Generating wandering subspaces for doubly commuting covariant representations

TL;DR

This paper extends the theory of wandering subspaces to covariant representations of C*-correspondences, introducing new decompositions and characterizations for doubly commuting cases.

Contribution

It develops a Halmos-Richter-type wandering subspace theorem and explores Cauchy duals and Shimorin's Wold-type decomposition for these representations.

Findings

Established a wandering subspace theorem for doubly commuting covariant representations.

Characterized generating wandering subspaces for product system representations.

Connected doubly commutativity with the structure of wandering subspaces.

Abstract

We obtain a Halmos-Richter-type wandering subspace theorem for covariant representations of C*-correspondences. Further the notion of Cauchy dual and a version of Shimorin's Wold-type decomposition for covariant representations of C*-correspondences is explored and as an application a wandering subspace theorem for doubly commuting covariant representations is derived. Using this wandering subspace theorem generating wandering subspaces are characterized for covariant representations of product systems in terms of the doubly commutativity condition.

Equations

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

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

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

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

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

$$\tilde{T}(\xi \otimes h):=T(\xi )h(\xi \in E,h\in \mathcal{H}),$$

$${\varphi }^n(a)({\xi }_1\otimes \cdots \otimes {\xi }_n):=\varphi (a){\xi }_1\otimes \cdots \otimes {\xi }_n.$$

$${\tilde{T}}_n({\xi }_1\otimes \cdots \otimes {\xi }_n\otimes h)=T({\xi }_1)\cdots T({\xi }_n)h$$

$${\tilde{T}}_n=\tilde{T}(I_E\otimes \tilde{T})\cdots (I_{E^{\otimes n-1}}\otimes \tilde{T}).$$

$${\varphi }_{\infty }(a)({\oplus }_{n\in {\mathbb{N}}_0}{\xi }_n):={\oplus }_{n\in {\mathbb{N}}_0}{\varphi }_n(a){\xi }_n,{\xi }_n\in E^{\otimes n}.$$

$$T_{\xi }(\eta ):=\xi \otimes \eta ,\eta \in E^{\otimes n},n\in {\mathbb{N}}_0.$$

$$\rho (a):$$

$$S(\xi ):$$

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

$$\mathcal{H}=\bigvee_{n\geq 0}\mathfrak{L}_{n}(\mathcal{W}),~{}\mbox{where $\mathcal{W}=\mbox{ker}\widetilde{T}^{*}.$}$$

$${\mathcal{H}}_2:={\left(\underset{n\in {\mathbb{N}}_0}{⨁}{\mathfrak{L}}_n(\mathcal{W})\right)}^{\perp }=\bigcap _{n\in {\mathbb{N}}_0}{\mathfrak{L}}_n(\mathcal{H}),$$

$$\Vert {\tilde{T}}_2(\eta \otimes h){\Vert }^2+\Vert \eta \otimes h{\Vert }^2\le 2\Vert (I_E\otimes \tilde{T})(\eta \otimes h){\Vert }^2$$

$${\tilde{T}}_2^{\ast }{\tilde{T}}_2-I_{E^{\otimes 2}\otimes \mathcal{H}}\le 2((I_E\otimes {\tilde{T}}^{\ast }\tilde{T})-I_{E^{\otimes 2}\otimes \mathcal{H}}).$$

$${\tilde{T}}_n^{\ast }{\tilde{T}}_n-I_{E^{\otimes n}\otimes \mathcal{H}}\le n((I_{E^{\otimes n-1}}\otimes {\tilde{T}}^{\ast }\tilde{T})-I_{E^{\otimes n}\otimes \mathcal{H}}).$$

$${\tilde{T}}_{n+1}^{\ast }{\tilde{T}}_{n+1}-I_{E^{\otimes n+1}\otimes \mathcal{H}}\le (n+1)((I_{E^{\otimes n}}\otimes {\tilde{T}}^{\ast }\tilde{T})-I_{E^{\otimes n+1}\otimes \mathcal{H}}),$$

$$(I_{E^{\otimes n}}\otimes {\tilde{T}}^{\ast }){\tilde{T}}_n^{\ast }{\tilde{T}}_n(I_{E^{\otimes n}}\otimes \tilde{T})-(I_{E^{\otimes n}}\otimes {\tilde{T}}^{\ast }\tilde{T})$$

$$\le n((I_{E^{\otimes n}}\otimes {\tilde{T}}^{\ast })(I_{E^{\otimes n-1}}\otimes {\tilde{T}}^{\ast }\tilde{T})(I_{E^{\otimes n}}\otimes \tilde{T})-(I_{E^{\otimes n}}\otimes {\tilde{T}}^{\ast }\tilde{T}))$$

$$=n((I_{E^{\otimes n-1}}\otimes {\tilde{T}}_2^{\ast }{\tilde{T}}_2)-(I_{E^{\otimes n}}\otimes {\tilde{T}}^{\ast }\tilde{T}))$$

$$\le n(2(I_{E^{\otimes n}}\otimes {\tilde{T}}^{\ast }\tilde{T})-I_{E^{\otimes n+1}\otimes \mathcal{H}}-(I_{E^{\otimes n}}\otimes {\tilde{T}}^{\ast }\tilde{T})),$$

$$\Vert \zeta \otimes h{\Vert }^2+n(\Vert (I_{E^{\otimes n-1}}\otimes \tilde{T})(\zeta \otimes h){\Vert }^2-\Vert \zeta \otimes h{\Vert }^2)\ge 0,$$

$$\Vert (I_{E^{\otimes n-1}}\otimes \tilde{T})(\zeta \otimes h){\Vert }^2\ge \frac{n-1}{n}\Vert \zeta \otimes h{\Vert }^2.$$

$$\Vert \tilde{T}(\xi \otimes h)\Vert \ge \frac{n-1}{n}\Vert \xi \otimes h\Vert \text{mbox}foralln\in \mathbb{N}.$$

$$\bigcap _{n\ge 1}{\tilde{T}}_n(E^{\otimes n}\otimes \mathcal{H})=\{0\}.$$

$$\mathcal{K}\ominus \tilde{T}(E\otimes \mathcal{K})$$

$$=\underset{n\ge 0}{\bigvee }{\mathfrak{L}}_n(\mathcal{W})\ominus \underset{n\ge 1}{\bigvee }{\mathfrak{L}}_n(\mathcal{W})=\mathcal{W}.$$

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

$$\mathcal{H}=\underset{n\in {\mathbb{N}}_0}{\bigvee }{\mathfrak{L}}_n(\mathcal{W}),\text{mbox}and\mathcal{W}=\text{mbox}ker{\tilde{T}}^{\ast }.$$

$$\underset{n\in {\mathbb{N}}_0}{\bigvee }{\mathfrak{L}}_n(\mathcal{W})\subseteq \mathcal{K}.$$

Full text

*corresponding author

1. Introduction

Beurling’s theorem [3] says that if $\mathcal{K}$ is a closed shift invariant subspace of the Hardy space $H^{2}(\mathbb{D})$ then $\mathcal{K}$ is the image of an inner function. Wold [36] gave a decomposition of isometries: “Every isometry on a Hilbert space is either a shift, or a unitary, or decomposes uniquely as a direct sum of a shift and a unitary.” In fact, the Beurling’s theorem is a corollary to the Wold decomposition (cf. [6]).

