The identity is isolated among composition operators
HTML articles powered by AMS MathViewer
- by C-H. Chu, R. V. Hügli and M. Mackey
- Proc. Amer. Math. Soc. 132 (2004), 3305-3308
- DOI: https://doi.org/10.1090/S0002-9939-04-07474-X
- Published electronically: May 21, 2004
- PDF | Request permission
Abstract:
Let $H^\infty (B)$ be the Banach algebra of bounded holomorphic functions on the open unit ball $B$ of a Banach space. We show that the identity operator is an isolated point in the space of composition operators on $H^\infty (B)$. This answers a conjecture of Aron, Galindo and Lindström.References
- Richard Aron, Pablo Galindo, and Mikael Lindström, Connected components in the space of composition operators in $H^\infty$ functions of many variables, Integral Equations Operator Theory 45 (2003), no. 1, 1–14. MR 1952340, DOI 10.1007/BF02789591
- Errett Bishop and R. R. Phelps, A proof that every Banach space is subreflexive, Bull. Amer. Math. Soc. 67 (1961), 97–98. MR 123174, DOI 10.1090/S0002-9904-1961-10514-4
- F. F. Bonsall and J. Duncan, Numerical ranges of operators on normed spaces and of elements of normed algebras, London Mathematical Society Lecture Note Series, vol. 2, Cambridge University Press, London-New York, 1971. MR 0288583
- Seán Dineen, The Schwarz lemma, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1989. Oxford Science Publications. MR 1033739
- P. Galindo and M. Lindström, Factorization of homomorphisms through $H^\infty (D)$, J. Math. Anal. Appl. 280 (2003) 375–386.
- Barbara MacCluer, Shûichi Ohno, and Ruhan Zhao, Topological structure of the space of composition operators on $H^\infty$, Integral Equations Operator Theory 40 (2001), no. 4, 481–494. MR 1839472, DOI 10.1007/BF01198142
- M. Mackey and P. Mellon, A Schwarz lemma and composition operators, Integr. Equ. Oper. Theory (to appear).
Bibliographic Information
- C-H. Chu
- Affiliation: School of Mathematical Sciences, Queen Mary College, University of London, London E1 4NS, England
- MR Author ID: 199837
- 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
- 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
- © Copyright 2004 American Mathematical Society
- 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
- MathSciNet review: 2073306