Properties of Beurling-Type Submodules via Agler Decompositions

TL;DR

This paper investigates the properties of Beurling-type submodules in the Hardy space over the bidisk using Agler decompositions, providing characterizations of operator commutators and reducing subspaces.

Contribution

It introduces new characterizations of operator properties on Beurling-type submodules via Agler decompositions, advancing understanding of multivariable operator theory.

Findings

Characterization of when commutators have finite rank

Conditions for subspaces to be reducing for shift operators

Open questions on operator-theoretic properties of submodules

Abstract

In this paper, we study operator-theoretic properties of the compressed shift operators $S_{z_1}$ and $S_{z_2}$ on complements of submodules of the Hardy space over the bidisk $H^2(\mathbb{D}^2)$. Specifically, we study Beurling-type submodules - namely submodules of the form $\theta H^2(\mathbb{D}^2)$ for $\theta$ inner - using properties of Agler decompositions of $\theta$ to deduce properties of $S_{z_1}$ and $S_{z_2}$ on model spaces $H^2(\mathbb{D}^2) \ominus \theta H^2(\mathbb{D}^2)$. Results include characterizations (in terms of $\theta$) of when a commutator $[S_{z_j}^*, S_{z_j}]$ has rank $n$ and when subspaces associated to Agler decompositions are reducing for $S_{z_1}$ and $S_{z_2}$. We include several open questions.