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Journal of Operator Theory

Volume 72, Issue 2, Fall 2014  pp. 371-385.

Jordan blocks of $H^2(\mathbb{D}^n)$

Authors:  Jaydeb Sarkar
Author institution: Indian Statistical Institute, Statistics and Mathematics Unit, 8th Mile, Mysore Road, Bangalore, 560059, India

Summary:  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 $H^2(\mathbb{D}^n)$ ($n \geqslant 2$) 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 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 $m \leqslant n$ and one variable inner functions $\{\Theta_i\}_{i=1}^m$.

Keywords:  Hilbert modules, Jordan blocks, doubly commuting quotient modules, Beurling's theorem, invariant subspaces

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