In [23], Rudin explained that the Beurling’s theorem fails in general in the multivariable case. However, using the Wold-type decomposition due to Słociński [30] for a pair of doubly commuting isometries, Mandrekar [12] proved a version of the Beurling’s theorem for the Hardy space over the bidisc $H^{2}(\mathbb{D}^{2})$. Sarkar, Sasane and Wick [24] proved a Beurling-type theorem in the polydisc case and their analysis is based on a generalization of the result of Słociński given in [25]. In [1], Aleman, Richter and Sundberg obtained a Beurling-type theorem for the Bergman space $A^{2}(\mathbb{D}).$ Shimorin [27] provided an elementary proof of this result by considering a Wold-type decomposition for operators close to isometries. Shimorin’s result is generalized in [20] for the Bergman space $A^{2}(\mathbb{D}^{2})$ and in [4] for the Bergman space $A^{2}(\mathbb{D}^{n}).$

In recent years, there has been several papers on the Wold decomposition in the multivariable Operator Theory, see [14, 19, 25, 28, 30, 34, 33]. Pimsner [18] extended the notion of Cuntz algebras [5] and showed that a covariant representation of a $C^{*}$-correspondence $E$ extends to the Toeplitz algebra of $E$ if and only if the covariant representation is isometric. Muhly and Solel [14] explored the Wold decomposition, for $C^{*}$-representations of tensor algebras of $C^{*}$-correspondences, and also analyzed the invariant subspace structure of certain subalgebras of Cuntz-Krieger algebras. The terminology of the tensor product system of Hilbert spaces has been used by Arveson [2] to classify $E_{0}$-semigroups. Discrete product system of $C^{*}$-correspondences is studied by Fowler [7]. Solel [32] defined the notion of doubly commuting covariant representations of product systems of $C^{*}$-correspondences and explored their regular dilations. Wold decomposition for the doubly commuting isometric covariant representations, which is a higher rank version of the Słociński decomposition, is due to Skalski-Zacharias [28].

One of the well-known implication of the Wold decomposition is that the wandering subspace of the shift is the kernel of its adjoint. Halmos [8] characterized all the shift invariant subspaces by proving a wandering subspace theorem which states that there is a one-to-one correspondence between the set of all wandering subspaces and the set of all invariant subspaces. Richter [21] obtained a wandering subspace theorem for the general case of analytic operators satisfying certain condition. In Section 2, a version of Richter’s wandering subspace theorem for covariant representations of $C^{*}$-correspondences is proved. Based on the computations of Section 2, in Section 3 we explore the notion of Cauchy dual and an analogue of Shimorin’s Wold-type decomposition. Using the results of Section 3, in the last section we generalize the main theorem of [4], which is a wandering subspace theorem for doubly commuting bounded operators, for the doubly commuting covariant representations.

1.1. Preliminaries and Notations

We assume the elementary theory of Hilbert $C^{*}$-modules (cf. [17, 11]). Moreover, in this subsection we survey elementary definitions and properties related to covariant representations of $C^{*}$-correspondences (cf. [18, 13]) and recall the Wold-type decomposition for isometric covariant representations from [9, 14].

Assume $E$ to be a Hilbert $C^{*}$-module over a $C^{*}$-algebra $\mathcal{M}.$ Let $\mathcal{L}(E)$ denote the $C^{*}$-algebra of all adjointable operators on $E$. We call $E$ a $C^{*}$-correspondence over $\mathcal{M}$ if $E$ has a left $\mathcal{M}$-module structure defined by a non-zero $*$-homomorphism $\phi:\mathcal{M}\to\mathcal{L}(E)$ in the following sense

[TABLE]

Every $*$-homomorphism considered in this paper is essential, that is, 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 corresponding linking algebra. If $F$ is a $C^{*}$-correspondence over $\mathcal{M},$ then we get the notion of tensor product $F\otimes_{\phi}E$ which has the properties

[TABLE]

[TABLE]

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

Unless it is necessary, the next section onwards we denote the tensor product $F\otimes_{\phi}E$ simply by $F\otimes E.$

Definition 1.1**.**

Assume $\mathcal{H}$ to be a Hilbert space, and $E$ to be a $C^{*}$-correspondence over a $C^{*}$-algebra $\mathcal{M}$. Consider a representation $\sigma:\mathcal{M}\to B(\mathcal{H})$ and a linear map $T:E\to B(\mathcal{H}).$ Then the tuple $(\sigma,T)$ is called a covariant representation of $E$ on $\mathcal{H}$ (cf. [13]) if

[TABLE]

We call the covariant representation $(\sigma,T)$ a completely bounded ( resp. completely contractive) if $T$ is completely bounded (resp. completely contractive). Further, it is called isometric or Toeplitz if

[TABLE]

Lemma 1.2**.**

([13, Lemma 3.5])* The map $(\sigma,T)\mapsto\widetilde{T}$ gives a bijection between the collection of all completely bounded (resp.completely contractive), covariant representations $(\sigma,T)$ of $E$ on $\mathcal{H}$ and the collection of all bounded (resp. contractive) linear maps \widetilde{T}:~{}\mbox{E\otimes_{\sigma}\mathcal{H}\to\mathcal{H}} defined by*

[TABLE]

and satisfying $\widetilde{T}(\phi(a)\otimes I_{\mathcal{H}})=\sigma(a)\widetilde{T}$, $a\in\mathcal{M}$. Moreover, $\widetilde{T}$ is an isometry if and only if the covariant representation $(\sigma,T)$ is isometric.

We say that the covariant representation $(\sigma,T)$ is fully co-isometric if $\widetilde{T}$ is co-isometric, that is, $\widetilde{T}\widetilde{T}^{*}=I_{\mathcal{H}}.$

Consider a $C^{*}$-correspondence $E$ over a $C^{*}$-algebra $\mathcal{M}.$ Then for every $n\in\mathbb{N}$, $E^{\otimes n}:=E\otimes_{\phi}\cdots\otimes_{\phi}E$ ($n$ fold tensor product) is the $C^{*}$-correspondence over the $C^{*}$-algebra $\mathcal{M}$, where the left action of $\mathcal{M}$ on $E^{\otimes n}$ is defined by

[TABLE]

We use $\mathbb{N}_{0}:=\mathbb{N}\cup\{0\}$ and $E^{\otimes 0}:=\mathcal{M}$.

For each $n\in\mathbb{N}$ define the map $\widetilde{T}_{n}:E^{\otimes n}\otimes_{\sigma}\mathcal{H}\to\mathcal{H}$ by

[TABLE]

where $\xi_{1},\ldots,\xi_{n}\in E,h\in\mathcal{H}$. We get,

[TABLE]

The Fock module $\mathcal{F}(E):=\bigoplus_{n\in\mathbb{N}_{0}}E^{\otimes n}$ is the $C^{*}$-correspondence over a $C^{*}$-algebra $\mathcal{M}$, where the left action of $\mathcal{M}$ on $\mathcal{F}(E)$ is denoted by $\phi_{\infty}:\mathcal{M}\longrightarrow L(\mathcal{F}(E))$ and it is defined by

[TABLE]

For $\xi\in E$, we define the creation operator $T_{\xi}$ on $\mathcal{F}(E)$ by

[TABLE]

The following definition of induced representation is a generalization of the multiplication operator $M_{z}\otimes I_{\mathcal{H}}$ on the vector valued Hardy spaces $H^{2}(\mathbb{D})\otimes\mathcal{H}$:

Definition 1.3**.**

Let $\pi$ denote a representation of a $C^{*}$-algebra $\mathcal{M}$ on a Hilbert space $\mathcal{H}$. Let $E$ be a $C^{*}$-correspondence over $\mathcal{M}.$ The isometric covariant representation $(\rho,S)$ of $E$ on the Hilbert space $\mathcal{F}(E)\otimes_{\pi}\mathcal{H}$ defined as

[TABLE]

is called an induced representation (cf. [22]) (induced by $\pi$).

