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

Published:2017-07-20

• Yanni Chen,
School of Mathematics and Information Science, Shaanxi Normal University, Xi'an, 710119, China
Department of Mathematics, University of New Hampshire, Durham, NH 03824, USA
• Zhe Liu,
Department of Mathematics, University of Central Florida, Orlando, FL 32816, USA
• Eric Nordgren,
Department of Mathematics, University of New Hampshire, Durham, NH 03824, USA
 Format: LaTeX MathJax PDF

## Abstract

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: Beurling theorem, invariant subspace, generalized Hardy space, gauge norm, multiply connected domain, Forelli projection, inner-outer factorization, affiliated operator
 MSC Classifications: 47L10 - Algebras of operators on Banach spaces and other topological linear spaces 30H10 - Hardy spaces

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