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A characterization of submodules via the Beurling-Lax-Halmos theorem


Authors: Yueshi Qin and Rongwei Yang
Journal: Proc. Amer. Math. Soc. 142 (2014), 3505-3510
MSC (2010): Primary 47A45; Secondary 47A10
DOI: https://doi.org/10.1090/S0002-9939-2014-12081-8
Published electronically: June 18, 2014
MathSciNet review: 3238425
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Abstract: Shift invariant subspaces in the vector-valued Hardy space $ H^2(E)$ play important roles in Nagy-Foias operator model theory. A theorem by Beurling, Lax and Halmos characterizes such invariant subspaces by operator-valued inner functions $ \Theta (z)$. When $ E=H^2(\mathbb{D})$, $ H^2(E)$ is the Hardy space over the bidisk $ H^2(\mathbb{D}^2)$. This paper shows that for some well-known examples of invariant subspaces in $ H^{2}({\mathbb{D}}^2)$, the function $ \Theta (z)$ turns out to be strikingly simple.


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Additional Information

Yueshi Qin
Affiliation: Department of Mathematics and Statistics, SUNY at Albany, Albany, New York 12222
Email: yqin@albany.edu

Rongwei Yang
Affiliation: Department of Mathematics and Statistics, SUNY at Albany, Albany, New York 12222
Email: ryang@albany.edu

DOI: https://doi.org/10.1090/S0002-9939-2014-12081-8
Keywords: Hardy space over the bidisk, operator inner function, submodule, spectrum
Received by editor(s): September 24, 2012
Received by editor(s) in revised form: October 19, 2012
Published electronically: June 18, 2014
Communicated by: Pamela B. Gorkin
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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