Definition 1.4**.**

• (i)

Let $(\sigma,T)$ be a completely bounded, covariant representation of $E$ on a Hilbert space $\mathcal{H},$ and let $\mathcal{K}$ be a closed subspace of $\mathcal{H}.$ The subspace $\mathcal{K}$ is called $(\sigma,T)$-invariant(resp. $(\sigma,T)$-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 $\mathcal{K}$ (resp. 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 $(\sigma,T)|_{\mathcal{K}}.$

• (ii)

For a closed $\sigma(\mathcal{M})$-invariant subspace $\mathcal{W},$ we define

[TABLE]

for $n\in\mathbb{N}$ and $\mathfrak{L}_{0}(\mathcal{W}):=\mathcal{W}$. Then $\mathcal{W}$ is called wandering (cf. [9]) for $(\sigma,T)$, if $\mathcal{W}$ is orthogonal to $\mathfrak{L}_{n}(\mathcal{W}),$ for all $n\in\mathbb{N}.$ We say that $(\sigma,T)$ has generating wandering subspace property if

[TABLE]

Theorem 1.5**.**

(Muhly and Solel)* Let $(\sigma,T)$ be an isometric, covariant representation of $E$ on a Hilbert space $\mathcal{H}$. Then the representation $(\sigma,T)$ decomposes into a direct sum $(\sigma_{1},T_{1})\bigoplus(\sigma_{2},T_{2})$ on $\mathcal{H}=\mathcal{H}_{1}\bigoplus\mathcal{H}_{2}$ where $(\sigma_{1},T_{1})=(\sigma,T)|_{\mathcal{H}_{1}}$ is an induced representation and $(\sigma_{2},T_{2})=(\sigma,T)|_{\mathcal{H}_{2}}$ is fully coisometric. The above decomposition is unique in the sense that if $\mathcal{K}$ reduces $(\sigma,T),$ and if the restriction $(\sigma,T)|_{\mathcal{K}}$ is induced (resp. fully coisometric), then $\mathcal{K}\subseteq\mathcal{H}_{1}$(resp. $\mathcal{K}\subseteq\mathcal{H}_{2}$). Moreover, $\mathcal{H}_{1}:=\bigoplus_{n\in\mathbb{N}_{0}}\mathfrak{L}_{n}(\mathcal{W}),$ and hence*

[TABLE]

where $\mathcal{W}:=\mbox{ran}(I_{\mathcal{H}}-\widetilde{T}\widetilde{T}^{*})$ and $T_{0}:=\sigma.$

2. Halmos-Richter-type Wandering subspace theorem for covariant representations of $C^{*}$-correspondences

Definition 2.1**.**

Let $(\sigma,T)$ be a completely bounded, covariant representation of $E$ on a Hilbert space $\mathcal{H}.$ If the inequality

[TABLE]

is satisfied for each $\eta\in E^{\otimes 2},h\in\mathcal{H};$ then we say that the covariant representation $(\sigma,T)$ is concave.

Lemma 2.2**.**

Let $(\sigma,T)$ be a concave completely bounded, covariant representation of $E$ on a Hilbert space $\mathcal{H}.$ Then

1. (1)

$\|\widetilde{T}(\xi\otimes h)\|\geq\|\xi\otimes h\|,\>~{}\mbox{for each}~{}\xi\in E,h\in\mathcal{H};$** 2. (2)

$\|\widetilde{T}_{n}(\zeta\otimes h)\|^{2}\leq\|\zeta\otimes h\|^{2}+n(\|(I_{E^{\otimes(n-1)}}\otimes\widetilde{T})(\zeta\otimes h)\|^{2}-\|\zeta\otimes h\|^{2}),~{}\mbox{for each}~{}$* $\zeta\in E^{\otimes n},n\in\mathbb{N}.$*

Proof.

First we shall prove the conclusion (2) by Mathematical induction. Note that Inequality (2.1) is equivalent to the operator inequality

[TABLE]

Assume for fixed $n\in\mathbb{N}$

[TABLE]

Then the inequality

[TABLE]

follows from

[TABLE]

where the last inequality follows from Inequality (2.2). Hence the conclusion (2) holds. It follows from (2) that

[TABLE]

for each $\zeta\in E^{\otimes n},h\in\mathcal{H},n\in\mathbb{N}.$ Then

[TABLE]

Using the properties of the creation operators it is evident that

[TABLE]

Letting $n\rightarrow\infty$ we obtain $\|\xi\otimes h\|\leq\|\widetilde{T}(\xi\otimes h)\|,\>\xi\in E.$ ∎

Definition 2.3**.**

Let $(\sigma,T)$ be a completely bounded, covariant representation of $E$ on a Hilbert space $\mathcal{H}.$ We say that $(\sigma,T)$ is analytic if

[TABLE]

Let $\mathcal{K}$ be a $(\sigma,T)$-invariant subspace of $\mathcal{H}$. Then $\mathcal{K}\ominus\widetilde{T}(E\otimes\mathcal{K})=\mbox{ker}\widetilde{T}^{*}|_{\mathcal{K}}$ is a wandering subspace for $(\sigma,T).$ Conversely, let $\mathcal{W}$ be a wandering subspace for $(\sigma,T)$. Then the subspace $\mathcal{K}:=\bigvee_{n\geq 0}\mathfrak{L}_{n}({\mathcal{W}})$ is $(\sigma,T)$-invariant and in this case $\mathcal{W}=\mathcal{K}\ominus\widetilde{T}(E\otimes\mathcal{K}).$ Indeed,

[TABLE]

Hence the wandering subspace $\mathcal{W}$ is uniquely determined by the invariant subspace $\mathcal{K}$. This combined with the following theorem, which is a generalization of Richter’s wandering subspace theorem for analytic operators [21, Theorem 1], gives us one-to-one correspondence between the set of all wandering subspaces of $\mathcal{H}$ to the set of all $(\sigma,T)$-invariant subspaces of $\mathcal{H}.$

Theorem 2.4**.**

Let $(\sigma,T)$ be an analytic, concave, completely bounded, covariant representation of $E$ on a Hilbert space $\mathcal{H}.$ If $\mathcal{K}$ is a $(\sigma,T)$-invariant subspace of $\mathcal{H}$, then there exists a wandering subspace $\mathcal{W}$ for $(\sigma,T)$ such that

[TABLE]

In particular, $(\sigma,T)$ has generating wandering subspace property, that is,

[TABLE]

Proof.

