{W}$ is orthogonal to the subspaces $\mathfrak{L}_{n}(\mathcal{W}),$ for all $n\in\mathbb{N}_{0}.$ We say that $(\sigma,T)$ has generating wandering subspace property if

[TABLE]

for some wandering subspace $\mathcal{W}$ of $\mathcal{H},$ and we call $\mathcal{W}$ as generating wandering subspace for $(\sigma,T).$

Note that if $(\sigma,T)$ is isometric covariant representation of $E$ on $\mathcal{H},$ then $\mathcal{W}$ is wandering subspace for $(\sigma,T)$ if and only if

[TABLE]

for all distinct $n,m\in\mathbb{N}_{0}.$

Definition 1.5**.**

Let $\mathcal{K}$ be a closed subspace of $\mathcal{H}.$ The subspace $\mathcal{K}$ is called $(\sigma,T)$-invariant (respectively, $(\sigma,T)$-reducing) (cf. [28]) if it is $\sigma(\mathcal{M})$-invariant (i.e. the projection onto $\mathcal{K}$, will be denoted throughout by $P_{\mathcal{K}}$, lies in $\sigma(\mathcal{M})^{\prime})$, and if $\mathcal{K}$ (respectively, both $\mathcal{K},\mathcal{K}^{\bot}$) is left invariant by each operator $T(\xi)$ for $\xi\in E.$ Then the natural restriction of this representation provides a new representation of $E$ on $\mathcal{K}$ and it will denoted by $(\sigma,T)|_{\mathcal{K}}.$

Definition 1.6**.**

Let $(\sigma,T)$ be a completely bounded, covariant representation of $E$ on a Hilbert space $\mathcal{H}.$ We say that the covariant representation $(\sigma,T)$ admits Wold-type decomposition if the representation $(\sigma,T)$ decomposes into a direct sum $(\sigma_{1},T_{1})\oplus(\sigma_{2},T_{2})$ on $\mathcal{H}=\mathcal{H}_{1}\oplus\mathcal{H}_{2}$ where $(\sigma_{1},T_{1})=(\sigma,T)|_{\mathcal{H}_{1}}$ has generating wandering subspace property and $(\sigma_{2},T_{2})=(\sigma,T)|_{\mathcal{H}_{2}}$ is isometric and fully co-isometric covariant representation.

The following Wold-type decomposition for a completely bounded, covariant representation of a $C^{*}$-correspondence which are close to isometric is from [25](see also [17]). We use symbol $I$ for $I_{\mathcal{H}}.$

Theorem 1.7**.**

Let $(\sigma,T)$ be a completely bounded, covariant representation of $E$ on $\mathcal{H},$ which satisfies any one of the following conditions:

1. (1)

$(\sigma,T)$* is concave, that is, *

$\|\widetilde{T}_{2}(\eta\otimes h)\|^{2}+\|\eta\otimes h\|^{2}\leq 2\|(I_{E}\otimes\widetilde{T})(\eta\otimes h)\|^{2},\eta\in E^{\otimes 2},h\in\mathcal{H};$** 2. (2)

$\|(I_{E}\otimes\widetilde{T})(\zeta)+\kappa\|^{2}\leq 2(\|\zeta\|^{2}+\|\widetilde{T}(\kappa)\|^{2}),\>\>\>\>\zeta\in E^{\otimes 2}\otimes\mathcal{H},\kappa\in E\otimes\mathcal{H};$** 3. (3)

$\|\widetilde{T}(\xi)\|^{2}+\|\widetilde{T}^{*}_{2}\widetilde{T}(\xi)\|^{2}\leq 2\|\widetilde{T}^{*}\widetilde{T}(\xi)\|^{2},\>\xi\in E\otimes\mathcal{H}.$**

Then $(\sigma,T)$ admits Wold-type decomposition. Moreover, the decomposition is unique and the corresponding reducing subspaces are given by

[TABLE]

where $\mathcal{W}=\mbox{ker}\widetilde{T}^{*}.$ In particular, if $(\sigma,T)$ is analytic (pure), that is, $\mathcal{H}_{2}=\{0\}$, then $\mathcal{W}=\mathcal{H}\ominus\widetilde{T}(E\otimes\mathcal{H})$ is the generating wandering subspace for $(\sigma,T),$ that is, $\mathcal{H}=\bigvee_{n\in\mathbb{N}_{0}}\mathfrak{L}_{n}(\mathcal{W}).$

In the case of isometric covariant representations of product systems Solel proved in [31] that the doubly commuting condition (2.5) is equivalent to Nica-covariance (see [18]). Our main theorem, Theorem 3.1, extends Theorem 1.7 for the doubly commuting completely bounded covariant representations which are close to isometric.

2. Notation and basic definitions for doubly commuting case

Throughout the paper $k\in\mathbb{N}.$ The central tool we require is a product system of $C^{*}$-correspondences (see [5, 30, 31, 28]): The product system $\mathbb{E}$ is defined by a family of $C^{*}$-correspondences $\{E_{1},\ldots,E_{k}\},$ and by the unitary isomorphisms $t_{i,j}:E_{i}\otimes E_{j}\to E_{j}\otimes E_{i}$ ($i>j$). Using these identifications, for all ${\bf n}=(n_{1},\cdots,n_{k})\in\mathbb{N}_{0}^{k}$ the correspondence $\mathbb{E}({\bf n})$ is identified with $E_{1}^{\otimes^{n_{1}}}\otimes\cdots\otimes E_{k}^{\otimes^{n_{k}}}.$ We use notations $t_{i,i}={\textrm{id}}_{E_{i}\otimes E_{i}}$ and $t_{i,j}=t_{j,i}^{-1}$ when $i<j.$

Definition 2.1**.**

Assume $\mathbb{E}$ to be a product system over $\mathbb{N}_{0}^{k}$. Let $\sigma$ be a representation of $\mathcal{M}$ on $\mathcal{H}$ and let $T^{(i)}:E_{i}\to B(\mathcal{H})$ be a linear map for $1\leq i\leq k.$ The tuple $(\sigma,T^{(1)},\dots,T^{(k)})$ is called a completely bounded (respectively, completely contractive) covariant representation of product system $\mathbb{E}$ on $\mathcal{H}$ if each pair $(\sigma,T^{(i)})$ is a completely bounded (respectively, completely contractive) covariant representation of $E_{i}$ on $\mathcal{H}$ and satisfy the commutation relation

[TABLE]

with $1\leq i,j\leq k$. The covariant representation $(\sigma,T^{(1)},\ldots,T^{(k)})$ is called isometric if every $(\sigma,T^{(i)})$ is isometric as a covariant representation of the $C^{*}$-correspondence $E_{i}$, and similarly fully coisometric representation is defined.

Example 2.2**.**

Let $T=(T_{1},\dots,T_{k})$ be a $k$-tuple of operators on Hilbert space $\mathcal{H}$ such that $T_{j}T_{i}=q_{ij}T_{i}T_{j}$ for all $1\leq i,j\leq k,$ where $q_{ij}$ are non-zero complex numbers, then we shall refer to $(T_{1},\dots,T_{k})$ as $q$-commuting (cf. [3, 10]). This implies that $q_{ij}=q^{-1}_{ji}$ when $T_{i}T_{j}\neq 0,$ where $i,j\in\{1,\dots,k\}.$ On the other hand, suppose $z_{ij}\in\mathbb{T}$ and $z_{ji}=\overline{z}_{ij}$ for all $i\neq j.$ For $1\leq i,j\leq k$ set $E_{i}:=\mathbb{C}$ and define $t_{ij}:E_{i}\otimes E_{j}\to E_{j}\otimes E_{i}$ as follows:

[TABLE]

Observe that $t_{ij}$’s are unitaries with $t_{ij}^{-1}=t_{ji}$ for $i\neq j$ and $t_{ii}=id.$ Hence $(\{E_{i}\}_{i=1}^{k},\{t_{ij}\}_{i,j=1}^{k})$ uniquely determines a product system $\mathbb{E}$ over $\mathbb{N}^{k}_{0}.$ Let $1\leq i\leq k,$ define $T^{(i)}:E_{i}\to B(\mathcal{H})$ by $T^{(i)}(x)h:=xT_{i}(h),x\in E_{i},h\in\mathcal{H}$ and $\sigma:\mathbb{C}\to B(\mathcal{H})$ by $\sigma(x):=xI_{\mathcal{H}},$ that is, $\sigma$ is identity representation of $\mathbb{C}$ on a Hilbert space $\mathcal{H}.$ It is easy to see that if $(T_{1},\dots,T_{k})$ is $q$-commuting, then $(\sigma,T^{(1)},\dots,T^{(k)})$ is completely bounded covariant representation of $\mathbb{E}$ on the Hilbert space $\mathcal{H}.$ Conversely, let $(\sigma,T^{(1)},\ldots,T^{(k)})$ be a completely bounded covariant representation of the product systems $\mathbb{E}$ over $\mathbb{N}^{k}_{0}$ on the Hilbert space $\mathcal{H},$ where $E_{i}=\mathbb{C},$ for all $1\leq i\leq n.$ Since $E_{i}=\mathcal{M}=\mathbb{C}$, $\sigma$ is the identity representation and for each $i\in\{1,\dots,k\},$ the completely bounded linear map $T^{(i)}$ may be considered as a bounded operator on $\mathcal{H.}$ Therefore, the map

[TABLE]

induces a bijection between the set of all completely bounded covariant representation of $\mathbb{E}$ (with $E_{i}=\mathbb{C}$ for all $1\leq i\leq k$) on a Hilbert space $\mathcal{H}$ over the $C^{*}$-algebra $\mathbb{C}$ onto the collection of all $q$-commuting operators $(T_{1},\dots,T_{k})$ on $\mathcal{H}.$

We say that two such a completely bounded covariant representations $(\sigma,T^{(1)},\ldots,T^{(k)})$ and $(\psi,V^{(1)},\ldots,V^{(k)})$ of the product system $\mathbb{E}$ over $\mathbb{N}^{k}_{0}$, respectively on Hilbert spaces $\mathcal{H}$ and $\mathcal{K}$, are isomorphic (cf. [28]) if we have a unitary $U:\mathcal{H}\to\mathcal{K}$ which gives the unitary equivalence of representations $\sigma$ and $\psi,$ and also for each $1\leq i\leq k$, $\xi\in E_{i}$ one has $S^{(i)}(\xi)=UT^{(i)}(\xi)U^{*}$.

