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

Authors:
P. Galindo, T. W. Gamelin and M. Lindström

Journal:
Trans. Amer. Math. Soc. **359** (2007), 2109-2121

MSC (2000):
Primary 46J10; Secondary 47B38, 47B48

DOI:
https://doi.org/10.1090/S0002-9947-06-04098-0

Published electronically:
November 22, 2006

MathSciNet review:
2276672

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be a uniform algebra, and let be a self-map of the spectrum of that induces a composition operator on . The object of this paper is to relate the notion of ``hyperbolic boundedness'' introduced by the authors in 2004 to the essential spectrum of . It is shown that the essential spectral radius of is strictly less than if and only if the image of under some iterate of 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:
https://doi.org/10.1090/S0002-9947-06-04098-0

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

Published electronically:
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)

Article copyright:
© Copyright 2006
American Mathematical Society