Generalized weighted composition operators on weighted Hardy spaces

TL;DR

This paper explores the complex symmetric properties of generalized weighted composition operators on weighted Hardy spaces, providing explicit conditions for symmetry, Hermitian, and normal operators, advancing understanding in operator theory.

Contribution

It introduces explicit criteria for complex symmetry, Hermitian, and normality of generalized weighted composition operators on weighted Hardy spaces, which is a novel contribution.

Findings

Explicit conditions for complex symmetry with conjugation J_w.

Necessary and sufficient conditions for Hermitian operators.

Criteria for normality of the operators.

Abstract

In this paper, we investigate the complex symmetric structure of generalized weighted composition operators $D_{m,\psi,\varphi}$ on the weighted Hardy space $H^2(\beta)$. We obtain explicit conditions for $ D_{m,\psi,\varphi}$ to be complex symmetric with the conjugation $J_w$. Under the assumption that $ D_{m,\psi,\varphi}$ is $J_w$-symmetric, some sufficient and necessary conditions for $D_{m,\psi,\varphi}$ to be Hermitian and normal are given.