For each $1\leq i\leq k$ and $l\in\mathbb{N}$ define $\widetilde{T}^{(i)}_{l}:E_{i}^{\otimes l}\otimes\mathcal{H}\to\mathcal{H}$ by

[TABLE]

where $\xi_{1},\ldots,\xi_{l}\in E_{i},h\in\mathcal{H}$. Then

[TABLE]

For $\mathbf{n}=(n_{1},\cdots,n_{k})\in\mathbb{N}_{0}^{k}$, we write $\widetilde{T}_{\mathbf{n}}:\mathbb{E}(\mathbf{n})\otimes_{\sigma}\mathcal{H}\to\mathcal{H}$ by

[TABLE]

The map $T_{\mathbf{n}}:\mathbb{E}(\mathbf{n})\to B(\mathcal{H})$ is then defined by $T_{\mathbf{n}}(\xi)h=\widetilde{T}_{\mathbf{n}}(\xi\otimes h),~{}\xi\in\mathbb{E}(\mathbf{n}),h\in\mathcal{H}.$

Definition 2.3**.**

Let $\mathcal{K}$ be a closed subspace of a Hilbert space $\mathcal{H}.$ The subspace $\mathcal{K}$ is called invariant( respectively, reducing) (cf. [28]) for a covariant representation $(\sigma,T^{(1)},\ldots,T^{(k)})$ on $\mathcal{H},$ if $\mathcal{K}$ is $(\sigma,T^{(i)})$-invariant(respectively, $(\sigma,T^{(i)})$-reducing) subspace for $1\leq i\leq k.$ Then it is evident that the natural ‘restriction’ of this representation to $\mathcal{K}$ provides a new representation of $\mathbb{E}$ on $\mathcal{K}$, which is called a summand of $(\sigma,T^{(1)},\ldots,T^{(k)})$ and will be denoted by $(\sigma,T^{(1)},\ldots,T^{(k)})|_{\mathcal{K}}$.

Let $A=\{i_{1},\dots i_{p}\}\subseteq I_{k},$ where $I_{k}:=\{1,2,\dots,k\}$, denote $\mathbb{N}_{0}^{A}:=\{\mathbf{m}=(m_{i_{1}},\cdots,m_{i_{p}})\>:\>m_{i_{j}}\in\mathbb{N}_{0},1\leq j\leq p\}.$ Let $\mathbf{m}=(m_{i_{1}},\cdots,m_{i_{p}})\in\mathbb{N}_{0}^{A}$, define $\widetilde{T}_{\mathbf{m}}^{A}:\mathbb{E}(\mathbf{m})\otimes_{\sigma}\mathcal{H}\to\mathcal{H}$ by

[TABLE]

Moreover, if $\mathcal{K}$ is a $\sigma(\mathcal{M})$-invariant subspace, then we denote

[TABLE]

Clearly $\mathfrak{L}_{\mathbf{m}}^{A}(\mathcal{K})=\overline{\widetilde{T}_{\mathbf{m}}^{A}(\mathbb{E}(\mathbf{m})\otimes_{\sigma}\mathcal{K})}.$ It is easy to see that $\mathfrak{L}_{\mathbf{m}}^{A}(\mathcal{K})$ is a smallest $(\sigma,T^{(i_{1})},\dots,T^{(i_{p})})$-invariant subspace which contains $\mathcal{K}.$

Definition 2.4**.**

1. (1)

We say that the $\sigma(\mathcal{M})$-invariant closed subspace $\mathcal{K}$ is said to be wandering for the covariant representation $(\sigma,T^{(i_{1})},\dots,T^{(i_{p})})$ if

[TABLE] 2. (2)

The covariant representation $(\sigma,T^{(i_{1})},\dots,T^{(i_{p})})$ is said to have the generating wandering subspace property if there exists a wandering subspace $\mathcal{K}\subseteq\mathcal{H}$ for $(\sigma,T^{(i_{1})},\dots,T^{(i_{n})})$ such that $[\mathcal{K}]_{T_{A}}=\mathcal{H},$ that is,

[TABLE]

Let $A=\{i_{1},\dots,i_{n}\}$ be a non-empty subset of $I_{k},$ define the closed subspace $\mathcal{W}_{A}$ of $\mathcal{H}$ by

[TABLE]

Again, if $A=\{i\}$ we simply write $\mathcal{W}_{i}:=\mathcal{H}\ominus\widetilde{T}^{(i)}(E_{i}\otimes\mathcal{H}).$ Therefore

[TABLE]

Definition 2.5**.**

A completely bounded, covariant representation $(\sigma,T^{(1)},\ldots,$ $T^{(k)})$ of $\mathbb{E}$ on a Hilbert space $\mathcal{H}$ is said to be doubly commuting (cf. [31]) if for each distinct $i,j\in\{1,\ldots,k\}$ we have

[TABLE]

Definition 2.6**.**

We say that a q-commuting operators $(T_{1},\dots,T_{k})$ is $q$-doubly commuting* (cf. [33, 13]) if*

[TABLE]

Example 2.7**.**

If $T:=(T_{1},\dots,T_{k})$ is $q$-doubly commuting tuple, the corresponding covariant representation $(\sigma,T^{(1)},$ $\dots,T^{(k)})$ defined in Example 2.2 is doubly commuting (for details see [31, Section 4]).

For distinct $i,j\in\{1,\ldots,k\}$, a simple calculation (cf. [28, p. 460]) using Equation (2.5) yields

[TABLE]

Thus the operators $\{I_{\mathcal{H}}-\widetilde{T}^{(i)^{*}}\widetilde{T}^{(i)}\}_{i=1}^{k}$ commute to each other. The following proposition is essential in order to extend Theorem 1.7 for the covariant representation $(\sigma,T^{(1)},\dots,T^{(k)})$ and it follows immediately from [7, Proposition 4.6].

Proposition 2.8**.**

Let $\mathbb{E}$ be a product system of $C^{*}$-correspondences over $\mathbb{N}_{0}^{k}$ and let $(\sigma,T^{(1)},\dots,$ $T^{(k)})$ be a doubly commuting completely bounded, covariant representation of $\mathbb{E}$ on a Hilbert space $\mathcal{H}.$ Then for each non-empty subset $\alpha\subseteq I_{k}$, the subspace $\mathcal{W}_{\alpha}$ is $(\sigma,T^{(j)})$-reducing, where $j\notin\alpha.$ Moreover

[TABLE]

3. Shimorin-Wold-type decomposition for a doubly commuting covariant representation of a product system

The following theorem is main result of this paper, we use it in the next section, to provide a complete classification of doubly commuting invariant subspaces.

Theorem 3.1**.**

Let $(\sigma,T^{(1)},\dots,T^{(k)})$ be a doubly commuting completely bounded, covariant representation of the product system $\mathbb{E}$ on a Hilbert space $\mathcal{H}$ satisfies one of the following properties:

1. (1)

$(\sigma,T^{(i)})$* is concave for each $i=1,\dots,k,$ that is, *

$\|\widetilde{T}_{2}^{(i)}(\eta_{i}\otimes h)\|^{2}+\|\eta_{i}\otimes h\|^{2}\leq 2\|(I_{E_{i}}\otimes\widetilde{T}^{(i)})(\eta_{i}\otimes h)\|^{2},\eta_{i}\in E^{\otimes 2}_{i},h\in\mathcal{H};$** 2. (2)

for any $\zeta_{i}\in E^{\otimes 2}_{i}\otimes\mathcal{H},\kappa_{i}\in E_{i}\otimes\mathcal{H}$

$\|(I_{{E}_{i}}\otimes\widetilde{T}^{(i)})(\zeta_{i})+\kappa_{i}\|^{2}\leq 2(\|\zeta_{i}\|^{2}+\|\widetilde{T}^{(i)}(\kappa_{i})\|^{2}).$** 3. (3)

$\|\widetilde{T}^{(i)}(\xi_{i})\|^{2}+\|\widetilde{T}^{(i)*}_{2}\widetilde{T}^{(i)}(\xi_{i})\|^{2}\leq 2\|\widetilde{T}^{(i)*}\widetilde{T}^{(i)}(\xi_{i})\|^{2},\>\xi_{i}\in E_{i}\otimes\mathcal{H}.$**

Then for $2\leq m\leq k,$ there exists $2^{m}$ $(\sigma,T^{(1)},\ldots,T^{(m)})$-reducing subspaces $\{\mathcal{H}_{A}\>:\>A\subseteq I_{m}\}$ such that $\mathcal{H}=\bigoplus_{A\subseteq I_{m}}\mathcal{H}_{A}$ and for each $A=\{i_{1},\dots,i_{p}\}\subseteq I_{m}:\>(\sigma,T^{(i_{1})},\dots,T^{(i_{p})})|_{\mathcal{H}_{A}}$ has generating wandering subspace property of the product system $\mathbb{E}_{A}$ over $\mathbb{N}^{A}_{0}$ given by the family of $C^{*}$-Correspondence $\{E_{i_{1}},\dots,E_{i_{p}}\}$ and $(\sigma,T^{(i)})|_{\mathcal{H}_{A}}$ is isometric and fully co-isometric covariant representation whenever $i\in I_{m}\setminus A.$ Moreover, the above decomposition is unique and

[TABLE]

Proof.

