Related papers: Submodules of the Hardy module over polydisc

We develop a several variables analog of the Jordan blocks of the Hardy space $H^2(\mathbb{D})$. In this consideration, we obtain a complete characterization of the doubly commuting quotient modules of the Hardy module $H^2(\mathbb{D}^n)$.…

In this paper, we investigate the structure of doubly commuting submodules and quotient modules of the Hardy space $H^2(\triangle_H)$ over the Hartogs triangle. We establish a complete classification of doubly commuting submodules. In…

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})…

In this note we establish a vector-valued version of Beurling's Theorem (the Lax-Halmos Theorem) for the polydisc. As an application of the main result, we provide necessary and sufficient conditions for the completion problem in…

Let $H^2(\mathbb{D}^n)$ denote the Hardy space over the polydisc $\mathbb{D}^n$, $n \geq 2$. A closed subspace $\mathcal{Q} \subseteq H^2(\mathbb{D}^n)$ is called Beurling quotient module if there exists an inner function $\theta \in…

We prove that the rank of a non-trivial co-doubly commuting submodule is $2$. More precisely, let $\varphi, \psi \in H^\infty(\mathbb{D})$ be two inner functions. If $\mathcal{Q}_{\varphi} = H^2(\mathbb{D})/ \varphi H^2(\mathbb{D})$ and…

Let $b(z) = \prod_{n=1}^\infty \frac{-\bar{\alpha}_n}{|\alpha_n|} \frac{z - \alpha_n}{1 - \bar{\alpha}_n z}$, where $\sum_{n=1}^\infty (1 - |\alpha_n|) <\infty$, be the Blaschke product with zeros at $\alpha_n \in \mathbb{D} \setminus…

Let $\mathbb{D}^n$ be the open unit polydisc in $\mathbb{C}^n$, $n \geq 1$, and let $H^2(\mathbb{D}^n)$ be the Hardy space over $\mathbb{D}^n$. For $n\ge 3$, we show that if $\theta \in H^\infty(\mathbb{D}^n)$ is an inner function, then the…

In this paper, we present a unified approach to problems of tensor product of quotient modules of Hilbert modules over $\mathbb{C}[z]$ and corresponding submodules of reproducing kernel Hilbert modules over $\mathbb{C}[z_1, \ldots, z_n]$…

Given any shift-invariant closed subspace $\mathcal{S}$ (aka submodule) of the Hardy space over the unit polydisc $H^2(\mathbb{D}^n)$ (where $n \geq 2$), let $R_{z_j}:=M_{z_j}|_{\mathcal{S}}$, and $E_{z_j}:=P_{\mathcal{S}}\circ ev_{z_j}$,…

An n-tuple (n \geq 2), T = (T_1, \ldots, T_n), of commuting bounded linear operators on a Hilbert space \mathcal{H} is doubly commuting if T_i T_j^* = T_j^* T_i for all $1 \leq i < j \leq n$. If in addition, each T_i \in C_{\cdot 0}, then…

Doubly commutativity of invariant subspaces of the Bergman space and the Dirichlet space over the unit polydisc $\mathbb{D}^n$ (with $ n \geq 2$) is investigated. We show that for any non-empty subset $\alpha=\{\alpha_1,\dots,\alpha_k\}$ of…

It is known that the structure of invariant subspaces of the Hardy space $H^2(\mathbb D^n)$ on the polydisc $\mathbb{D}^n$ is very complicated; hence, we need good examples help us to understand the structure of invariant subspaces of…

In this article, we prove that any pair of doubly commuting $2$-isometries on a Hilbert space has a Wold-type decomposition. Moreover, the analytic part of the pair is unitary equivalent to the pair of multiplication by coordinate function…

Recent work in Dynamical Sampling has been centered on characterizing frames obtained by the orbit of a vector under a bounded operator. We prove a necessary and sufficient condition for a pair of bounded commuting operators on a separable…

In this paper, we study operator-theoretic properties of the compressed shift operators $S_{z_1}$ and $S_{z_2}$ on complements of submodules of the Hardy space over the bidisk $H^2(\mathbb{D}^2)$. Specifically, we study Beurling-type…

We give a complete characterization of invariant subspaces for $(M_{z_1}, \ldots, M_{z_n})$ on the Hardy space $H^2(\mathbb{D}^n)$ over the unit polydisc $\mathbb{D}^n$ in $\mathbb{C}^n$, $n >1$. In particular, this yields a complete set of…

Let $\mathcal{E}$ be a Hilbert space and $H^2_{\mathcal{E}}(\mathbb{D})$ be the $\mathcal{E}$-valued Hardy space over the unit disc $\mathbb{D}$ in $\mathbb{C}$. The well known Beurling-Lax-Halmos theorem states that every shift invariant…

A closed subspace $\mathcal{M}$ of the Hardy space $H^2(\mathbb{D}^2)$ over the bidisk is called a submodule if it is invariant under multiplication by coordinate functions $z_1$ and $z_2$. Whether every finitely generated submodule is…

The Hardy space on the unit ball in C^n provides examples of a quasi-free, finite rank Hilbert module which contains a pure submodule isometrically isomorphic to the module itself. For n=1 the submodule has finite codimension. In this note…