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.