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2007 The Carath\'{e}odory--Cartan--Kaup--Wu theorem on an infinite-dimensional Hilbert space
Joseph A. Cima, Ian Graham, Kang Tae Kim, Steven G. Krantz
Nagoya Math. J. 185: 17-30 (2007).


This paper treats a holomorphic self-mapping $f: \Omega \rightarrow \Omega$ of a bounded domain $\Omega$ in a separable Hilbert space ${\cal H}$ with a fixed point $p$. In case the domain is convex, we prove an infinite-dimensional version of the Cartan-Carath\'eodory-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.


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Joseph A. Cima. Ian Graham. Kang Tae Kim. Steven G. Krantz. "The Carath\'{e}odory--Cartan--Kaup--Wu theorem on an infinite-dimensional Hilbert space." Nagoya Math. J. 185 17 - 30, 2007.


Published: 2007
First available in Project Euclid: 23 March 2007

zbMATH: 1143.46024
MathSciNet: MR2301456

Primary: 46G20
Secondary: 32H50

Rights: Copyright © 2007 Editorial Board, Nagoya Mathematical Journal


Vol.185 • 2007
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