Continuity with respect to symbols of composition operators on the weighted Bergman space

Abstract

Let α > -1, U be the open unit disk in C and denote by H(U) the set of all holomorphic functions on U. Let Cφ be a composition operator induced by an analytic self-map φ of U. Composition operators Cφ on the weighted Hilbert Bergman space A2α (U) = {f ∈ H(U) |∫U | (z)|2(1 - |z|2)αdm(z) < ∞} are considered. We investigate when convergence of sequences (φn) φ, implies the convergence of the induced composition operators. We give a necessary and sufficient condition for a sequence of Hilbert-Schmidt composition operators (Cφn) to converge in Hilbert-Schmidt norm to Cφ, and we obtain a sufficient condition for convergence in operator norm.