Jordan Blocks of H^2(D^n)

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 $H^2({\mathbb{D}}^n)$ ($n\ge 2$) is doubly commuting if and only if \[\clq = \clq_{Θ_1} \otimes \cdots \otimes \clq_{Θ_n},\]where each $\clq_{Θ_i}$ 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 $i=1,\dots ,n$. 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 \[\cls = \mathop{\sum}_{i=1}^m Θ_i H^2(\mathbb{D}^n),\]for some integer $m\le n$ and one variable inner functions $\{{\Theta }_i{\}}_{i=1}^m$.

14 pages. Revised. To appear in the Journal of Operator Theory