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A Beurling Theorem for Generalized Hardy Spaces on a Multiply Connected Domain

Published online by Cambridge University Press:  20 November 2018

Yanni Chen ,
Don Hadwin ,
Zhe Liu  and
Eric Nordgren
Yanni Chen
Affiliation:
School of Mathematics and Information Science, Shaanxi Normal University, Xi'an, 710119, China email: yanni.chen@snnu.edu.cn
Don Hadwin
Affiliation:
Department of Mathematics, University of New Hampshire, Durham, NH 03824, USA email: don@unh.edu, ean@unh.edu
Zhe Liu
Affiliation:
Department of Mathematics, University of Central Florida, Orlando, FL 32816, USA email: zhe.liu@ucf.edu
Eric Nordgren
Affiliation:
Department of Mathematics, University of New Hampshire, Durham, NH 03824, USA email: don@unh.edu, ean@unh.edu
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Abstract

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The object of this paper is to prove a version of the Beurling–Helson–Lowdenslager invariant subspace theorem for operators on certain Banach spaces of functions on a multiply connected domain in $\mathbb{C}$. The norms for these spaces are either the usual Lebesgue and Hardy space norms or certain continuous gauge norms. In the Hardy space case the expected corollaries include the characterization of the cyclic vectors as the outer functions in this context, a demonstration that the set of analytic multiplication operators is maximal abelian and reflexive, and a determination of the closed operators that commute with all analytic multiplication operators.

Keywords

47L1030H10Beurling theoreminvariant subspacegeneralized Hardy spacegauge normmultiply connected domainForelli projectioninner-outer factorizationaffiliated operator
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 70 , Issue 3 , 01 June 2018 , pp. 515 - 537
Copyright
Copyright © Canadian Mathematical Society 2018

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