We shall prove this result by Mathematical induction.

Suppose $m=2:$ Apply Wold-type decomposition, Theorem 1.7 to the covariant representation $(\sigma,T^{(1)}),$ we get

[TABLE]

Since $\mathcal{W}_{1}$ is a $(\sigma,T^{(2)})$- reducing subspace, by applying the Wold-type decomposition to $(\sigma,T^{(2)})|_{\mathcal{W}_{1}}$, we have

[TABLE]

where the second equality follows from Proposition 2.8. Therefore

[TABLE]

where $A=\{1,2\}.$ The last equality follows that

[TABLE]

[TABLE]

Applying Theorem 1.7 again for the covariant representation $(\sigma,T^{(2)}),$ we obtain

[TABLE]

yields

[TABLE]

since $\mathcal{W}_{2}$ is $(\sigma,T^{(1)})$-reducing and $\bigcap_{\mathbf{n}\in\mathbb{N}_{0}^{2}}\mathfrak{L}_{\mathbf{n}}^{A}(\mathcal{H})\subseteq\bigcap_{n_{2}\in\mathbb{N}_{0}}\mathfrak{L}_{n_{2}}^{(2)}(\mathcal{H}).$ Now applying Equations (3) and (3.3), we get

[TABLE]

that is,

[TABLE]

where $\mathcal{H}_{A}$ as in Equation (3.1) for each $A\subseteq I_{2}$.

For the case $m+1\leq n$. Let us assume that for each $m<n$, we have $\mathcal{H}=\bigoplus_{A\subseteq I_{m}}\mathcal{H}_{A}$, where for each non-empty subset $A$ of $I_{m}$

[TABLE]

and when $A$ is an empty set,

[TABLE]

We want to prove this result for $m+1\leq k,$ that is,

[TABLE]

Since $\mathcal{W}_{A}$ is $(\sigma,T^{(m+1)})$-reducing subspace for all non-empty subset $A\subseteq I_{m},$ Theorem 1.7 for $(\sigma,T^{(m+1)})|_{\mathcal{W}_{A}}$ provides us

[TABLE]

Note that $\bigcap_{\mathbf{j}\in\mathbb{N}_{0}^{I_{m}\setminus A}}\mathfrak{L}_{\mathbf{J}}^{I_{m}\setminus A}(\mathcal{W}_{B})\subseteq\mathcal{W}_{m+1},$ where $B=A\cup\{m+1\}.$ Then

[TABLE]

[TABLE]

and hence it follows that

[TABLE]

Using Theorem 1.7, for the covariant representation $(\sigma,T^{(m+1)}),$ we have

[TABLE]

When $A$ is an empty set,

[TABLE]

It follows from the above orthogonal decomposition of $\mathcal{H}$ that the covariant representation $(\sigma,T^{(i_{1})},\dots,T^{(i_{p})})|_{\mathcal{H}_{A}}$ has generating wandering subspace property of the product system $\mathbb{E}_{A}$ over $\mathbb{N}^{A}_{0}$ and $(\sigma,T^{(i)})|_{\mathcal{H}_{A}}$ is an isometric and fully co-isometric covariant representation for all $i\in I_{m}\setminus A$ (cf. Theorem 1.7). The uniqueness also follows immediately from the uniqueness of Theorem 1.7. ∎

Remark 3.2**.**

1. (1)

Let $(\sigma,T^{(1)},\dots,T^{(k)})$ be a completely bounded covariant representation of $\mathbb{E}$ on the Hilbert space $\mathcal{H}.$ Let $\mathcal{W}$ be a wandering subspace for $(\sigma,T^{(1)},\dots,T^{(k)})$, that is, $\mathcal{W}\>\bot\>\mathfrak{L}_{\mathbf{n}}^{k}(\mathcal{W}),\>\>\mathbf{n}\in\mathbb{N}_{0}^{k}\backslash\{0\}.$ Let $\mathcal{K}$ be the smallest closed $(\sigma,T^{(1)},\dots,T^{(k)})$-invariant subspace containing $\mathcal{W},$ that is,

[TABLE]

Then the wandering subspace is unique. Indeed, the uniqueness of the wandering subspace follows from

[TABLE]

Therefore $\mathcal{W}=\bigcap_{i=1}^{k}\mbox{ker}(I_{E_{i}}\otimes P_{\mathcal{K}}){\widetilde{T}}^{(i)^{*}}|_{\mathcal{K}},$ where $P_{\mathcal{K}}$ is projection of $\mathcal{H}$ on $\mathcal{K}.$ 2. (2)

Suppose $(\sigma,T^{(1)},\dots,T^{(k)})$ be a doubly commuting, isometric covariant representation of $\mathbb{E}$ on the Hilbert space $\mathcal{H}.$ Let $\mathbf{n}=(n_{1},\cdots,n_{k})$ and $\mathbf{m}=(m_{1},\cdots,m_{k})\in\mathbb{N}^{k}_{0}$ with $\mathbf{n}\neq\mathbf{m},\>n_{i}\neq m_{i}$ for some $i\in I_{k}.$ Let $\xi_{\mathbf{n}}\in E(\mathbf{n}),\eta_{\mathbf{m}}\in E(\mathbf{m})$ and $h_{1},h_{2}\in\mathcal{W}.$ Assume $n_{i}<m_{i}$ and if $\mathcal{K}$ is in the above equation such that $\mathcal{K}$ is $(\sigma,T^{(1)},\dots,T^{(k)})$-reducing subspace, then we have

[TABLE]

Thus, $\mathfrak{L}_{\mathbf{n}}^{k}(\mathcal{W})\bot\mathfrak{L}_{\mathbf{m}}^{k}(\mathcal{W})$ for all distinct $\mathbf{n},\mathbf{m}\in\mathbb{N}^{k}_{0}.$ Hence

[TABLE]

The following statement is an analogue of generating wandering subspace property of the covariant representations and we provide a different proof of [7, Corollary 4.8], which is a generalization of [6, Corollary 2.4].

Corollary 3.3**.**

Let $\mathbb{E}$ be a product system of $C^{*}$-correspondences over $\mathbb{N}_{0}^{k}.$ Let $(\sigma,T^{(1)},\ldots,T^{(k)})$ be as in Theorem 3.1 such that $(\sigma,T^{(j)})$ is analytic, for $1\leq i\leq k.$ Let $\mathcal{K}$ be a $(\sigma,T^{(1)},\ldots,T^{(k)})$-reducing subspace. Then $(\sigma,T^{(1)},\ldots,T^{(k)})|_{\mathcal{K}}$ has generating wandering subspace property,

[TABLE]

for some wandering subspace $\mathcal{W}$ for $(\sigma,T^{(1)},\ldots,T^{(k)})|_{\mathcal{K}}.$ Moreover, $\mathcal{W}$ is unique, in fact $\mathcal{W}=\bigcap_{i=1}^{k}\mbox{ker}{\widetilde{T}}^{(i)^{*}}|_{\mathcal{K}}.$ In particular,

[TABLE]

is a generating wandering subspace for $(\sigma,T^{(1)},\ldots,T^{(k)}),$ that is,

[TABLE]

Proof.

Since $(\sigma,T^{(1)},\ldots,T^{(k)})$ is analytic, then for each $i\in I_{k}$

[TABLE]

This implies that, $(\sigma,T^{(1)},\ldots,T^{(k)})|_{\mathcal{K}}$ is analytic and it satisfies hypothesis of the Theorem 3.1. Since $\mathcal{K}$ is $(\sigma,T^{(1)},\ldots,T^{(k)})$-reducing subspace, without loss of generality we can assume that $\mathcal{K}=\mathcal{H}.$ Let $A$ be the proper subset of $I_{k}$ and let $i\in A\setminus I_{k},$ we have

[TABLE]

where $\mathcal{W}_{A}$ as in Equation (2.4). Apply previous equation to Equation (3.1) in Theorem 3.1, we get $\mathcal{H}_{A}=\{0\}.$ Therefore, if $A$ is empty set then $\mathcal{H}_{A}=\mathcal{H}.$ This completes the proof. ∎

The following definition of induced representation is a generalization of the multiplication operators $M_{z_{i}}\otimes I_{\mathcal{H}}$ on the vector valued Hardy space $H^{2}_{\mathcal{H}}(\mathbb{D}^{k}).$

Definition 3.4**.**

Let $\mathbb{E}$ be the product system over $\mathbb{N}^{k}_{0},$ and let $\pi$ be a representation of $\mathcal{M}$ on a Hilbert space $\mathcal{K}.$ Define the Fock module of $\mathbb{E},$

[TABLE]

Note that $\mathcal{F}(\mathbb{E})$ is a $C^{*}$ correspondence over $\mathcal{M}.$ Define a completely contractive covariant representation $(\rho,S^{(i)})$ of $E_{i}$ on the Hilbert space $\mathcal{F}(\mathbb{E})\otimes_{\pi}\mathcal{K}$ (cf. [28]) by

[TABLE]

and

[TABLE]

where $T_{\xi_{i}}$ denotes the creation operator on $\mathcal{F}(\mathbb{E})$ determined by $\xi_{i},$ that is, $T_{\xi_{i}}(\eta)=\xi_{i}\otimes\eta,$ where $\eta\in\mathcal{F}(\mathbb{E})$ and $\phi_{\infty}$ denotes the canonical left action of $\mathcal{M}$ on $\mathcal{F}(\mathbb{E}).$ It is easy to see that the above representation $(\rho,S^{(1)},\dots,S^{(k)})$ is doubly commuting isometric covariant representation of $\mathbb{E}$ on the Hilbert space $\mathcal{F}(\mathbb{E})\otimes_{\pi}\mathcal{K}$ and it is called induced representation of $\mathbb{E}$ induced by $\pi.$ Any covariant representation of $\mathbb{E}$ which is isomorphic to $(\rho,S^{(1)},\dots,S^{(k)})$ is called an induced representation.