Let $\mathcal{W}=\mathcal{K}\ominus\widetilde{T}(E\otimes\mathcal{K})(=\mathcal{K}\cap\mbox{ker}\widetilde{T}^{*})$. Then clearly $\mathcal{W}$ is a wandering subspace for $(\sigma,T)$, and $\mathcal{W}\subseteq\mathcal{K}.$ Since $\mathcal{K}$ is $(\sigma,T)$-invariant,

[TABLE]

Now we shall prove the reverse inclusion. Define $(\rho,V):=(\sigma,T)|_{\mathcal{K}}$. Then the covariant representation $(\rho,V)$ is analytic and satisfy Inequality (2.1). Apply Lemma 2.2 (1) to $(\rho,V),$ then $\widetilde{V}^{*}\widetilde{V}-I_{E\otimes\mathcal{K}}$ is a positive operator. Let $\mathcal{D}_{V}:=(\widetilde{V}^{*}\widetilde{V}-I_{E\otimes\mathcal{K}})^{1/2}.$ Then

[TABLE]

where $\xi\in E,h\in\mathcal{K}.$ Using Lemma 2.2 (2) we get

[TABLE]

Since $\widetilde{V}$ is bounded below, it is easy to check that $\widetilde{V}^{*}\widetilde{V}$ is invertible and hence $L:=(\widetilde{V}^{*}\widetilde{V})^{-1}\widetilde{V}^{*}$ is a left inverse of $\widetilde{V}.$ Let $Q:=\widetilde{V}L$ and $P:=I_{\mathcal{K}}-Q$. Then $P$ and $Q$ are orthogonal projections with $\mbox{ran}P=\mbox{ker}Q=\mbox{ker}\widetilde{V}^{*}=\mathcal{W}.$ For each $n\in\mathbb{N}$, we define $L^{n}:\mathcal{K}\rightarrow E^{\otimes n}\otimes\mathcal{K}$ by

[TABLE]

Fix $h\in\mathcal{K}$, we have

[TABLE]

Since $\bigvee_{n\in\mathbb{N}_{0}}\mathfrak{L}_{n}(\mathcal{W})$ is weakly closed, it is enough to prove that the sequence $\{(I_{\mathcal{K}}-\widetilde{V}_{n}L^{n})h\}$ has a weakly convergent subsequence and it converges to $h$(weakly). First using Mathematical induction we shall show that for each $n\in\mathbb{N}$,

[TABLE]

Note that the $n=1$ case follows from

[TABLE]

Assume $n\geq 1,$ it follows that

[TABLE]

Using the previous equation and the induction hypothesis for $n\in\mathbb{N},$ we see that

[TABLE]

Hence the claim has been proved for each $n\in\mathbb{N}$.

Let $l,m\in\mathbb{N}$ with $l\leq m,$ then

[TABLE]

Using Equation (2) we get that $\{\|L^{n}h\|^{2}\}$ is a non-increasing sequence of non-negative numbers. Hence $\lim_{n\rightarrow\infty}\|L^{n}h\|^{2}$ exists and

[TABLE]

Then there exists a weakly convergent subsequence of $\{\widetilde{V}_{n}L^{n}h\}$, say $\{\widetilde{V}_{n_{j}}L^{n_{j}}h\}$ which converges to $w$(weakly) for some $w\in\mathcal{K}$. Let $N\in\mathbb{N}$ be such that $n_{j}\geq N$, implies $\widetilde{V}_{n_{j}}L^{n_{j}}h\in\widetilde{V}_{n_{j}}(E^{\otimes n_{j}}\otimes\mathcal{K})\subseteq\widetilde{V}_{N}(E^{\otimes N}\otimes\mathcal{K}).$ Since $(\sigma,T)$ is analytic and $\widetilde{V}_{N}(E^{\otimes N}\otimes\mathcal{K})$ is weakly closed (by Lemma 2.2(a)), it follows that $w=0.$ Hence $(I_{\mathcal{K}}-\widetilde{V}_{n_{j}}L^{n_{j}})h\rightarrow h$(weakly). ∎

Remark 2.5**.**

We shall see that $(I_{\mathcal{K}}-\widetilde{V}_{n_{j}}L^{n_{j}})h\rightarrow h$ in norm. This will follow, if $\|L^{n}h\|\rightarrow 0.$ Let $h\in\mathcal{K},$ by the previous theorem there exists a sequence $h_{n}$ converging to $h$ in the norm such that

[TABLE]

Let $\varepsilon>0.$ Then there exists $N\in\mathbb{N}$ such that $\|h-h_{N}\|<\varepsilon.$ For each $n>N,$ it follows that

[TABLE]

Hence $\|L^{n}h\|\rightarrow 0.$

Using the previous remark and Equation (2.6) we see that for every $h\in\mathcal{K}$ we have

[TABLE]

3. Cauchy Dual and Shimorin-type decomposition for covariant representations of $C^{*}$-correspondences

The main theorem of this section, Theorem 3.13, establishes two sufficient conditions for the covariant representation $(\sigma,T)$ for the existence of the following Wold-type decomposition:

Definition 3.1**.**

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 there exists a wandering subspace $\mathcal{W}$ for $(\sigma,T)$ which decomposes $\mathcal{H}$ into the following direct sum of two $(\sigma,T)$-reducing subspaces

[TABLE]

such that the restriction of $(\sigma,T)$ on the subspace $\bigcap_{n\geq 1}\widetilde{T}_{n}(E^{\otimes n}\otimes\mathcal{H})$ is isometric as well as fully co-isometric covariant representation.

Remark 3.2**.**

Since the wandering subspace is unique, $\mathcal{W}=\mbox{ker}\widetilde{T}^{*}$. Throughout this section we use notations $\mathcal{H}_{\infty}$ and $\mathcal{W}$ for

[TABLE]

Let $(\sigma,T)$ be a completely bounded, covariant representation of $E$ on a Hilbert space $\mathcal{H},$ such that $\widetilde{T}$ is left invertible. Suppose that $\widetilde{T}$ is invertible, then $\mathcal{H}_{\infty}=\mathcal{H}$ and $\mathcal{W}=\{0\}.$ If $\widetilde{T}$ is not invertible, then the subspace $\mathcal{H}_{\infty}$ is $(\sigma,T)$-invariant and $\mathcal{W}$ is a non-trivial wandering subspace for $(\sigma,T).$ Since $\widetilde{T}$ is left invertible, $\widetilde{T}^{*}\widetilde{T}$ is invertible, define the operator $L:\mathcal{H}\rightarrow E\otimes\mathcal{H}$ by

[TABLE]

It is easy to see that $\mbox{ker}L=\mathcal{W}$ and $P_{\mathcal{W}}=I_{\mathcal{H}}-\widetilde{T}L,$ where $P_{\mathcal{W}}$ denote the orthogonal projection on $\mathcal{W}.$ We denote this orthogonal projecton on $\mathcal{W}$ by $P$(instead of $P_{\mathcal{W}}).$ Let $n\in\mathbb{N}$, define $L^{n}:\mathcal{H}\rightarrow E^{\otimes n}\otimes\mathcal{H}$ by

[TABLE]

Note that

[TABLE]

Lemma 3.3**.**

$\mbox{ker}L^{n}=\bigvee\{\widetilde{T}_{j}(\eta_{j}\otimes w):\>\eta_{j}\in E^{\otimes j},w\in\mathcal{W},j=0,1,\dots,n-1\}.$**

Proof.

If $L^{n}w=0,$ then

[TABLE]

and it shows the inclusion $\subseteq.$ Since $L\widetilde{T}=I_{\mathcal{H}},$ we have $\mbox{ker}L=\mbox{ker}\widetilde{T}^{*}=\mathcal{W},$ and the reverse inclusion follows. ∎

Let $(\sigma,T)$ be a concave completely bounded, covariant representation of $E$ on $\mathcal{H}.$ Define the operator $U:\mathcal{H}\longrightarrow\mathcal{F}(E)\otimes\mathcal{W}$ by

[TABLE]

where $P=I_{\mathcal{H}}-Q,\>Q=\widetilde{T}L.$ Then the operator $U$ is well defined contraction. Indeed,

[TABLE]

Lemma 3.4**.**

$\mbox{ker}U=\mathcal{H}_{\infty}.$**

Proof.

