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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