Compact Weighted Composition Operators and Fixed Points in Convex Domains

We extend a classical result of Caughran/H. Schwartz and another recent result of Gunatillake by showing that if D is a bounded, convex domain in C, ψ : D → C is analytic and bounded away from zero toward the boundary of D, and φ : D → D is a holomorphic map such that the weighted composition operator Wψ,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, then φ has a unique fixed point in D. We apply this result by making a reasonable conjecture about the spectrum ofWψ,φ based on previous one-variable and multivariable results concerning compact weighted and unweighted composition operators.