Suppose that for all $n\in\mathbb{N},$ $(I_{E^{\otimes n}}\otimes P)L^{n}h=0.$ It follows that

[TABLE]

and we get $h\in\mathcal{H}_{\infty}.$ Conversely, assume that $h\in\mathcal{H}_{\infty},$ then

[TABLE]

Hence

[TABLE]

By Remark 2.5, if $(\sigma,T)$ is analytic then for all $h\in\mathcal{H},$ we have $L^{n}h\rightarrow 0$ as $n\rightarrow\infty.$ Therefore, $U$ is unitary if and only if $(\sigma,T)$ is analytic.

Let $(\sigma,T)$ be a completely bounded, covariant representation of $E$ on a Hilbert space $\mathcal{H},$ such that $\widetilde{T}$ is left invertible. Define $\widetilde{T}^{\prime}:E\otimes\mathcal{H}\longrightarrow\mathcal{H}$ by

[TABLE]

Since $\widetilde{T}^{*}\widetilde{T}(\phi(a)\otimes I_{\mathcal{H}})=\widetilde{T}^{*}\sigma(a)\widetilde{T}=(\phi(a)\otimes I_{\mathcal{H}})\widetilde{T}\widetilde{T}^{*}$ for each $a\in\mathcal{M},$ it is easy to observe that $(\widetilde{T}^{*}\widetilde{T})^{-1}(\phi(a)\otimes I_{\mathcal{H}})=(\phi(a)\otimes I_{\mathcal{H}})(\widetilde{T}^{*}\widetilde{T})^{-1}$ and hence

[TABLE]

Therefore the corresponding covariant representation $(\sigma,T^{\prime})$ of $E$ on the Hilbert space $\mathcal{H}$ is completely bounded by Lemma 1.2.

Definition 3.5**.**

We say that the covariant representation $(\sigma,T^{\prime})$ defined above is Cauchy dual of $(\sigma,T).$

Notation 3.6**.**

$\mathcal{H}^{\prime}_{\infty}:=\bigcap_{n\geq 1}\widetilde{T}^{\prime}_{n}(E^{\otimes n}\otimes\mathcal{H})\>\>\mbox{and}\>\>\mathcal{W}^{\prime}:=\mbox{ker}\widetilde{T}^{\prime*}.$

Obviously, $\mathcal{W^{\prime}}=\mbox{ker}\widetilde{T}^{*}=\mathcal{W}.$

Proposition 3.7**.**

*(1)$(\mathcal{H}^{\prime}_{\infty})^{\bot}=\bigvee_{n\geq 0}\mathfrak{L}_{n}(\mathcal{W});$

(2)$\mathcal{H}_{\infty}^{\>\bot}=\bigvee_{n\geq 0}\mathfrak{L}_{n}^{\prime}(\mathcal{W}),$ where*

[TABLE]

Proof.

First we shall prove (1). Note that

[TABLE]

The last equality follows from Lemma 3.3. Note that

[TABLE]

that is, the Cauchy dual of $(\sigma,T^{\prime})$ is $(\sigma,T).$ Hence (2) follows. ∎

Corollary 3.8**.**

1. (1)

$(\sigma,T)$* is analytic if and only if $(\sigma,T^{\prime})$ has the generating wandering subspace property;* 2. (2)

$(\sigma,T)$* has the generating wandering subspace property if and only if $(\sigma,T^{\prime})$ is analytic.*

Corollary 3.9**.**

$(\sigma,T)$* admits Wold-type decomposition if and only if $(\sigma,T^{\prime})$ does. Indeed, $\mathcal{H}_{\infty}=\mathcal{H}_{\infty}^{\prime}$ and $\bigvee_{n\geq 0}\mathfrak{L}_{n}(\mathcal{W})=\bigvee_{n\geq 0}\mathfrak{L}^{\prime}_{n}(\mathcal{W}).$*

Proposition 3.10**.**

If $\mathcal{H}_{\infty}^{\prime}$ is $(\sigma,T^{\prime})$-reducing, then $\mathcal{H}_{\infty}^{\prime}\subseteq\mathcal{H}_{\infty}.$

Proof.

Suppose $\mathcal{H}_{\infty}^{\prime}$ is $(\sigma,T^{\prime})$-reducing. From Equation (3.4), it follows that $\widetilde{T}=\widetilde{T^{\prime}}(\widetilde{T^{\prime}}^{*}\widetilde{T^{\prime}})^{-1}$. Thus $\mathcal{H}_{\infty}^{\prime}$ is also $(\sigma,T)$-reducing. Note that $\widetilde{T}|_{E\otimes\mathcal{H}_{\infty}^{\prime}}=\widetilde{T^{\prime}}_{\infty}(\widetilde{T^{\prime}}^{*}_{\infty}\widetilde{T^{\prime}}_{\infty})^{-1},$ where $\widetilde{T^{\prime}}_{\infty}:=\widetilde{T^{\prime}}|_{E\otimes\mathcal{H}_{\infty}^{\prime}}.$ Therefore, $\mathcal{H}_{\infty}^{\prime}$ is $(\sigma,T)|_{\mathcal{H}_{\infty}^{\prime}}$-reducing. Since $\widetilde{T^{\prime}}_{\infty}$ is invertible, we conclude that $\widetilde{T}|_{E\otimes\mathcal{H}_{\infty}^{\prime}}$ is also invertible, hence $\mathcal{H}_{\infty}^{\prime}=\widetilde{T}(E\otimes\mathcal{H}_{\infty}^{\prime})\subseteq\mathcal{H}_{\infty},$ which proves the proposition. ∎

Corollary 3.11**.**

Let $(\sigma,T)$ be analytic and $\mathcal{H}_{\infty}^{\prime}$ be $(\sigma,T^{\prime})$-reducing. Then $(\sigma,T)$ has the generating wandering subspace property.

The following theorem gives a sufficient condition for the wandering subspace property for an analytic covariant representation $(\sigma,T).$

Theorem 3.12**.**

Let $(\sigma,T)$ be a concave, completely bounded, covariant representation of $E$ on $\mathcal{H}.$ Then $\mathcal{H}_{\infty}$ is $(\sigma,T)$-reducing and $(\sigma,T)|_{\mathcal{H}_{\infty}}$ is an isometric and fully co-isometric, covariant representation of $E$ on the Hilbert space $\mathcal{H}_{\infty}.$

Proof.

Let $h\in\mathcal{H}_{\infty},$ then by Lemma 3.4, $h=\widetilde{T}_{n}L^{n}h,n\in\mathbb{N}.$ Since $(\sigma,T)$ is concave, by Equation (2) $\|L^{n+1}h\|\leq\|L^{n}h\|\leq\|h\|.$ Also, by Equation (2.8), we obtain $\lim_{n\rightarrow\infty}\|L^{n}h\|=\|h\|,$ therefore $\|L^{n}h\|=\|h\|,$ for each $n\geq 0.$ In particular, $\|Lh\|=\|h\|,$ that is, $L$ is an isometry. For every $\xi\in E,$

[TABLE]

Hence $\widetilde{T}|_{E\otimes\mathcal{H}_{\infty}}$ is an isometry.

