Fixed Point Theory and Applications
Volume 2007 (2007), Article ID 28750, 8 pages
doi:10.1155/2007/28750

Compact weighted composition operators and fixed points in convex domains

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.