Compact and Hilbert-Schmidt weighted composition operators on weighted Bergman spaces

Abstract

Let u and φ be two analytic functions on the unit disk D such that φ(D)⊂D . A weighted composition operator uCφ induced by u and φ is defined on A2α , the weighted Bergman space of D, by uCφf:=u⋅f∘φ for every f∈A2α . We obtain sufficient conditions for the compactness of uCφ in terms of function-theoretic properties of u and φ . We also characterize when uCφ on A2α is Hilbert–Schmidt. In particular, the characterization is independent of α when φ is an automorphism of D. Furthermore, we investigate the Hilbert–Schmidt difference of two weighted composition operators on A2α.