Composition operators on uniform algebras, essential norms, and hyperbolically bounded sets
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- by P. Galindo, T. W. Gamelin and M. Lindström PDF
- Trans. Amer. Math. Soc. 359 (2007), 2109-2121 Request permission
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.References
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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
- 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) - © Copyright 2006 American Mathematical Society
- 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
- MathSciNet review: 2276672