Composition-Differentiation Operator on Weighted Bergman Spaces

Abstract: In this paper, we study the complex symmetry of weighted composition-differentiation operator $D_{n, \psi, \phi}$ on weighted Bergman spaces $\mathcal{A}^2_{\alpha}$ 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 $D_{n, \psi, \phi}$ is Hermitian and normal. We also characterize the complex symmetric weighted composition-differentiation operator for derivative Hardy spaces.