Journal of the
Korean Mathematical Society
JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

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J. Korean Math. Soc. 2004; 41(1): 175-192

Printed January 1, 2004

Copyright © The Korean Mathematical Society.

Fixed point theorems for infinite dimensional holomorphic functions

Lawrence A. Harris

University of Kentucky

Abstract

This talk discusses conditions on the numerical range of a holomorphic function defined on a bounded convex domain in a complex Banach space that imply that the function has a unique fixed point. In particular, extensions of the Earle-Hamilton Theorem are given for such domains. The theorems are applied to obtain a quantitative version of the inverse function theorem for holomorphic functions and a distortion form of Cartan's uniqueness theorem.

Keywords: Banach space, Frechet derivative, convex domain, holomorphic numerical range, Bloch radii, Cartan uniqueness theorem

MSC numbers: 46G20, 46T25, 47H10, 47J07