The following corollary is a generalization of [28, Theorem 2.4]:

Corollary 3.5**.**

Let $\mathbb{E}$ be a product system of $C^{*}$-correspondences over $\mathbb{N}_{0}^{k}.$ Let $(\sigma,T^{(1)},\ldots,T^{(k)})$ be a doubly commuting isometric covariant representation of $\mathbb{E}$ on a Hilbert space $\mathcal{H}.$ Then for $2\leq m\leq k,$ there exists $2^{m}$ $(\sigma,T^{(1)},\ldots,T^{(m)})$-reducing subspaces $\{\mathcal{H}_{\mathcal{A}}\>:\>A\subseteq I_{m}\}$ such that $\mathcal{H}=\bigoplus_{A\subseteq I_{m}}\mathcal{H}_{A}$ and for each $A=\{i_{1},\dots,i_{p}\}\subseteq I_{m}:\>(\sigma,T^{(i_{1})},\dots,T^{(i_{p})})|_{\mathcal{H}_{A}}$ is an induced representation of the product system $\mathbb{E}_{A}$ over $\mathbb{N}^{A}_{0}$ given by the family of $C^{*}$-correspondence $\{E_{i_{1}},\dots,E_{i_{p}}\}$ and $(\sigma,T^{(i)})|_{\mathcal{H}_{A}}$ is isometric and fully co-isometric covariant representation whenever $i\in I_{m}\setminus A.$ Moreover, the above decomposition is unique and

[TABLE]

Proof.

If $(\sigma,T^{(1)},\dots,T^{(k)})$ be an isometric doubly commuting covariant representation of $\mathbb{E}$ on the Hilbert space $\mathcal{H},$ then it satisfies hypothesis of the Theorem 3.1. By (2) of the Remark 3.2, then $\mathcal{H}_{A}$ as in Theorem 3.1 can be written as

[TABLE]

Moreover, $\bigcap_{\mathbf{j}\in\mathbb{N}_{0}^{I_{m}\setminus A}}\mathfrak{L}_{\mathbf{J}}^{I_{m}\setminus A}(\mathcal{W}_{A})=\bigcap_{i\in A}\mbox{ker}\widetilde{T}^{(i)^{*}}|_{\mathcal{H}_{A}}$ and is $\sigma$-invariant. Define a new representation $\sigma_{1}$ of $\mathcal{M}$ on the Hilbert space $\mathcal{K}$ by $\sigma_{1}(a)=\sigma(a)|_{\mathcal{K}},$ where $\mathcal{K}=\bigcap_{i\in A}\mbox{ker}\widetilde{T}^{(i)^{*}}|_{\mathcal{H}_{A}}.$ Let $A=\{i_{1},i_{2},\dots,i_{p}\}\subseteq I_{m}$ and let $(\rho,S^{(i_{1})},\dots,S^{(i_{p})})$ be an induced representation of the product system $\mathbb{E}_{A}$ over $\mathbb{N}^{A}_{0}$ induced by $\sigma_{1}.$ Define the operator $U:\mathcal{F}(\mathbb{E}_{A})\otimes_{\sigma_{1}}\mathcal{K}\to\mathcal{H}_{A}$ by $U(\xi_{\mathbf{n}}\otimes h)=\widetilde{T}^{A}_{\mathbf{n}}(\xi_{\mathbf{n}}\otimes h),\>\xi_{\mathbf{n}}\in\mathbb{E}_{A}(\mathbf{n})$ and $h\in\mathcal{K}.$ This shows that $U$ is unitary operator and

[TABLE]

where $\phi_{\mathbf{n}}$ is left action of $E_{A}({\mathbf{n}})$ and, for all $\eta_{i_{j}}\in E_{i_{j}},\>\xi_{\mathbf{n}}\in E_{A}(\mathbf{n}),h\in\mathcal{K}$ and $1\leq j\leq p.$ Hence $(\sigma,T^{(i_{1})},\dots,T^{(i_{p})})|_{\mathcal{H}_{A}}$ is an induced representation. ∎

From the above Corollary, if $(\sigma,T)$ be an isometric covariant representation of $E$ on the Hilbert space $\mathcal{H}$. Then $(\sigma,T)$ is analytic if and only if $(\sigma,T)$ is an induced representation. Let $(\sigma,T^{(1)},\ldots,T^{(k)})$ be an induced representation of $\mathbb{E}$, then for each $i,\>1\leq i\leq k,$ $(\sigma,T^{(i)})$ is also an induced representation. But, the converse not true. The following theorem gives a conceptual characterization of induced representation and generating wandering subspace for the isometric covariant representation of the product system $\mathbb{E}$ over $\mathbb{N}^{k}_{0},\>k\geq 2.$

Theorem 3.6**.**

Let $(\sigma,T^{(1)},\ldots,T^{(k)})$ be an isometric covariant representation of the product systems $\mathbb{E}$ over $\mathbb{N}^{k}_{0}$ on the Hilbert space $\mathcal{H}.$ Then the following conditions are equivalent:

1. (1)

There exists a wandering subspace $\mathcal{W}$ for $(\sigma,T^{(1)},\ldots,T^{(k)})$ such that

[TABLE] 2. (2)

For every $j\in I_{k},$ $(\sigma,T^{(j)})$ is an induced representation of $E_{j}$ and $(\sigma,T^{(1)},\ldots,T^{(k)})$ is doubly commuting. 3. (3)

There exists $j\in I_{k}$ such that $(\sigma,T^{(j)})$ is an induced representation and the wandering subspace for $(\sigma,T^{(j)})$ is

[TABLE] 4. (4)

$\mathcal{W}_{I_{k}}$* is a wandering subspace for $(\sigma,T^{(1)},\ldots,T^{(k)})$ and *

$\mathcal{H}=\bigoplus_{\mathbf{n}\in\mathbb{N}^{k}_{0}}\mathfrak{L}_{\mathbf{n}}^{I_{k}}(\mathcal{W}_{I_{k}}).$** 5. (5)

$(\sigma,T^{(1)},\ldots,T^{(k)})$* is isomorphic to an induced representation $(\rho,S^{(1)},$ $\cdots,S^{(k)})$ induced by some representation $\pi$ on $\mathcal{K}$ with $dim\mathcal{K}=dim\mathcal{W}_{I_{k}}$.*

Proof.

(1) $\implies$ (2): Observe that, for each $j,1\leq j\leq k$,

[TABLE]

It implies that $(\sigma,T^{(j)})$ is an induced representation. Let $h\in\mathcal{H}$ such that

[TABLE]

Then for all $i\neq j$ and $\xi_{i}\in E_{i}$, we have

[TABLE]

Hence $(\sigma,T^{(1)},\ldots,T^{(k)})$ is doubly commuting.

(2)$\implies$ (3): By Corollary 3.3 we obtain

[TABLE]

and hence (3) follows.

(3) $\implies$ (4): Given that $(\sigma,T^{(j)})$ is an induced representation with the wandering subspace

[TABLE]

It follows that

[TABLE]

and hence (4) follows.

(4) $\implies$(5): Define $\sigma_{0}:={\sigma}|_{\mathcal{W}_{I_{k}}}$ and define the unitary operator

[TABLE]

by

[TABLE]

Then it is easy to see that

[TABLE]

for every $\xi_{j}\in E_{j},a\in\mathcal{M},j\in I_{k},$ where $(\rho,S^{(1)},\ldots,S^{(k)})$ is an induced representation induced by $\sigma_{0}.$

(5)$\implies$ (1): is obvious. ∎

Definition 3.7**.**

Let $T=(T_{1},\dots,T_{k})$ be a $q$-commuting operators on Hilbert space $\mathcal{H}$ and $1\leq m\leq k.$ Let $A=\{i_{1},\dots i_{p}\}\subseteq I_{m}.$ We denote by $T_{A}$ the $|A|$- tuple of $q$-commuting operators $(T_{i_{1}},\cdots,T_{i_{p}})$ and also denote $T_{i_{1}}^{n_{i_{1}}}\cdots T_{i_{p}}^{n_{i_{p}}}$ by $T^{\mathbf{n}}_{A},$ where $\mathbf{n}=(n_{i_{1}},\cdots,n_{i_{p}})\in\mathbb{N}_{0}^{A}.$ For a closed subspace $\mathcal{K}$ of $\mathcal{H},$ we write $[\mathcal{K}]_{T_{A}}$ to denote the smallest closed joint $T_{A}$-invariant subspace of the Hilbert space $\mathcal{H}$ containing $\mathcal{K},$ that is, $[\mathcal{K}]_{T_{A}}=\bigvee_{n\in\mathbb{N}_{0}^{A}}T_{A}^{\mathbf{n}}\mathcal{K}.$ A closed subspace $\mathcal{W}$ of $\mathcal{H}$ is said to be generating wandering subspace for an $A$-tuple $T_{A}=(T_{i_{1}},\cdots,T_{i_{p}})$ of $q$-commuting operators on $\mathcal{H}$ if $\mathcal{W}\>\bot\>T_{A}^{\mathbf{n}}(\mathcal{W})$ for all $\mathbf{n}\in\mathbb{N}_{0}^{A}$ and $\mathcal{H}=[\mathcal{W}]_{T_{A}}.$ If there exist a closed subspace $\mathcal{W}$ of $\mathcal{H}$ such that $\mathcal{W}$ is generating subspace for $T,$ the tuple $T$ is said to have the generating wandering subspace property.

