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 $\{\Theta_i\}_{i=1}^m$ .

DOI: http://dx.doi.org/10.7900/jot.2013mar16.1980
Keywords: Hilbert modules, Jordan blocks, doubly commuting quotient modules, Beurling's theorem, invariant subspaces

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