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.