Let $(\sigma,T^{(1)},\ldots,T^{(k)})$ be an completely bounded covariant representation of the product systems $\mathbb{E}$(with $E_{i}=\mathbb{C}$ for all $1\leq i\leq k$) over $\mathbb{N}^{k}_{0}$ on the Hilbert space $\mathcal{H}.$ From the Equation (2.2) it implies that the $k$-tuple $T:=(T^{(1)}(1),T^{(2)}(1),\dots,T^{(k)}(1))$ is $q$-commuting. Let $\mathcal{K}$ be a closed subspace of $\mathcal{H},$ then it is easy to see that

[TABLE]

where $A\subseteq I_{m}$ and $\mathbf{n}\in\mathbb{N}_{0}^{A}.$ This combined with the following corollary, which is a generalization of [13, Theorem 3.4] and [24, Theorem 2.4].

Corollary 3.8**.**

Let $T=(T_{1},\dots,T_{k})$ be a $k$-tuple of $q$-doubly commuting operators on Hilbert space $\mathcal{H}$ such that $T$ satisfies one of the following properties:

1. (1)

$T_{i}$* is concave for each $i=1,\dots,k,$ that is,*

$\|T_{i}^{2}h\|^{2}+\|h\|^{2}\leq 2\|T_{i}h\|^{2},\>h\in\mathcal{H},$** 2. (2)

$\|T_{i}h+g\|^{2}\leq 2(\|h\|^{2}+\|T_{i}g\|^{2}),$*   $h,g\in\mathcal{H}$ and for $i=1,\dots,k,$* 3. (3)

$\|T_{i}h\|^{2}+\|T_{i}^{*2}T_{i}h\|^{2}\leq 2\|T_{i}^{*}T_{i}h\|^{2},\>h\in\mathcal{H}$* and for $i=1,\dots,k.$*

For $2\leq m\leq k,$ there exist $2^{m}$ joint $(T_{1},\dots,T_{m})$-reducing subspaces $\{\mathcal{H}_{A}\>:\>A\subseteq I_{m}\>\}$ such that $\mathcal{H}=\bigoplus_{A\subseteq I_{m}}\mathcal{H}_{A}$ and for all $A=\{i_{1},\dots,i_{p}\}\subseteq I_{m}$ and $\mathcal{H}_{A}\neq\{0\};\>\>(T_{i_{1}},\dots,T_{i_{p}})|_{{\mathcal{H_{A}}}}$ has generating wandering subspace property and $T_{i}|_{{\mathcal{H_{A}}}}$ is unitary whenever $i\in I_{m}\setminus A.$ Moreover, the above decomposition is unique and

[TABLE]

where $\mathcal{W}_{A}=\bigcap_{i\in A}(\mathcal{H}\ominus T_{i}\mathcal{H}).$

Using the above corollary we can easily prove the results [6, Corollary 2.4], [13, Theorem 3.4] and [24, Theorem 2.4] in the following remark:

Remark 3.9**.**

(1) Let $T=(T_{1},\dots,T_{k})$ be as in the above Corollary 3.8 such that each $T_{i}$ is analytic, that is, $\bigcap_{n_{i}=1}^{\infty}T_{i}^{n_{i}}\mathcal{H}=\{0\}$ for $1\leq i\leq k.$ From the Corollary 3.3, we have that

[TABLE]

*that is, $T$ has a generating wandering subspace property.

(2) If $T=(T_{1},\dots,T_{k})$ such that each $T_{i}$ is an isometry which satisfy $T_{i}^{*}T_{j}=\overline{{z}_{ij}}T_{j}T_{i}^{*},$ where $z_{ij}\in\mathbb{T}$ with $i\neq j,\>1\leq i,j\leq k.$ Then it satisfy hypothesis of the Corollary 3.8. Using (2) of the Remark 3.2, we conclude that*

[TABLE]

where $\mathcal{H}_{A}$ and $\mathcal{W}_{A}$ as in the Corollary 3.8 and $T_{i}|_{\mathcal{H}_{A}}$ is pure isometry if $i\in A.$ In particular, if all $T_{i}$ is analytic, then $\mathcal{H}=\bigoplus_{\mathbf{n}\in\mathbb{N}^{k}_{0}}T^{\mathbf{n}}\mathcal{W}_{I_{k}}.$ Moreover, the $k$-tuple $(T_{1},\dots,T_{k})$ of operators on $\mathcal{H}$ is isomorphic to the multiplication operator $M_{\mathbf{z}}:=(M_{z_{1}},\dots,M_{z_{k}})$ on $\mathcal{H}_{\mathcal{W}}^{2}(\mathbb{D}^{k}),$ where $\mathcal{H}_{\mathcal{W}}^{2}(\mathbb{D}^{k})$ denote the Hardy space of $\mathcal{W}(:=\mathcal{W}_{I_{k}})$-valued analytic functions on $\mathbb{D}^{k}$ and $\mathbb{D}$ is an open unit disk in $\mathbb{C}.$ Indeed, define a unitary operator

[TABLE]

by $U(\sum_{\mathbf{n}\in\mathbb{N}_{0}^{k}}\mathbf{z}^{\mathbf{n}}h_{\mathbf{n}})=\sum_{\mathbf{n}\in\mathbb{N}_{0}^{k}}T^{\mathbf{n}}h_{\mathbf{n}}$ for all $h_{\mathbf{n}}\in\mathcal{W}.$ It is easy to verify that, if $z_{ij}=1$ for all $1\leq i,j\leq k,$ then $UM_{z_{i}}=T_{i}U$ for $1\leq i\leq k.$ Conversely, if the multiplication operator $M_{\mathbf{z}}$ and the doubly commuting isometries $T$ are isomorphic then each $T_{i}$ is analytic, $1\leq i\leq k$.

4. Beurling-Lax type decomposition for Doubly commuting invariant subspaces

In this section we completely characterize the doubly commuting subspaces of covariant representations of the product systems $\mathbb{E}$ on a Hilbert space $\mathcal{H}.$ This result is a generalization of a version of the Beurling’s theorem for the doubly commuting shift on the bidisc by Mandrekar [14] and the polydisc case proved by Sarkar, Sasane and Wick [23]. We first prove Beurling’s theorem for the covariant representation of a $C^{*}$-correspondence which extends Beurling-Lax type theorem due to Popescu in [22](see also a relevant tensor algebra version of Beurling’s theorem by Muhly and Solel [17]).

Definition 4.1**.**

Let $\mathbb{E}$ be a product system of $C^{*}$-correspondences over $\mathbb{N}_{0}^{k}$. Let $(\sigma,T^{(1)},\dots,T^{(k)})$ and $(\psi,V^{(1)},\dots,V^{(k)})$ be a completely bounded covariant representations of $\mathbb{E}$ on the Hilbert spaces $\mathcal{H}$ and $\mathcal{K}$ respectively. A bounded operator $A:\mathcal{H}\to\mathcal{K}$ is called multi-analytic if it satisfies the following condition

[TABLE]

where $\xi_{i}\in E_{i},\>h\in\mathcal{H}$ and $a\in\mathcal{M},1\leq i\leq k.$

Throughout this section, we assume $(\sigma,T^{(1)},\dots,T^{(k)})$ and $(\psi,V^{(1)},\dots,V^{(k)})$ to be the doubly commuting isometric covariant representations of $\mathbb{E}$ on the Hilbert spaces $\mathcal{H}$ and $\mathcal{K}$ such that for each $1\leq i\leq k,$ $(\sigma,T^{(i)})$ and $(\psi,V^{(i)})$ are analytic. Then by Corollary 3.3, we have

[TABLE]

where $\mathcal{W}_{\mathcal{H}}$ and $\mathcal{W}_{\mathcal{K}}$ are the generating wandering subspaces for $(\sigma,T^{(1)},\dots,T^{(k)})$ and $(\psi,V^{(1)},$ $\dots,V^{(k)}).$

Notation 4.2**.**

If $A:\mathcal{H}\to\mathcal{K}$ is multi-analytic operator, then $A$ is uniquely determined by the operator $\theta:\mathcal{W}_{\mathcal{H}}\to\mathcal{K},$ where $\theta:=A|_{\mathcal{W}_{\mathcal{H}}}.$ This follows because for every $\xi_{\mathbf{n}}\in E(\mathbf{n}),h\in\mathcal{W}_{\mathcal{H}}$ we have $AT_{\mathbf{n}}(\xi_{\mathbf{n}})h=V_{\mathbf{n}}(\xi_{\mathbf{n}})\theta h$ and $\mathcal{H}=\bigoplus_{\mathbf{n}\in\mathbb{N}_{0}^{k}}\widetilde{T}_{\mathbf{n}}(E(\mathbf{n})\otimes\mathcal{W}_{\mathcal{H}}).$

Now, let us consider an operator $\theta:\mathcal{W}_{\mathcal{H}}\to\mathcal{K}\left(=\bigoplus_{\mathbf{n}\in\mathbb{N}_{0}^{k}}\widetilde{V}_{\mathbf{n}}(E(\mathbf{n})\otimes\mathcal{W}_{\mathcal{K}})\right).$ We define the operator $M_{\theta}:\mathcal{H}\to\mathcal{K}$ by the relation

[TABLE]

In this section we only work with $\theta$ such that $M_{\theta}$ is a contraction. It is easy to see that

[TABLE]

Definition 4.3**.**

An operator $\theta:\mathcal{W}_{\mathcal{H}}\to\mathcal{K}$ will be called

1. (1)

inner* if $M_{\theta}$ is an isometry,* 2. (2)

outer* if $\overline{M_{\theta}\mathcal{H}}=\mathcal{K}$.*

Proposition 4.4**.**

Let $\theta:\mathcal{W}_{\mathcal{H}}\to\mathcal{K}$ be an operator such that $M_{\theta}$ is a contraction.