Since $\widetilde{T}:E\otimes\mathcal{H}_{\infty}\longrightarrow\mathcal{H}_{\infty}$ is onto, there exists $\kappa\in E\otimes\mathcal{H}_{\infty}$ such that $\widetilde{T}(\kappa)=h.$ Then $\widetilde{T}^{*}(h)=\widetilde{T}^{*}\widetilde{T}(\kappa)=\kappa\in E\otimes\mathcal{H}_{\infty}$, and it follows that $\mathcal{H}_{\infty}$ is a reducing subspace for $(\sigma,T).$ Hence $(\sigma,T)|_{\mathcal{H}_{\infty}}$ is an isometric and fully co-isometric covariant representation. ∎

The following result is a generalization of Shimorin’s Wold-type decomposition [27, Theorem 3.6] and Theorem 1.5:

Theorem 3.13**.**

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)

for any $\zeta\in E^{\otimes 2}\otimes\mathcal{H},\kappa\in E\otimes\mathcal{H}$

[TABLE]

Then $(\sigma,T)$ admits Wold-type decomposition. In particular, if $(\sigma,T)$ is analytic, then $\mathcal{W}=\mathcal{H}\ominus\widetilde{T}(E\otimes\mathcal{H})$ is the generating wandering subspace for $(\sigma,T),$ that is, $\mathcal{H}=\bigvee_{n\in\mathbb{N}_{0}}\mathfrak{L}_{n}(\mathcal{W}).$

Proof.

First assume that $(\sigma,T)$ is concave. By Theorem 3.12, $(\sigma,T)|_{\mathcal{H}_{\infty}}$ is an isometric and fully co-isometric covariant representation. It remains to prove the relation $\mathcal{H}\ominus\mathcal{H}_{\infty}=\bigvee_{n\geq 0}\mathfrak{L}_{n}(\mathcal{W}).$ Since $\mathcal{H}\ominus\mathcal{H}_{\infty}$ is reducing for $(\sigma,T)$, then it is easy to check that $(\sigma,T)|_{\mathcal{H}\ominus\mathcal{H}_{\infty}}$ is analytic, and therefore $\mathcal{W}=\mbox{ker}(\widetilde{T}|_{E\otimes(\mathcal{H}\ominus\mathcal{H}_{\infty})})^{*}.$ Now apply Theorem 2.4 for the $(\sigma,T)$-invariant subspace $\mathcal{H}\ominus\mathcal{H}_{\infty},$ then $\mathcal{W}$ is a generating wandering subspace for $(\sigma,T)|_{\mathcal{H}\ominus\mathcal{H}_{\infty}},$ and from the uniqueness of the wandering subspaces we get the desired relation.

Next, assume that $(\sigma,T)$ satisfies the condition $(2)$ of the statement. Taking $\zeta=0$ in the Inequality (3.5), we get $\widetilde{T}$ is left invertible. Let $\kappa\in E\otimes\mathcal{H}.$ Then $\kappa=(\widetilde{T}^{*}\widetilde{T})^{-1}\xi$ for some $\xi\in E\otimes\mathcal{H}$, and substituting $\kappa$ in the same inequality gives

[TABLE]

Define the operator $X:(E^{\otimes 2}\otimes\mathcal{H})\oplus(E\otimes\mathcal{H})\longrightarrow E\otimes\mathcal{H}$ by

[TABLE]

Then from the previous inequality, $\|X\|\leq\sqrt{2},$ which gives $XX^{*}\leq 2I_{E\otimes\mathcal{H}},$ and we obtain

[TABLE]

Now, substitute $\widetilde{T}=\widetilde{T^{\prime}}(\widetilde{T^{\prime}}^{*}\widetilde{T^{\prime}})^{-1}$ in the previous inequality and use Equation (3.4), we have

[TABLE]

Multiply this inequality by $I_{E}\otimes\widetilde{T^{\prime}}^{*}$ on the left side and by $I_{E}\otimes\widetilde{T^{\prime}}$ on the right, we obtain

[TABLE]

This shows that the Cauchy dual $(\sigma,T^{\prime})$ of $(\sigma,T)$ is concave. Therefore $(\sigma,T^{\prime})$ admits Wold-type decomposition. Hence, using Corollary 3.9, we get the desired conclusion. ∎

Corollary 3.14**.**

Let $(\sigma,T)$ be an analytic completely bounded, covariant representation of $E$ on a Hilbert space $\mathcal{H}$ which satisfies the Inequality (3.5). Then $(\sigma,T)$ has the generating wandering subspace property.

Corollary 3.15**.**

Let $(\sigma,T)$ be a completely bounded, covariant representation of $E$ on $\mathcal{H}$ which satisfies

[TABLE]

and let $\widetilde{T}$ be bounded below. Then $(\sigma,T)$ admits Wold-type decomposition. In particular, if $(\sigma,T)$ is analytic, then $(\sigma,T)$ has the generating wandering subspace property.

Proof.

We have that

[TABLE]

where $\xi\in E\otimes\mathcal{H}.$ Since $\widetilde{T}$ is left invertible, $\xi=(\widetilde{T}^{*}\widetilde{T})^{-1}(\eta)$ for some $\eta\in E\otimes\mathcal{H},$ substituting it in the above inequality, we get

[TABLE]

that is,

[TABLE]

In fact, Equation (3.7) is equivalent to Equation (3.6).

Define the operator $X:(E^{\otimes 2}\otimes\mathcal{H})\oplus(E\otimes\mathcal{H})\longrightarrow E\otimes\mathcal{H}$ as in the previous theorem. Inequality (3.7) yields

[TABLE]

Further, substituting $\eta=(\widetilde{T}^{*}\widetilde{T})^{-1/2}\xi_{2}$ we have

[TABLE]

This shows that Inequality (3.7) is equivalent to Inequality (3.5). Hence from Theorem 3.13 $(\sigma,T)$ admits Wold-type decomposition. ∎

4. Generating wandering subspaces for doubly commuting covariant representations of product systems

Wandering subspaces for covariant representations of subproduct systems of $C^{*}$-correspondences is studied in [26]. In this section we recall several notions related to product systems of $C^{*}$-correspondences from (see [7, 31, 32, 28]) and explore wandering subspaces for doubly commuting covariant representations of product systems.

Definition 4.1**.**

• (1)

Let $k\in\mathbb{N}.$ A product system (cf. [7]) $\mathbb{E}$ is defined as a family of $C^{*}$-correspondences $\{E_{1},\ldots,E_{k}\},$ along with the unitary isomorphisms $t_{i,j}:E_{i}\otimes E_{j}\to E_{j}\otimes E_{i}$ ($i>j$). Using these isomorphisms, 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 define maps $t_{i,i}:={\textrm{id}}_{E_{i}\otimes E_{i}}$ and $t_{i,j}:=t_{j,i}^{-1}$ when $i<j.$

• (2)

Assume $\mathbb{E}$ to be a product system over $\mathbb{N}_{0}^{k}$. A completely bounded, covariant representation (cf. [32]) of $\mathbb{E}$ on a Hilbert space $\mathcal{H}$ is defined as a tuple $(\sigma,T^{(1)},\ldots,T^{(k)})$, where $\sigma$ is a representation of the $C^{*}$-algebra $\mathcal{M}$ on $\mathcal{H}$, $T^{(i)}:E_{i}\to B(\mathcal{H})$ are completely bounded linear maps satisfying

[TABLE]

and satisfy the commutation relation

[TABLE]

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

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]

