Products of composition and differentiation operators on the weighted Bergman space

Abstract

Motivated by a recent paper by S. Ohno we calculate Hilbert-Schmidt norms of products of composition and differentiation operators on the Bergman space A2α, α > - 1 and the Hardy space H2 on the unit disk. When the convergence of sequences (φn) of symbols to a given symbol φ implies the convergence of product operators CφnDk is also studied. Finally, the boundedness and compactness of the operator CφDk : A2α → A2α are characterized in terms of the generalized Nevanlinna counting function.