1. (1)

$\theta$* is inner if and only if $\theta$ is an isometry and $\theta\mathcal{W}_{\mathcal{H}}$ is a wandering subspace for $(\psi,V^{(1)},\dots,V^{(k)}).$* 2. (2)

$\theta$* is outer if and only if $\theta\mathcal{W}_{\mathcal{H}}$ is cyclic for $(\psi,V^{(1)},\dots,V^{(k)})$, i.e.*

[TABLE] 3. (3)

$\theta$* is inner and outer if and only if $\theta$ is a unitary operator from $\mathcal{W}_{\mathcal{H}}$ to $\mathcal{W}_{\mathcal{K}}.$*

Proof.

(1) Assume $M_{\theta}$ is an isometry. Let $h_{1},h_{2}\in\mathcal{W}_{\mathcal{H}},$ and $\xi_{\mathbf{n}}\in E(\mathbf{n}),\mathbf{n}\in\mathbb{N}^{k}_{0}\setminus\{0\},$ we have

[TABLE]

since $\mathcal{W}_{\mathcal{H}}$ is wandering subspace for $(\sigma,T^{(1)},\dots,T^{(k)}).$ Therefore, $\theta\mathcal{W}_{\mathcal{H}}$ is wandering subspace for $(\psi,V^{(1)},\dots,V^{(k)}).$

Conversely, let $h,k\in\mathcal{H}\left(=\bigoplus_{\mathbf{n}\in\mathbb{N}_{0}^{k}}\widetilde{T}_{\mathbf{n}}(E(\mathbf{n})\otimes\mathcal{W}_{\mathcal{H}})\right),$ where

[TABLE]

for some $\xi_{\mathbf{n}}\in E(\mathbf{n}),\eta_{\mathbf{m}}\in E(\mathbf{m})$ and $h_{\mathbf{n}},k_{\mathbf{n}}\in\mathcal{W}_{\mathcal{H}}.$ Since $\theta$ is an isometry and $\theta\mathcal{W}_{\mathcal{H}}$ is wandering subspace

[TABLE]

Hence $M_{\theta}$ is an isometry.

The statement (2) is easy to prove and (3) follows from Theorem 3.6. ∎

The following theorem for a covariant representation of a $C^{*}$-correspondence is a generalization of Popescu’s version of Beurling-Lax theorem [22, Theorem 2.2].

Theorem 4.5**.**

Let $(\sigma,T)$ be a analytic, isometric covariant representation of $E$ on the Hilbert space $\mathcal{H}$ and let $\mathcal{K}$ be closed subspace of $\mathcal{H}.$ Then $\mathcal{K}$ is $(\sigma,T)$-invariant if and only if there exists a Hilbert space $\mathcal{W},$ a representation $\sigma_{1}$ of $\mathcal{M}$ on $\mathcal{W}$ and inner operator $\theta:\mathcal{W}_{\mathcal{F}(E)\otimes\mathcal{W}}\to\mathcal{H}$ such that

[TABLE]

where $\mathcal{W}_{\mathcal{F}(E)\otimes\mathcal{W}}$ is generating wandering subspace for the induced representation $(\rho,V)$ induced by $\sigma_{1}.$ In particular, if $\mathcal{K}=\mathcal{H},$ then $\theta$ is outer.

Proof.

Let $\mathcal{K}$ be a $(\sigma,T)$-invariant subspace. It was shown in [7, Theorem 2.4], there exists a wandering subspace $\mathcal{W}_{\mathcal{K}}$ for $(\sigma,T)|_{\mathcal{K}}$ such that

[TABLE]

Since $\mathcal{W}_{\mathcal{K}}$ is $\sigma(\mathcal{M})$-invariant, consider the induced representations $(\rho,V)$ on $\mathcal{H}_{1}:=\mathcal{F}(E)\otimes\mathcal{W}_{\mathcal{K}}$ induced by $\sigma|_{\mathcal{W}_{\mathcal{K}}}.$ Define the operator $A:\mathcal{F}(E)\otimes\mathcal{W}_{\mathcal{K}}\to\mathcal{H}$ by

[TABLE]

then $A$ satisfies $AV(\xi)=T(\xi)A,A\rho(a)=\sigma(a)A$ for $\xi\in E,a\in\mathcal{M},$ that is, $A$ is multi-analytic operator. It is easy to see that $A=M_{\theta}$ and $M_{\theta}$ is an isometry, where $\theta:\mathcal{W}_{\mathcal{H}_{1}}\to\mathcal{H},\mathcal{W}_{\mathcal{H}_{1}}$ is generating wandering subspace for $(\rho,V).$ Finally, it follows from the Equation (4.1) that $M_{\theta}\mathcal{H}_{1}=\mathcal{K}.$ The converse part is immediate. ∎

Let us recall a setup due to Popescu in [22]: Let $\Lambda$ be either the set $\{1,2,\dots,m\}$ where $m\in\mathbb{N},$ or $\mathbb{N}.$ For every $k\in\mathbb{N},$ let $F(k,\Lambda)$ be the set of all functions from the set $\{1,2,\dots,k\}$ to $\Lambda.$ Define $\mathfrak{F}:=\bigcup_{k=0}^{\infty}F(k,\Lambda),$ where $F(0,\Lambda)=\{0\}.$ A closed subspace $\mathcal{W}\subseteq\mathcal{H}$ is said to be generating wandering subspace for the sequence of isometries $\{T_{n}\}_{n=1}^{m}$ on the Hilbert space $\mathcal{H}$ if for any distinct element $f,g\in\mathfrak{F}$ we have $T_{f}\mathcal{W}\bot T_{g}\mathcal{W},$ where $T_{f}$ stands for the product $T_{f(1)}T_{f(2)}\cdots T_{f(k)}$ and $T_{0}=I_{\mathcal{H}},f\in\mathfrak{F},$ and

[TABLE]

Let $\mathcal{H}$ be a Hilbert space. Define the Hilbert space of all formal power series with noncommuting indeterminates $X_{\lambda}(\lambda\in\Lambda)$ by

[TABLE]

with the inner product is

[TABLE]

where $X_{f}=X_{f(1)}X_{f(2)}\cdots X_{f(k)}$ for any $f\in F(k,\Lambda).$ Define the $\Lambda$-orthogonal shift $\mathfrak{S}=\{S_{\lambda}\}_{\lambda\in\Lambda}$ on $\mathfrak{L}^{2}(\mathfrak{F},\mathcal{H})$ by

[TABLE]

It is easy to see that, the noncommuting operators $S_{\lambda}(\lambda\in\Lambda)$ is an isometry, $\mathcal{H}$ is generating wandering subspace for the $\Lambda$-orthogonal shift $\mathfrak{S}$ and $\sum_{\lambda\in\Lambda}S_{\lambda}S_{\lambda}^{*}\leq I.$ As a consequence of Theorem 4.5 we obtain the following corollary (which is [22, Theorem 2.2]).

Corollary 4.6**.**

Let $\{T_{n}\}$ be a finite (respectively, infinite )sequence of noncommuting isometries on the Hilbert space $\mathcal{H}$ such that $\sum T_{n}T^{*}_{n}\leq I$ and $\{T_{n}\}$ has generating wandering subspace property. Let $\mathcal{K}$ be a closed subspace of $\mathcal{H}.$ Then $\mathcal{K}$ is joint $\{T_{n}\}$-invariant if and only if there exists a Hilbert space $\mathcal{H}_{1},$ a finite (respectively, infinite) sequence of orthogonal shift $\{S_{\lambda}\}$ on $\mathfrak{L}^{2}(\mathfrak{F},\mathcal{H}_{1})$ and inner operator $\theta:\mathcal{H}_{1}\to\mathcal{H}$ such that $\mathcal{K}=M_{\theta}\mathfrak{L}^{2}(\mathfrak{F},\mathcal{H}).$

In the following proposition we characterize the reducing subspace of the covariant representations $(\sigma,T^{(1)},\dots,T^{(k)})$ of $\mathbb{E}$ on the Hilbert space $\mathcal{H}.$

Proposition 4.7**.**

Let $(\sigma,T^{(1)},\dots,T^{(k)})$ be a doubly commuting isometric covariant representations of $\mathbb{E}$ on the Hilbert spaces $\mathcal{H}$ such that each $(\sigma,T^{(i)})$ is analytic. Then $\mathcal{K}$ is a $(\sigma,T^{(1)},\dots,T^{(k)})$- reducing subspace of $\mathcal{H}$ if and only if

[TABLE]

where $\mathcal{W}=\bigcap_{i=1}^{k}\mbox{ker}\widetilde{T}^{(i)^{*}}|_{\mathcal{K}}$

Proof.