Similarly for $\mathbf{n}=(n_{1},\cdots,n_{k})\in\mathbb{N}_{0}^{k}$, we use notation $\widetilde{T}_{\mathbf{n}}:\mathbb{E}(\mathbf{n})\otimes\mathcal{H}\longrightarrow\mathcal{H}$ for

[TABLE]

Let us define the linear map $T_{\mathbf{n}}:\mathbb{E}(\mathbf{n})\longrightarrow B(\mathcal{H})$ (cf. [32]) by

[TABLE]

We use $I_{k}$ for $\{1,2,\dots,k\}.$ Let $\alpha=\{\alpha_{1},\dots,\alpha_{n}\}\subseteq I_{k}$, define

[TABLE]

For each $\mathbf{m}=(m_{\alpha_{1}},\cdots,m_{\alpha_{n}})\in\mathbb{N}_{0}^{\alpha}$, the map $\widetilde{T}_{\mathbf{m}}^{\alpha}:\mathbb{E}(\mathbf{m})\otimes\mathcal{H}\longrightarrow\mathcal{H}$ is defined by

[TABLE]

Definition 4.2**.**

Let $\mathcal{K}$ be a closed subspace of a Hilbert space $\mathcal{H}.$ The subspace $\mathcal{K}$ is called invariant( resp. reducing) (cf. [28]) for a covariant representation $(\sigma,T^{(1)},\ldots,T^{(k)})$ on $\mathcal{H},$ if $\mathcal{K}$ is $(\sigma,T^{(i)})$-invariant(resp. $(\sigma,T^{(i)})$-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 $(\sigma,T^{(1)},\ldots,T^{(k)})|_{\mathcal{K}}$.

Moreover, for a closed $\sigma(\mathcal{M})$-invariant subspace $\mathcal{K},$ we use symbol

[TABLE]

Clearly $\mathfrak{L}_{\mathbf{m}}^{\alpha}(\mathcal{K})=\overline{\widetilde{T}_{\mathbf{m}}^{\alpha}(\mathbb{E}(\mathbf{m})\otimes\mathcal{K})}.$ We use $[\mathcal{K}]_{T_{\alpha}}$ for the smallest closed

$(\sigma,T^{(\alpha_{1})},\dots,T^{(\alpha_{n})})$-invariant subspace of $\mathcal{H}$ containing $\mathcal{K},$ that is,

[TABLE]

Further, we simply write $[\mathcal{K}]_{T^{(i)}}$ if $\alpha$ is a singleton set $\{i\}.$

Definition 4.3**.**

1. (1)

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

[TABLE] 2. (2)

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

[TABLE]

Notation 4.4**.**

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

[TABLE]

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

[TABLE]

Definition 4.5**.**

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. [32]) if for each distinct $i,j\in\{1,\ldots,k\}$ we have

[TABLE]

For distinct $i,j\in\{1,\ldots,k\}$, a simple calculation (cf. [28, p. 460]) using Equation (4.5) 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 3.13 for the covariant representation $(\sigma,T^{(1)},\dots,T^{(k)}).$

Proposition 4.6**.**

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 $j\notin\alpha.$

Proof.

Let $\alpha=\{\alpha_{1},\dots,\alpha_{n}\}$ be a non-empty subset of $I_{k}$ and let $j\notin\alpha.$ Since $\mathcal{W}_{l}=\mbox{ker}\widetilde{T}^{(l)^{*}}$ for all $1\leq l\leq k,$ we get $\mathcal{W}_{\alpha}=\bigcap_{i=1}^{n}\mbox{ker}\widetilde{T}^{(\alpha_{i})^{*}}.$ Let $\eta_{j}\in E_{j},\xi_{\alpha_{i}}\in E_{\alpha_{i}},w_{\alpha}\in\mathcal{W}_{\alpha}$ and $h\in\mathcal{H}.$ By doubly commutativity of $(\sigma,T^{(1)},\dots,T^{(k)}),$ we obtain

[TABLE]

It follows that $\widetilde{T}^{(j)}(E_{j}\otimes\mathcal{W}_{\alpha})\bot\widetilde{T}^{(\alpha_{i})}(E_{\alpha_{i}}\otimes\mathcal{H})$ and hence $\widetilde{T}^{(j)}(E_{j}\otimes\mathcal{W}_{\alpha})\subseteq\mathcal{W}_{\alpha_{i}}$ for all $\alpha_{i}\in\alpha$ and $j\notin\alpha.$ Therefore $\mathcal{W}_{\alpha}$ is $(\sigma,T^{(j)})$-invariant. Also,

[TABLE]

for all $\alpha_{i}\in\alpha$ and $h\in\mathcal{H}.$ Note that $\mbox{ker}(I_{E}\otimes\widetilde{T}^{(\alpha_{i})^{*}})=E\otimes\mbox{ker}\widetilde{T}^{(\alpha_{i})^{*}}$ (see proof of [29, Lemma 1.6 ]), then $\widetilde{T}^{(j)^{*}}w_{\alpha}\in((I_{E_{j}}\otimes\widetilde{T}^{(\alpha_{i})})(E_{j}\otimes E_{\alpha_{i}}\otimes\mathcal{H}))^{\bot}=\mbox{ker}(I_{E_{j}}\otimes\widetilde{T}^{(\alpha_{i})^{*}})=E_{j}\otimes\mathcal{W}_{\alpha_{i}}$ and then $\widetilde{T}^{(j)^{*}}\mathcal{W}_{\alpha}\subseteq E_{j}\otimes\mathcal{W}_{\alpha_{i}}$ for all $\alpha_{i}\in\alpha.$ It follows that $\widetilde{T}^{(j)^{*}}\mathcal{W}_{\alpha}\subseteq E_{j}\otimes\mathcal{W}_{\alpha}$ for $j\notin\alpha.$ This completes the proof. ∎

We denote the cardinality of $\alpha$ by $\#\alpha.$

Theorem 4.7**.**

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 for any $(\sigma,T^{(i)})$-reducing subspace $\mathcal{K}_{i}$, the subspace

[TABLE]

is a generating wandering subspace for $(\sigma,T^{(i)})|_{\mathcal{K}_{i}}~{}\mbox{where}~{}i=1,\ldots,k.$ Then for each non-empty set $\alpha=\{\alpha_{1},\ldots,\alpha_{n}\}\subseteq I_{k},$ the covariant representation $(\sigma,T^{(\alpha_{1})},\dots,T^{(\alpha_{n})})$ has a generating wandering subspace property. Moreover, the corresponding generating wandering subspace is given by

[TABLE]

Proof.

