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Composition operators on uniform algebras, essential norms, and hyperbolically bounded sets

Author(s): P. Galindo; T. W. Gamelin; M. Lindström
Journal: Trans. Amer. Math. Soc. 359 (2007), 2109-2121.
MSC (2000): Primary 46J10; Secondary 47B38, 47B48
Posted: November 22, 2006
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Abstract: Let $ A$ be a uniform algebra, and let $ \phi$ be a self-map of the spectrum $ M_A$ of $ A$ that induces a composition operator $ C_\phi$ on $ A$. The object of this paper is to relate the notion of ``hyperbolic boundedness'' introduced by the authors in 2004 to the essential spectrum of $ C_\phi$. It is shown that the essential spectral radius of $ C_\phi$ is strictly less than $ 1$ if and only if the image of $ M_A$ under some iterate $ \phi^n$ of $ \phi$ is hyperbolically bounded. The set of composition operators is partitioned into ``hyperbolic vicinities" that are clopen with respect to the essential operator norm. This partition is related to the analogous partition with respect to the uniform operator norm.

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Additional Information:

P. Galindo
Affiliation: Departamento de Análisis Matemático, Universidad de Valencia, 46.100, Burjasot, Valencia, Spain
Email: galindo@uv.es

T. W. Gamelin
Affiliation: Department of Mathematics, University of California Los Angeles, Los Angeles, California 90095-1555
Email: twg@math.ucla.edu

M. Lindström
Affiliation: Department of Mathematics, Abo Akademi University, FIN-20500 Abo, Finland
Email: mlindstr@abo.fi

DOI: 10.1090/S0002-9947-06-04098-0
PII: S 0002-9947(06)04098-0
Keywords: Composition operator, hyperbolically bounded, Gleason part, essential norm
Received by editor(s): February 5, 2004
Received by editor(s) in revised form: February 17, 2005
Posted: November 22, 2006
Additional Notes: The first author was supported by Projects AE-2003-0392 (Universidad de Valencia) and BFM-FEDER 2003-07540 (DGI, Spain)
The second author was supported partially by the Academy of Finland Project 51096 and Project BFM-FEDER 2003-07540 (DGI, Spain)
Copyright of article: Copyright 2006, American Mathematical Society

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