The necessary part follows from Corollary 3.5. Conversely, it is obvious that $\mathcal{K}$ is $(\sigma,T^{(1)},\dots,T^{(k)})$-invariant. Let $i\in I_{k}$ be fixed and $\mathbf{n}=(n_{1},\cdots,n_{k})\in\mathbb{N}^{k}_{0}.$ If $n_{i}=0,$ then

[TABLE]

where $t_{\mathbf{n},i}:\mathbb{E}(\mathbf{n})\otimes E_{i}\to E_{i}\otimes\mathbb{E}(\mathbf{n})$ is the isomorphism which is a composition of the isomorphism $t_{i,j}:E_{i}\otimes E_{j}\to E_{j}\otimes E_{i}.$ Since $\mathcal{W}=\bigcap_{j=1}^{k}\mbox{ker}\widetilde{T}^{(i)^{*}}|_{\mathcal{K}},$ $\widetilde{T}^{(i)^{*}}\mathcal{W}=\{0\}.$ Thus, it follows from the above equation we get $\widetilde{T}^{(i)*}\widetilde{T}_{\mathbf{n}}(\mathbb{E}(\mathbf{n})\otimes\mathcal{W})=\{0\},$ for $n\in\mathbb{N}^{k}_{0}$ with $n_{i}=0.$ Suppose that $n_{i}>0,$ $\widetilde{T}^{(i)*}\widetilde{T}_{\mathbf{n}}(\mathbb{E}(\mathbf{n})\otimes\mathcal{W})=\widetilde{T}_{(n_{i}-1)}^{(i)}(I_{E_{i}^{\otimes(n_{i}-1)}}\otimes\widetilde{T}^{(i)}_{\mathbf{n}-n_{i}})(\mathbb{E}(\mathbf{n})\otimes\mathcal{W})\subset\mathcal{K}$ because $\mathcal{K}$ is $(\sigma,T^{(1)},\dots,T^{(k)})$-invariant. Therefore $\widetilde{T}^{(i)*}\widetilde{T}_{\mathbf{n}}(E(\mathbf{n})\otimes\mathcal{W})\subset\mathcal{K}$ for all $\mathbf{n}\in\mathbb{N}^{k}_{0}$ and hence we get the desired result. ∎

Remark 4.8**.**

Let $\pi$ be the representation of $\mathcal{M}$ on the Hilbert space $\mathcal{H}_{2}$ and let $(\rho,S^{(1)},\dots,S^{(k)})$ be the induced representation on $\mathcal{H}_{1}:=\mathcal{F}(\mathbb{E})\otimes\mathcal{H}_{2}$ induced by $\pi.$ Let $\mathcal{K}$ be the subspace of $\mathcal{H}_{1}$ such that

[TABLE]

for some closed subspace $\mathcal{W}$ of $\mathcal{H}_{1}.$ For $1\leq i\leq k,$ we obtain

[TABLE]

*Thus $\bigcap_{i=1}^{k}\mbox{ker}\widetilde{S}^{(i)^{*}}|_{\mathcal{K}}=\mathcal{W}.$ Therefore, the sufficient part of Proposition 4.7, $\mathcal{W}=\bigcap_{i=1}^{k}\mbox{ker}\widetilde{T}^{(i)^{*}}|_{\mathcal{K}}$ is not necessary to prove $\mathcal{K}$ is $(\sigma,T^{(1)},\dots,T^{(k)})$-reducing for the case of induced representation $(\rho,S^{(1)},\dots,S^{(k)}).$

From Proposition 4.7, $\mathcal{W}$ is $\sigma$-invariant. Let $(\rho,S^{(1)},\dots,S^{(k)})$ be a induced representations on $\mathcal{H}_{1}:=\mathcal{F}(\mathbb{E})\otimes\mathcal{W}$ induced by $\sigma|_{\mathcal{W}}.$ We define the operator $A:\mathcal{F}(\mathbb{E})\otimes\mathcal{W}\to\mathcal{K}$ by*

[TABLE]

where $\xi_{\mathbf{n}}\in E(\mathbf{n}),w_{\mathbf{n}}\in\mathcal{W}.$ Then it is easy to see that $A$ is multi-Analytic and hence $A=M_{\theta},$ where $\theta:=A|_{\mathcal{W}_{\mathcal{H}_{1}}}:\mathcal{W}_{\mathcal{H}_{1}}\to\mathcal{H},\>\mathcal{W}_{\mathcal{H}_{1}}$ is generating wandering subspace for $(\rho,S^{(1)},\dots,S^{(k)}),$ in fact $\mathcal{W}_{\mathcal{H}_{1}}=\mathcal{W}.$ From Equation (4.4), $M_{\theta}$ is an isometry and

[TABLE]

Next we explore the invariant subspaces for the isometric covariant representations $(\sigma,T^{(1)},$ $\dots,T^{(k)}),$ when $k\geq 2.$

Definition 4.9**.**

Let $(\sigma,T^{(1)},\dots,T^{(k)})$ be a completely bounded covariant representation of $\mathbb{E}$ on the Hilbert space $\mathcal{H}.$ An $(\sigma,T^{(1)},\dots,T^{(k)})$- invariant subspace $\mathcal{K}$ is said to doubly commuting subspace if the covariant representation $(\sigma,T^{(1)},\dots,T^{(k)})|_{\mathcal{K}}$ is doubly commuting of $\mathbb{E}$ on the Hilbert space $\mathcal{K},$ that is,

[TABLE]

where $i,j\in\{1,2,\dots,k\}$ with $i\neq j.$

Now we present a wandering subspace theorem concerning doubly commuting subspaces of $\mathcal{H}.$

Theorem 4.10**.**

Let $(\sigma,T^{(1)},\dots,T^{(k)})$ be a doubly commuting completely bounded, covariant representation of the product system $\mathbb{E}$ on a Hilbert space $\mathcal{H}$ such that $(\sigma,T^{(1)},\dots,$ $T^{(k)})$ is analytic and satisfies one of the following properties:

1. (1)

$(\sigma,T^{(i)})$* is concave for each $i=1,\dots,k,$* 2. (2)

$\|(I_{E_{i}}\otimes\widetilde{T}^{(i)})(\zeta)+\xi\|^{2}\leq 2(\|\zeta\|^{2}+\|\widetilde{T}^{(i)}(\xi)\|^{2}),$* for each $\zeta\in E_{i}^{\otimes 2}\otimes\mathcal{H},\xi\in E_{i}\otimes\mathcal{H},i=1,\dots,k.$* 3. (3)

$\|\widetilde{T}^{(i)}(\xi_{i})\|^{2}+\|\widetilde{T}^{(i)*}_{2}\widetilde{T}^{(i)}(\xi_{i})\|^{2}\leq 2\|\widetilde{T}^{(i)*}\widetilde{T}^{(i)}(\xi_{i})\|^{2},\>\xi_{i}\in E_{i}\otimes\mathcal{H}.$**

Let $\mathcal{K}$ be a $(\sigma,T^{(1)},\dots,T^{(k)})$- doubly commuting subspace. Then

[TABLE]

for some wandering subspace $\mathcal{W}^{\prime}_{\mathcal{K}}$ for $(\sigma,T^{(1)},\dots,$ $T^{(k)})|_{\mathcal{K}}.$

Proof.

Since $\mathcal{K}$ is $(\sigma,T^{(i)})$-invariant and $(\sigma,T^{(i)})$ is analytic for each $1\leq i\leq k,$ then $(\sigma,T^{(i)})|_{\mathcal{K}}$ is analytic. Also $(\sigma,T^{(1)},\dots,T^{(k)})|_{\mathcal{K}}$ is doubly commuting covariant representation on $\mathbb{E}$, therefore the desired result follows from Theorem 3.1. ∎

Remark 4.11**.**

Let $(\sigma,T^{(1)},\dots,T^{(k)})$ be a doubly commuting isometric covariant representations of $\mathbb{E}$ on the Hilbert space $\mathcal{H}$ such that each $(\sigma,T^{(i)})$ is analytic. From the above Theorem 4.10,

[TABLE]

for some wandering subspace $\mathcal{W}.$ Moreover, the corresponding generating wandering subspace is $\mathcal{W}=\bigcap_{i=1}^{k}(\mathcal{K}\ominus\widetilde{T}^{(i)}(E_{i}\otimes\mathcal{K})).$

The following theorem gives necessary and sufficient for doubly commuting and generating wandering subspace for isometric covariant representations.

Theorem 4.12**.**

Let $(\sigma,T^{(1)},\dots,T^{(k)})$ be a doubly commuting isometric covariant representations of $\mathbb{E}$ on a Hilbert space $\mathcal{H}$ such that each $(\sigma,T^{(i)})$ is analytic. Let $\mathcal{K}$ be a $(\sigma,T^{(1)},\dots,T^{(k)})$-invariant. Then $\mathcal{K}$ is $(\sigma,T^{(1)},\dots,T^{(k)})$- doubly commuting subspace if and only if there exist a doubly commuting isometric covariant representations $(\rho,V^{(1)},\dots,V^{(k)})$ of $\mathbb{E}$ on the Hilbert space $\mathcal{H}_{1}$, $(\sigma,V^{(i)})$ is analytic and an inner operator $\theta:\mathcal{W}_{\mathcal{H}_{1}}\to\mathcal{H}$ such that

[TABLE]

In particular, if $\mathcal{K}=\mathcal{H},$ then $\theta$ is outer.

