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

Let u and be two analytic functions on the unit disk D such that. A weighted composition operator induced by u and is defined on, the weighted Bergman space of D, by for every. We obtain sufficient conditions for the compactness of in terms of function-theoretic properties of u and. We also characterize when on is Hilbert-Schmidt. In particular, the characterization is independent of when is an automorphism of D. Furthermore, we investigate the Hilbert-Schmidt difference of two weighted composition operators on.