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arXiv · 1411.5759

Properties of Beurling-Type Submodules via Agler Decompositions

Kelly Bickel·Constanze Liaw
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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 $θH^2(\mathbb{D}^2)$ for $θ$ inner - using properties of Agler decompositions of $θ$ to deduce properties of $S_{z_1}$ and $S_{z_2}$ on model spaces $H^2(\mathbb{D}^2) \ominus θH^2(\mathbb{D}^2)$. Results include characterizations (in terms of $θ$) 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.

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Kelly Bickel, Constanze Liaw. 2014-11-21. Properties of Beurling-Type Submodules via Agler Decompositions. https://doi.org/10.1016/j.jfa.2016.10.007

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