Products of composition and differentiation operators on the weighted Bergman space

Motivated by a recent paper by S. Ohno we calculate Hilbert-Schmidt norms of products of composition and differentiation operators on the Bergman space A(alpha)(2), alpha > -1 and the Hardy space H-2 on the unit disk. When the convergence of sequences (phi(n)) of symbols to a given symbol phi implies the convergence of product operators C phi nDk is also studied. Finally, the boundedness and compactness of the operator C phi Dk : A(alpha)(2) -> A(alpha)(2) are characterized in terms of the generalized Nevanlinna counting function.