# Complex symmetric weighted composition operators on Dirichlet spaces and Hardy spaces in the unit ball

## TL;DR

This paper characterizes when weighted composition operators are complex symmetric, Hermitian, or unitary on Dirichlet and Hardy spaces in the unit ball, providing new examples and discussing their normality.

## Contribution

It offers new characterizations and conditions for complex symmetry, Hermitian, and unitary properties of weighted composition operators on these spaces.

## Key findings

- Characterization of complex symmetric weighted composition operators on Dirichlet spaces.
- Necessary and sufficient conditions for unitarity and Hermitian properties on Hardy spaces.
- Examples of complex symmetric weighted composition operators and their normality.

## Abstract

In this paper, we investigate when weighted composition operators acting on Dirichlet spaces $\mathcal{D}(\mathbb{B}_{N})$ are complex symmetric with respect to some special conjugations, and provide some characterizations of Hermitian weighted composition operators on $\mathcal{D}(\mathbb{B}_{N})$. Furthermore, we give a sufficient and necessary condition for $J$-symmetric weighted composition operators on Hardy spaces $H^2(\mathbb{B}_{N})$ to be unitary or Hermitian, then some new examples of complex symmetric weighted composition operators on $H^2(\mathbb{B}_{N})$ are obtained. We also discuss the normality of complex symmetric weighted composition operators on $H^2(\mathbb{B}_{N})$.