Properties of Beurling-type submodules via Agler decompositions

In this paper, we study operator-theoretic properties of the compressed shift operators Sz1 and Sz2 on complements of submodules of the Hardy space over the bidisk H2(D2). Specifically, we study Beurling-type submodules – namely submodules of the form θH2(D2) for θ inner – using properties of Agler...

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Bibliographic Details
Published in:Journal of functional analysis 2017-01, Vol.272 (1), p.83-111
Main Authors: Bickel, Kelly, Liaw, Constanze
Format: Article
Language:English
Subjects:
Agler decomposition
Compressed shifts
Essential normality
Two-variable model spaces
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Summary:In this paper, we study operator-theoretic properties of the compressed shift operators Sz1 and Sz2 on complements of submodules of the Hardy space over the bidisk H2(D2). Specifically, we study Beurling-type submodules – namely submodules of the form θH2(D2) for θ inner – using properties of Agler decompositions of θ to deduce properties of Sz1 and Sz2 on model spaces H2(D2)⊖θH2(D2). Results include characterizations (in terms of θ) of when a commutator [Szj⁎,Szj] has rank n and when subspaces associated to Agler decompositions are reducing for Sz1 and Sz2. We include several open questions.
ISSN:0022-1236
1096-0783
DOI:10.1016/j.jfa.2016.10.007