We prove that the rank of a non-trivial co-doubly commuting submodule is $2$. More precisely, let $φ, ψ\in H^\infty(\mathbb{D})$ be two inner functions. If $\mathcal{Q}_φ = H^2(\mathbb{D})/ φH^2(\mathbb{D})$ and $\mathcal{Q}_ψ = H^2(\mathbb{D})/ ψH^2(\mathbb{D})$, then \[ \mbox{rank~}(\mathcal{Q}_φ \otimes \mathcal{Q}_ψ)^\perp = 2. \] An immediate consequence is the following: Let $\mathcal{S}$ be a co-doubly commuting submodule of $H^2(\mathbb{D}^2)$. Then $\mbox{rank~} \mathcal{S} = 1$ if and only if $\mathcal{S} = ΦH^2(\mathbb{D}^2)$ for some one variable inner function $Φ\in H^\infty(\mathbb{D}^2)$. This answers a question posed by R. G. Douglas and R. Yang.
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.
We obtain a complete characterization for doubly commuting mixed invariant subspaces of the Hardy space over the unit polydisc. We say a closed subspace $\mathcal{Q}$ of $H^2(\mathbb{D}^n)$ is mixed invariant if $M_{z_{j}}(\mathcal{Q}) \subseteq \mathcal{Q}$ for $1 \leq j \leq k$ and $M_{z_{j}}^*(\mathcal{Q}) \subseteq \mathcal{Q}$, $k+1 \leq j \leq n$ for some integer $k \in \{1, 2, \ldots, n-1 \}$. We prove that a mixed invariant subspace $\mathcal{Q}$ of $H^2(\mathbb{D}^n)$ is doubly commuting if and only if \[ \mathcal{Q} = ΘH^2(\mathbb{D}^k) \otimes \mathcal{Q}_{θ_1} \otimes \cdots \otime…
The goal of the paper is to study the structure of the k-tuples of doubly $Λ$-commuting row isometries and the $C^*$-algebras they generate from the point of view of noncommutative multivariable operator theory. We obtain Wold decompositions, in this setting, and use them to classify the $k$-tuples of doubly $Λ$-commuting row isometries up to a unitary equivalence. We introduce a universal model in this setting, describe its invariant subspaces, and develop a dilation theory on $Λ$-polyballs.
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.
Sz.-Nagy and Foias proved that each $C_{\cdot0}$-contraction has a dilation to a Hardy shift and thus established an elegant analytic functional model for contractions of class $C_{\cdot0}$. This has motivated lots of further works on model theory and generalizations to commuting tuples of $C_{\cdot0}$-contractions. In this paper, we focus on doubly commuting sequences of $C_{\cdot0}$-contractions, and establish the dilation theory and the analytic model theory for these sequences of operators. These results are applied to generalize the Beurling-Lax theorem and Jordan blocks in the multivaria…
For a tuple of operators $\boldsymbol{A}= (A_1, \ldots, A_d)$, $\text{dist}(\boldsymbol{A}, \mathbb C^d \boldsymbol{I})$ is defined as $\min\limits_{\boldsymbol{z} \in \mathbb C^d} \|\boldsymbol{A-zI}\|$ and $\text{var}_x (\boldsymbol{A})$ as $\|\boldsymbol{A} x\|^2-\sum_{j=1}^d {\big|}\langle x| A_j x\rangle{\big|}^2.$ For a tuple $\boldsymbol{A}$ of commuting normal operators, it is known that $$\text{dist}(\boldsymbol{A}, \mathbb C^d \boldsymbol{I})^2=\sup_{\|x\|=1}\text{var}_x (\boldsymbol{A}).$$ We give an expression for the maximal joint numerical range of a tuple of doubly commuting mat…
This survey aims to give a brief introduction to operator theory in the Hardy space over the bidisc $H^2(\mathbb D^2)$. As an important component of multivariable operator theory, the theory in $H^2(\mathbb D^2)$ focuses primarily on two pairs of commuting operators that are naturally associated with invariant subspaces (or submodules) in $H^2(\mathbb D^2)$. Connection between operator-theoretic properties of the pairs and the structure of the invariant subspaces is the main subject. The theory in $H^2(\mathbb D^2)$ is motivated by and still tightly related to several other influential theorie…