Proof.

By Corollary 3.6 we have

[TABLE]

where $\mathcal{W}=\bigcap_{i=1}^{k}(\mathcal{K}\ominus\widetilde{T}^{(i)}(E_{i}\otimes\mathcal{K})).$ Since $\mathcal{W}$ is $\sigma(\mathcal{M})$-invariant, we define the representation $\rho$ of $\mathcal{M}$ on the Hilbert space $\mathcal{W}$ by $\rho(a):=\sigma(a)|_{\mathcal{W}},a\in\mathcal{M}.$ Consider the induced representation $(\rho,S^{(1)},\dots,S^{(k)})$ induced by $\rho,$ now we define the Hilbert space $\mathcal{H}_{1}$ by $\mathcal{H}_{1}=\mathcal{F}(\mathbb{E})\otimes_{\rho}\mathcal{W}$ and the bounded operator $V:\mathcal{H}_{1}\to\mathcal{H}$ by

[TABLE]

Observe that

[TABLE]

thus $V$ is an isometry. Moreover, for all $\xi_{\mathbf{n}}\in E(\mathbf{n}),\eta_{\mathbf{m}}\in E(\mathbf{m}),w\in\mathcal{W},\mathbf{n},\mathbf{m}\in\mathbb{N}^{k}_{0}$ we have

[TABLE]

that is, $VS_{\mathbf{n}}(\xi_{\mathbf{n}})=T_{\mathbf{n}}(\xi_{\mathbf{n}})V$ for all $\xi_{\mathbf{n}}\in E(\mathbf{n}),\>\mathbf{n}\in\mathbb{N}^{k}_{0}.$ Therefore $V=M_{\theta},$ $\theta:\mathcal{W}_{\mathcal{H}_{1}}\to\mathcal{H}$ is inner, in fact $\mathcal{W}_{\mathcal{H}_{1}}=\mathcal{W}.$ it is easy to see that

[TABLE]

that is, $M_{\theta}\mathcal{H}_{1}=\mathcal{K}.$ To prove the converse part, let $\mathcal{K}=M_{\theta}\mathcal{H}_{1}$ be a $(\sigma,T^{(1)},\dots,T^{(k)})$-invariant such that $\theta$ is inner.

Then

[TABLE]

where $P_{\mathcal{K}}$ is orthogonal projection oh $\mathcal{H}$ onto $\mathcal{K}.$ Then for all $i\neq j,$ we have

[TABLE]

since $(I_{E_{i}}\otimes M_{\theta})\widetilde{V}^{(i)^{*}}=(I_{E_{i}}\otimes P_{\mathcal{K}})\widetilde{T}^{(i)^{*}}.$ Hence $(\sigma,T^{(1)},\dots,T^{(k)})|_{\mathcal{K}}$ is doubly commuting covariant representations of $\mathbb{E}$ on $\mathcal{H}.$ ∎

Let $\Lambda$ be a set $\{1,\dots,k\}(k\in\mathbb{N})$ and let $\mathcal{H}$ be a Hilbert space. Define the Hilbert space $\mathfrak{L}^{2}(\mathfrak{F},\mathcal{H})$ as in Equation (4.2) with commuting indeterminates $X_{\lambda}(\lambda\in\Lambda).$ Similarly, define the $\Lambda$-orthogonal shift $\mathfrak{S}=\{S_{\lambda}\}_{\lambda\in\Lambda}$ as in Equation (4.3) with commuting indeterminates $X_{\lambda}(\lambda\in\Lambda)$ on $\mathfrak{L}^{2}(\mathfrak{F},\mathcal{H}).$ It is easy to see that, the operators $\{S_{\lambda}:1\leq\lambda\leq k\}$ with commuting indeterminates $X_{\lambda}$ is doubly commuting isometry and $\mathcal{H}$ is generating wandering subspace for the $\Lambda$-orthogonal shift $\mathfrak{S}.$ This combined with the following corollary which characterize the doubly commuting subspace.

Corollary 4.13**.**

Let $T=(T_{1},\dots,T_{k})$ be an $q$-doubly commuting isometries on $\mathcal{H}$ such that $T_{i}$ is analytic for $1\leq i\leq k.$ Let $\mathcal{K}$ be an joint $T$-invariant subspace. Then $\mathcal{K}$ is doubly commuting subspace for $T$ if and only if there exists a Hilbert subspace $\mathcal{W}$ of $\mathcal{H}$, a $\Lambda$-orthogonal shift $\{S_{\lambda}\}$ on $\mathfrak{L}^{2}(\mathfrak{F},\mathcal{W})$ with commuting indeterminates $X_{\lambda}(\lambda\in\Lambda)$ and an inner operator $\theta:\mathcal{W}\to\mathcal{H}$ such that $\mathcal{K}=M_{\theta}\mathfrak{L}^{2}(\mathfrak{F},\mathcal{W}).$

Let $\mathcal{H}_{1}$ and $\mathcal{H}_{2}$ be the Hilbert spaces and $B(\mathcal{H}_{1},\mathcal{H}_{2})$ be set of all bounded operator from $\mathcal{H}_{1}$ to $\mathcal{H}_{2}.$ Denote $H^{\infty}_{\mathcal{H}_{1}\to\mathcal{H}_{2}}(\mathbb{D}^{k})$ by the set of all $B(\mathcal{H}_{1},\mathcal{H}_{2})$-valued bounded holomorphic functions on $\mathbb{D}^{n}$, that is,

[TABLE]

We call the operator valued function $\Theta\in H^{\infty}_{\mathcal{W}_{1}\to\mathcal{W}}(\mathbb{D}^{n})$ is inner if

[TABLE]

for almost all $\mathbf{z}\in\mathbb{T}^{k},$ where $\mathbb{T}$ is boundary of $\mathbb{D}.$

Corollary 4.14**.**

Let $M_{\mathbf{z}}:=(M_{z_{1}},\dots,M_{z_{k}})$ be a multiplication operators on Hilbert valued Hardy space $\mathcal{H}_{\mathcal{W}}^{2}(\mathbb{D}^{k})$ and $\mathcal{K}$ be a closed subspace of $\mathcal{H}_{\mathcal{W}}^{2}(\mathbb{D}^{k}).$ Then $\mathcal{K}$ is doubly commuting subspace for $M_{\mathbf{z}}$ if and only if there exists a closed subspace $\mathcal{W}_{1}$ of $\mathcal{W}$ and an inner function $\Theta\in H^{\infty}_{\mathcal{W}_{1}\to\mathcal{W}}(\mathbb{D}^{n})$ such that

[TABLE]

Proof.

Since $M_{\mathbf{z}}$ is doubly commuting isometries on $\mathcal{H}_{\mathcal{W}}^{2}(\mathbb{D}^{k})$ and $M_{z_{i}}$ is analytic for $1\leq i\leq k.$ Then by Corollary 4.13, there exists a closed subspace $\mathcal{W}_{1}$ of $\mathcal{W}$, a orthogonal shift $\{S_{i}:1\leq i\leq k\}$ on $\mathfrak{L}^{2}(\mathfrak{F},\mathcal{W}_{1})$ with commuting indeterminates $X_{i}(\lambda\in\Lambda)$ and an inner operator $\theta:\mathcal{W}\to\mathcal{H}$ such that $\mathcal{K}=M_{\theta}\mathfrak{L}^{2}(\mathfrak{F},\mathcal{W}).$ Since $\{S_{i}:1\leq i\leq k\}$ is commuting operator, define the unitary operator

[TABLE]

by $U_{1}(\sum_{\mathbf{n}\in\mathbb{N}_{0}^{k}}\mathbf{z}^{n}h_{\mathbf{n}})=\sum_{\mathbf{n}\in\mathbb{N}_{0}^{k}}S^{\mathbf{n}}h_{\mathbf{n}}$ for all $h_{\mathbf{n}}\in\mathcal{W}_{1}.$ It follows that $U_{1}M_{z_{i}}=S_{i}U_{1}$ for all $1\leq i\leq k.$ Now, we define the isometry $V:\mathcal{H}_{\mathcal{W}_{1}}^{2}(\mathbb{D}^{k})\to\mathcal{H}_{\mathcal{W}}^{2}(\mathbb{D}^{k})$ by $V=M_{\theta}U_{1}.$ By the definition of multi-analytic operator $M_{\theta},$ we have $VM_{z_{i}}=M_{\theta}S_{i}U_{1}=M_{z_{i}}M_{\theta}U_{1}=M_{z_{i}}V.$ Hence the isometry $V$ is an bimodule map and it follows from [2], $V=M_{\Theta}$ for some inner function $\Theta\in\mathcal{H}_{\mathcal{W}_{1}}^{2}(\mathbb{D}^{k}).$ Also, $M_{\Theta}\mathcal{H}_{\mathcal{W}}^{2}(\mathbb{D}^{k})=M_{\theta}U_{1}\mathcal{H}_{\mathcal{W}}^{2}(\mathbb{D}^{k})=M_{\theta}\mathfrak{L}^{2}(\mathfrak{F},\mathcal{W})=\mathcal{K}.$ Converse part follows from the proof of the converse of Theorem 4.12. ∎

Acknowledgment

Shankar V. is grateful to The LNM Institute of Information Technology for providing research facility and warm hospitality during a visit in December 2019. Shankar V. is supported by CSIR Fellowship (File No: 09/115(0782)/2017-EMR-I).