The identity is isolated among composition operators

Authors:
C-H. Chu, R. V. Hügli and M. Mackey

Journal:
Proc. Amer. Math. Soc. **132** (2004), 3305-3308

MSC (2000):
Primary 47B38, 46J15, 46G20, 32A10

DOI:
https://doi.org/10.1090/S0002-9939-04-07474-X

Published electronically:
May 21, 2004

MathSciNet review:
2073306

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be the Banach algebra of bounded holomorphic functions on the open unit ball of a Banach space. We show that the identity operator is an isolated point in the space of composition operators on . This answers a conjecture of Aron, Galindo and Lindström.

**1.**R. ARON, P. GALINDO AND M. LINDSTRÖM,*Connected components in the space of composition operators*, Integr. Equ. Oper. Theory,**45**(2003) 1-14. MR**2003m:47041****2.**E. BISHOP AND R. R. PHELPS,*A proof that every Banach space is subreflexive*, Bull. Amer. Math. Soc.,**67**(1961) 97-98.MR**23:A503****3.**F. F. BONSALL AND J. DUNCAN,*Numerical Ranges of Operators on Normed Spaces and of Elements of Normed Algebras*, Vol. 2, London Math. Soc. Lect. Note Ser., Cambridge Univ. Press (1971) Cambridge.MR**44:5779****4.**S. DINEEN,*The Schwarz Lemma*, Oxford Univ. Press (1989) Oxford.MR**91f:46064****5.**P. GALINDO AND M. LINDSTRÖM,*Factorization of homomorphisms through*, J. Math. Anal. Appl.**280**(2003) 375-386.**6.**B. MACCLUER, S. OHNO AND R. ZHAO,*Topological structure of the space of composition operators on*, Integr. Equ. Oper. Theory,**40**(2001) 481-494. MR**2002d:47039****7.**M. MACKEY AND P. MELLON,*A Schwarz lemma and composition operators*, Integr. Equ. Oper. Theory (to appear).

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Additional Information

**C-H. Chu**

Affiliation:
School of Mathematical Sciences, Queen Mary College, University of London, London E1 4NS, England

Email:
c.chu@qmul.ac.uk

**R. V. Hügli**

Affiliation:
Department of Mathematics, University of California, Irvine, California 92697

Email:
remo.huegli@stat.unibe.ch

**M. Mackey**

Affiliation:
Department of Mathematics, University College, Dublin 4, Ireland

Email:
michael.mackey@ucd.ie

DOI:
https://doi.org/10.1090/S0002-9939-04-07474-X

Keywords:
Composition operator,
holomorphic function on Banach space,
Carath\'eodory distance

Received by editor(s):
May 7, 2003

Received by editor(s) in revised form:
August 12, 2003

Published electronically:
May 21, 2004

Communicated by:
Joseph A. Ball

Article copyright:
© Copyright 2004
American Mathematical Society