Composition-Differentiation Operator on Weighted Bergman Spaces

TL;DR

This paper investigates the complex symmetry properties of weighted composition-differentiation operators on weighted Bergman and Hardy spaces, providing explicit conditions for Hermitian, normal, and complex symmetric cases.

Contribution

It offers new explicit criteria for when these operators are Hermitian, normal, or complex symmetric on specific function spaces.

Findings

Operators are Hermitian under certain conditions.

Operators are normal when specific criteria are met.

Characterization of complex symmetry on derivative Hardy 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.