Fixed Point Theory and Applications
Volume 2007 (2007), Article ID 28750, 8 pages
doi:10.1155/2007/28750
Research Article
Compact Weighted Composition Operators and Fixed Points in Convex Domains
Department of Mathematics, University of California, Riverside 92521, CA, USA
Received 18 April 2007; Accepted 24 June 2007
Academic Editor: Fabio Zanolin
Copyright © 2007 Dana D. Clahane. 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.
Abstract
Let D be a bounded, convex domain in ℂn, and suppose
that φ:D→D is holomorphic. Assume that ψ:D→ℂ
is analytic, bounded away from zero toward the boundary of D,
and not identically zero on the fixed point set of D. Suppose also
that the weighted composition operator Wψ,φ given by Wψ,φ(f)=ψ(f∘φ) is compact on a holomorphic, functional Hilbert space
(containing the polynomial functions densely) on D
with reproducing
kernel K satisfying K(z,z)→∞ as z→∂D. 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 φ has a unique fixed point in D. We apply
this result by making a reasonable conjecture about the spectrum
of Wψ,φ based on previous results.