First, we shall show that $\mathcal{W}_{\alpha}=\bigcap_{i=1}^{n}\mathcal{W}_{\alpha_{i}},$ where $\mathcal{W}_{\alpha_{i}}=\mbox{ker}\widetilde{T}^{(\alpha_{i})^{*}}$ is the wandering subspace for the covariant representation $(\sigma,T^{(\alpha_{1})},\dots,T^{(\alpha_{n})}).$ Let $w_{1},w_{2}\in\mathcal{W}_{\alpha}$ and $\mathbf{m}=(m_{\alpha_{1}},\cdots,m_{\alpha_{n}})\in\mathbb{N}_{0}^{\alpha}\setminus\{0\}.$ Without loss of generality we can assume that $m_{\alpha_{1}}\neq 0.$ Then

[TABLE]

where $\beta=\alpha\setminus\{\alpha_{1}\},$ $\eta_{\alpha}\in\mathbb{E}(\mathbf{m})$ and $\mathbf{e_{i}}\in\mathbb{N}_{0}^{\alpha}$ whose $\alpha_{i}^{\mbox{th}}$ entry is 1 and all other entries are $0.$ Next, we need to show that $\mathcal{H}=\bigvee_{\mathbf{m}\in\mathbb{N}_{0}^{\alpha}}\mathfrak{L}^{\alpha}_{\mathbf{m}}(\mathcal{W}_{\alpha}).$ Suppose that $\alpha=\{i\},$ then by hypothesis $\mathcal{W}_{i}$ is a generating wandering subspace for $(\sigma,T^{(i)}),i=1,\ldots,k.$ Now for $\#\alpha\geq 2,$ it is enough to show that $[\mathcal{W}_{\alpha}]_{T^{(\alpha_{i})}}=\mathcal{W}_{\alpha\setminus\{\alpha_{i}\}}$ for any $\alpha_{i}\in\alpha.$ Because for $\alpha_{i},\alpha_{j}\in\alpha,$ one can repeat the procedure and obtain

[TABLE]

and continue this procedure until the set $\alpha\setminus\{\alpha_{i},\alpha_{j}\}$ becomes a singleton set and apply the assumption for the singleton set.

To this end, let $\alpha\subseteq I_{k},\>\#\alpha\geq 2$ and $\alpha_{i}\in\alpha.$ Consider the subspace $F=\mathcal{W}_{\alpha\setminus\{\alpha_{i}\}}\ominus\widetilde{T}^{(\alpha_{i})}(E_{\alpha_{i}}\otimes\mathcal{W}_{\alpha\setminus\{\alpha_{i}\}}).$ Now by Proposition 4.6, $\mathcal{W}_{\alpha\setminus\{\alpha_{i}\}}$ is $(\sigma,T^{(\alpha_{i})})$-reducing and therefore $F=\mathcal{W}_{\alpha\setminus\{\alpha_{i}\}}\cap\mathcal{W}_{\alpha_{i}}=\mathcal{W}_{\alpha}.$ On the other hand, since $\mathcal{W}_{\alpha\setminus\{\alpha_{i}\}}$ is $(\sigma,T^{(\alpha_{i})})$-reducing, by hypothesis $F=\mathcal{W}_{\alpha}$ is the wandering subspace for $(\sigma,T^{(\alpha_{i})}).$ Hence

[TABLE]

This completes the proof. ∎

Corollary 4.8**.**

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 $\zeta\in E_{i}^{\otimes 2}\otimes\mathcal{H},\xi\in E_{i}\otimes\mathcal{H},i=1,\dots,k.$*

Then for any non-empty set $\alpha=\{\alpha_{1},\dots,\alpha_{k}\}\subseteq I_{k},\>\mathcal{W}_{\alpha}$ is a generating wandering subspace for $(\sigma,T^{(\alpha_{1})},\dots,T^{(\alpha_{k})}).$

Proof.

Let $\mathcal{K}_{i}$ be a $(\sigma,T^{(i)})$-reducing subspace of $\mathcal{H}$ for $1\leq i\leq k.$ Then the covariant representation $(\sigma,T^{(i)})|_{\mathcal{K}_{i}}$ also satisfies one of the conditions (1) and (2), respectively. Thus by Theorem 3.13, $\mathcal{K}_{i}\ominus\widetilde{T}^{(i)}(E\otimes\mathcal{K}_{i})$ is a generating wandering subspace for $(\sigma,T^{(i)})|_{\mathcal{K}_{i}}.$ Hence from Theorem 4.7 the proof follows. ∎

The following main theorem of the paper is a generalization of [4, Theorem 2.5]:

Theorem 4.9**.**

*Let $(\sigma,T^{(1)},\dots,T^{(k)})$ be a completely bounded, covariant representation of the product system $\mathbb{E}$ on a Hilbert space $\mathcal{H}$ such that

$\|(I_{E_{i}}\otimes\widetilde{T}^{(i)})(\zeta)+\xi\|^{2}\leq 2(\|\zeta\|^{2}+\|\widetilde{T}^{(i)}(\xi)\|^{2}),\>\zeta\in E_{i}^{\otimes 2}\otimes\mathcal{H},\xi\in E_{i}\otimes\mathcal{H}$

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

1. (1)

$(\sigma,T^{(1)},\dots,T^{(k)})$* is doubly commuting, and* 2. (2)

$(\sigma,T^{(i)})$* is analytic for all $i=1,\dots,k.$*

if and only if

1. (a)

For any non-empty set $\alpha=\{\alpha_{1},\dots,\alpha_{n}\}\subseteq I_{k},\>\mathcal{W}_{\alpha}$ is a generating wandering subspace for $(\sigma,T^{(\alpha_{1})},\dots,T^{(\alpha_{n})}),$ and for $n\geq 2,\>[\mathcal{W}_{\alpha}]_{{T}^{(\alpha_{i})}}=\mathcal{W}_{\alpha\setminus\{\alpha_{i}\}},$ for all $\alpha_{i}\in\alpha,$ 2. (b)

$(I_{E_{j}}\otimes\widetilde{T}^{(i)})(t_{i,j}\otimes I_{\mathcal{H}})(I_{E_{i}}\otimes\widetilde{T}^{(j)^{*}}\widetilde{T}^{(j)})=\widetilde{T}^{(j)^{*}}\widetilde{T}^{(j)}(I_{E_{j}}\otimes\widetilde{T}^{(i)})(t_{i,j}\otimes I_{\mathcal{H}})$* for all $1\leq i<j\leq k.$*

Proof.

The necessary part follows from Theorem 4.7. Indeed, for $1\leq i<j\leq k,$ by the assumption of the doubly commutativity, we have

[TABLE]

Conversely, suppose that (a) and (b) hold. First we shall show that $(\sigma,T^{(1)},\dots,$ $T^{(k)})$ is doubly commuting. Let $1\leq i\leq k$ be fixed. By assumption(a), $\mathcal{W}_{i}=[\mathcal{W}_{\{i,j\}}]_{T^{(j)}}$ for all $1\leq i\neq j\leq k.$ It follows that $\mathcal{W}_{i}$ is $(\sigma,T^{(j)})$-invariant subspace for all $1\leq j\neq i\leq k.$ Let $h\in\mathcal{H},$ since $[\mathcal{W}_{i}]_{T^{(i)}}=\mathcal{H},$ thus there exists a sequence $h_{n}$ converging to $h$ such that

[TABLE]

Then for any $1\leq j\neq i\leq k\>\mbox{and}\>\eta_{j}\in E_{j},$ we obtain

[TABLE]

since $\mathcal{W}_{i}=\mbox{ker}\widetilde{T}^{(i)^{*}}.$ On the other hand, using notation $\eta\otimes h$ for $\eta_{m,n}\otimes h_{m,n}$ we have

[TABLE]

where the $3^{rd}$ equality follows from (b). It implies

[TABLE]

and then by taking limit we obtain

[TABLE]

Thus we have (1).

Finally, from Theorem 3.13 we have that

[TABLE]

for all $1\leq i\leq k.$ But by part (a), $[\mathcal{W}_{i}]_{T^{(i)}}=\mathcal{H}$ for all $1\leq i\leq k.$ Thus $\bigcap_{m\geq 1}\widetilde{T}_{m}^{(i)}(E_{i}^{\otimes m}\otimes\mathcal{H})$={0} for all $1\leq i\leq k.$ ∎

Acknowledgment

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