Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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
Published electronically: November 22, 2006
MathSciNet review: 2276672
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [AGL] R. Aron, P. Galindo, and M. Lindström, Connected components in the space of composition operators on analytic functions of many variables, Int. Eq. Op. Theory 45 (2003), 1-14. MR 1952340 (2003m:47041)
  • [CHM] C. H. Chu, R. V. Hügli, and M. Mackey, The identity is an isolated composition operator in $ H^\infty(B),$ Proc. Amer. Math. Soc. 132 (2004), no. 11, 3305-3308 (electronic). MR 2073306 (2005b:46091)
  • [Da] A. Davie, Linear extension operators for spaces and algebras of functions, Amer. J. Math. 94 (1972), 156-172. MR 0300093 (45:9141)
  • [GGL] P. Galindo, T. W. Gamelin, and M. Lindström, Composition operators on uniform algebras and the pseudohyperbolic metric, J. Korean Math. Soc. 41 (2004), 1-20. MR 2048697 (2004m:47047)
  • [GL] P. Galindo and M. Lindström, Factorization of homomorphisms through $ H^\infty(D)$, J. of Math. Anal. and Appl. 280 (2003), 375-386. MR 1977918 (2004g:46067)
  • [Ga1] T. Gamelin, Uniform Algebras, second edition, AMS Chelsea Publishing, 1984. MR 0410387 (53:14137)
  • [Ga2] T. Gamelin, Embedding Riemann surfaces in maximal ideal spaces, J. Functional Anal. 2 (1968), 123-146. MR 0223894 (36:6941)
  • [Ga3] T. Gamelin, Uniform algebras on plane sets, in Approximation Theory, edited by G.G. Lorentz, Academic Press, 1973, 101-149. MR 0338784 (49:3548)
  • [Ga4] T. Gamelin, Homomorphisms of uniform algebras, in Recent Progress in Functional Analysis, edited by Bierstedt et al., Elsevier/North-Holland, 2001, 95-105. MR 1861749 (2002h:46084)
  • [GM] P. Gorkin and R. Mortini, Norms and essential norms of linear combinations of endomorphisms, Trans. Amer. Math. Soc. 358 (2006), 553-571. MR 2177030
  • [HIZ] T. Hosokava, K. Izuchi, and D. Zheng, Isolated points and essential components of composition operators on $ H^\infty$ functions, Proc. Amer. Math. Soc. 130 (2002), 1765-1773. MR 1887024 (2003d:47033)
  • [Ka] H. Kamowitz, Compact endomorphisms of Banach algebras, Pac. J. Math. 89 (1980), 313-325. MR 0599123 (82c:46063)
  • [Kl] U. Klein, Kompakte multiplikative Operatoren auf uniformen Algebren, Mitt. Math. Sem. Giessen No. 232, 1997, iv + 120 pp. MR 1479364 (99b:47043)
  • [Ko] H. König, Zur abstrakten theorie der analytischen funktionen II, Math. Ann. 163 (1966), 9-17. MR 0190792 (32:8202)
  • [Zh] L. Zheng, The essential norms and spectra of composition operators on $ H^\infty$, Pacific J. Math. 203 (2002), 503-510. MR 1897912 (2003e:47052)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 46J10, 47B38, 47B48

Retrieve articles in all journals with MSC (2000): 46J10, 47B38, 47B48

Additional Information

P. Galindo
Affiliation: Departamento de Análisis Matemático, Universidad de Valencia, 46.100, Burjasot, Valencia, Spain

T. W. Gamelin
Affiliation: Department of Mathematics, University of California Los Angeles, Los Angeles, California 90095-1555

M. Lindström
Affiliation: Department of Mathematics, Abo Akademi University, FIN-20500 Abo, Finland

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

American Mathematical Society