Jordan Blocks of H^2(D^n)

TL;DR

This paper extends the concept of Jordan blocks to several variables in the Hardy space, providing a complete characterization of doubly commuting quotient modules and a Beurling-like theorem for co-doubly commuting submodules.

Contribution

It introduces a multivariable analog of Jordan blocks and characterizes doubly commuting quotient modules and co-doubly commuting submodules in Hardy spaces.

Findings

Doubly commuting quotient modules are tensor products of one-variable Jordan blocks or Hardy modules.

A Beurling-like theorem describes co-doubly commuting submodules as sums of inner function multiples.

Complete characterization of doubly commuting quotient modules in multivariable Hardy spaces.

Abstract

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 $H^2(\mathbb{D}^n)$. We prove that a quotient module $\clq$ of $H^2(\mathbb{D}^n)$ ($n \geq 2$) is doubly commuting if and only if \[\clq = \clq_{\Theta_1} \otimes \cdots \otimes \clq_{\Theta_n},\]where each $\clq_{\Theta_i}$ is either a one variable Jordan block $H^2(\mathbb{D})/\Theta_i H^2(\mathbb{D})$ for some inner function $\Theta_i$ or the Hardy module $H^2(\mathbb{D})$ on the unit disk for all $i = 1, \ldots, n$. We say that a submodule $\cls$ of $H^2(\mathbb{D}^n)$ is a co-doubly commuting if the quotient module $H^2(\mathbb{D}^n)/\cls$ is doubly commuting. We obtain a Beurling like theorem for the class of co-doubly commuting submodules of $H^2(\mathbb{D}^n)$. We prove that a submodule $\cls$ of $H^2(\mathbb{D}^n)$ is co-doubly commuting if and only if \[\cls = \mathop{\sum}_{i=1}^m \Theta_i H^2(\mathbb{D}^n),\]for some integer $m \leq n$ and one variable inner functions $\{\Theta_i\}_{i=1}^m$.