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 $A^2_\alpha,$ $\alpha>-1$ and the Hardy space $H^2$ on the unit disk. When the convergence of sequences $(\varphi_n)$ of symbols to a given symbol $\varphi$ implies the convergence of product operators $C_{\varphi_n}D^k$ is also studied. Finally, the boundedness and compactness of the operator $C_{\varphi}D^k: A^2_\alpha\to A^2_\alpha$ are characterized in terms of the generalized Nevanlinna counting function.