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The Carathéodory–Cartan–Kaup–Wu Theorem on an Infinite-Dimensional Hilbert Space

Published online by Cambridge University Press:  11 January 2016

Joseph A. Cima ,
Ian Graham ,
Kang Tae Kim  and
Steven G. Krantz
Joseph A. Cima
Affiliation:
Department of Mathematics, University of North Carolina, Chapel Hill, North Carolina 27514, USA, cima@math.unc.edu
Ian Graham
Affiliation:
Department of Mathematics, University of Toronto, Toronto, CANADA M5S 3G3, graham@math.toronto.edu
Kang Tae Kim
Affiliation:
Department of Mathematics, Pohang University of Science and Technology, Pohang 790-784, Koreakimkt@postech.ac.kr
Steven G. Krantz
Affiliation:
Department of Mathematics, Campus Box 1146 Washington University in St. Louis St. Louis, Missouri 63130, USAsk@math.wustl.edu
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Abstract

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This paper treats a holomorphic self-mapping f: Ω → Ω of a bounded domain Ω in a separable Hilbert space with a fixed point p. In case the domain is convex, we prove an infinite-dimensional version of the Cartan-Carathéodory-Kaup-Wu Theorem. This is basically a rigidity result in the vein of the uniqueness part of the classical Schwarz lemma. The main technique, inspired by an old idea of H. Cartan, is iteration of the mapping f and its derivative. A normality result for holomorphic mappings in the compact-weak-open topology, due to Kim and Krantz, is used.

Keywords

46G2032H50
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 185 , 2007 , pp. 17 - 30
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2007

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