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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Invertible weighted composition operators

Author: Paul S. Bourdon
Journal: Proc. Amer. Math. Soc. 142 (2014), 289-299
MSC (2010): Primary 47B33, 30J99
Published electronically: October 3, 2013
MathSciNet review: 3119203
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Abstract: Let $ X$ be a set of analytic functions on the open unit disk $ \mathbb{D}$, and let $ \phi $ be an analytic function on $ \mathbb{D}$ such that $ \phi (\mathbb{D})\subseteq \mathbb{D}$ and $ f\mapsto f\circ \phi $ takes $ X$ into itself. We present conditions on $ X$ ensuring that if $ f\mapsto f\circ \phi $ is invertible on $ X$, then $ \phi $ is an automorphism of $ \mathbb{D}$, and we derive a similar result for mappings of the form $ f\mapsto \psi \cdot (f\circ \phi )$, where $ \psi $ is some analytic function on $ \mathbb{D}$. We obtain as corollaries of this purely function-theoretic work new results concerning invertibility of composition operators and weighted composition operators on Banach spaces of analytic functions such as $ S^p$ and the weighted Hardy spaces $ H^2(\beta )$.

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Paul S. Bourdon
Affiliation: Department of Mathematics, Washington and Lee University, Lexington, Virginia 24450

Received by editor(s): September 12, 2011
Received by editor(s) in revised form: March 11, 2012
Published electronically: October 3, 2013
Communicated by: Richard Rochberg
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.