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Shift-invariant subspaces invariant for composition operators on the Hardy-Hilbert space


Authors: Carl C. Cowen and Rebecca G. Wahl
Journal: Proc. Amer. Math. Soc. 142 (2014), 4143-4154
MSC (2010): Primary 47B33; Secondary 47B38, 47A15
DOI: https://doi.org/10.1090/S0002-9939-2014-12132-0
Published electronically: July 29, 2014
MathSciNet review: 3266985
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Abstract: If $ \varphi $ is an analytic map of the unit disk $ \mathbb{D}$ into itself, the composition operator $ C_{\varphi }$ on a Hardy space $ H^{2}$ is defined by $ C_{\varphi }(f)=f\circ \varphi $. The unilateral shift on $ H^{2}$ is the operator of multiplication by $ z$. Beurling (1949) characterized the invariant subspaces for the shift. In this paper, we consider the shift-invariant subspaces that are invariant for composition operators. More specifically, necessary and sufficient conditions are provided for an atomic inner function with a single atom to be invariant for a composition operator, and the Blaschke product invariant subspaces for a composition operator are described. We show that if $ \varphi $ has Denjoy-Wolff point $ a$ on the unit circle, the atomic inner function subspaces with a single atom at $ a$ are invariant subspaces for the composition operator $ C_{\varphi }$.


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Additional Information

Carl C. Cowen
Affiliation: Department of Mathematical Sciences, IUPUI (Indiana University–Purdue University, Indianapolis), Indianapolis, Indiana 46202-3216
Email: ccowen@math.iupui.edu

Rebecca G. Wahl
Affiliation: Department of Mathematics, Butler University, Indianapolis, Indiana 46208-3485
Email: rwahl@butler.edu

DOI: https://doi.org/10.1090/S0002-9939-2014-12132-0
Keywords: Composition operator, shift-invariant subspace
Received by editor(s): March 26, 2012
Received by editor(s) in revised form: January 3, 2013
Published electronically: July 29, 2014
Communicated by: Richard Rochberg
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.