Normal complex symmetric weighted composition operators on the Hardy space

TL;DR

This paper characterizes when certain weighted composition operators on the Hardy space are normal and symmetric, providing conditions based on the properties of the symbol functions and their fixed points.

Contribution

It offers new criteria for the normality of symmetric weighted composition operators on the Hardy space, including cases with specific types of symbol functions.

Findings

Conditions for normality of symmetric weighted composition operators

Characterization when symbols have interior fixed points

Analysis of operators with hyperbolic or parabolic type symbols

Abstract

In this paper, we investigate the normal weighed composition operators $W_{\psi,\varphi}$ which is $\mathcal{J}-$symmetric, $\mathcal{C}_1-$symmetric and $\mathcal{C}_2-$symmetric on the Hardy space $H^2(\mathbb{D})$ respectively. Firstly, equivalent conditions of the normality of $\mathcal{C}_1-$symmetric and $\mathcal{C}_2-$symmetric weighted composition operators on $H^2(\mathbb{D})$ is given. Furthermore, the normal $\mathcal{J}-$symmetric, $\mathcal{C}_1-$symmetric and $\mathcal{C}_2-$symmetric weighted composition operators on $H^2(\mathbb{D})$ when $\varphi$ has an interior fixed point, $\varphi$ is of hyperbolic type or parabolic type are respectively investigated.