Composition-Differentiation Operator on Weighted Bergman Spaces

Abstract

In this paper, we study the complex symmetry of weighted composition-differentiation operator $D_{n,\psi ,\varphi }$ on weighted Bergman spaces ${\mathcal{A}}_{\alpha }^2$ with respect to the conjugation $C_{\mu ,\eta }$ for $\mu ,\eta \in \{z\in \mathbb{C}:|z|=1\}$. We obtain explicit conditions for which the operator $D_{n,\psi ,\varphi }$ is Hermitian and normal. We also characterize the complex symmetric weighted composition-differentiation operator for derivative Hardy spaces.

14 pages