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The Caratheodory-Cartan-Kaup-Wu theorem on an infinite dimensional Hilbert space

Research paper by Joseph Cima, Ian Graham, Kang-Tae Kim, Steven G. Krantz

Indexed on: 21 Feb '07Published on: 21 Feb '07Published in: Mathematics - Complex Variables



Abstract

This paper treats a holomorphic self-mapping f: Omega --> Omega of a bounded domain Omega in a separable Hilbert space H with a fixed point p. In case the domain is convex, we prove an infinite-dimensional version of the Cartan-Caratheodory-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.