Open Access

Compact Weighted Composition Operators and Fixed Points in Convex Domains

Fixed Point Theory and Applications20072007:028750

DOI: 10.1155/2007/28750

Received: 18 April 2007

Accepted: 24 June 2007

Published: 17 October 2007

Abstract

Let be a bounded, convex domain in , and suppose that is holomorphic. Assume that is analytic, bounded away from zero toward the boundary of , and not identically zero on the fixed point set of . Suppose also that the weighted composition operator given by is compact on a holomorphic, functional Hilbert space (containing the polynomial functions densely) on with reproducing kernel satisfying as . We extend the results of J. Caughran/H. Schwartz for unweighted composition operators on the Hardy space of the unit disk and B. MacCluer on the ball by showing that has a unique fixed point in . We apply this result by making a reasonable conjecture about the spectrum of based on previous results.

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Authors’ Affiliations

(1)
Department of Mathematics, University of California

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Copyright

© Dana D. Clahane. 2007

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.