Submodules of the Hardy module over polydisc

Abstract

We say that a submodule $\cls$ of $H^2({\mathbb{D}}^n)$ ($n>1$) is co-doubly commuting if the quotient module $H^2(\mathbb{D}^n)/\cls$ is doubly commuting. We show that a co-doubly commuting submodule of $H^2({\mathbb{D}}^n)$ is essentially doubly commuting if and only if the corresponding one variable inner functions are finite Blaschke products or that $n=2$. In particular, a co-doubly commuting submodule $\cls$ of $H^2({\mathbb{D}}^n)$ is essentially doubly commuting if and only if $n=2$ or that $\cls$ is of finite co-dimension. We obtain an explicit representation of the Beurling-Lax-Halmos inner functions for those submodules of $H_{H^2({\mathbb{D}}^{n-1})}^2(\mathbb{D})$ which are co-doubly commuting submodules of $H^2({\mathbb{D}}^n)$. Finally, we prove that a pair of co-doubly commuting submodules of $H^2({\mathbb{D}}^n)$ are unitarily equivalent if and only if they are equal.

Revised. 15 pages. To appear in Israel Journal of Mathematics

References in corpus (3)

• p-Summable Commutators in Dimension d
• Essentially Reductive Hilbert Modules II
• Jordan Blocks of H^2(D^n)

Cited by in corpus (2)

• Jordan Blocks of H^2(D^n)
• On Quotient modules of $H^2({\mathbb{D}}^n)$: Essential Normality and Boundary Representations