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

  title={The Carath{\'e}odory–Cartan–Kaup–Wu Theorem on an Infinite-Dimensional Hilbert Space},
  author={Joseph A. Cima and Ian R. Graham and Kang-Tae Kim and Steven G. Krantz},
  journal={Nagoya Mathematical Journal},
  pages={17 - 30}
Abstract 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… 
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