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

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 $H^2({\mathbb{B}}_N)$.

18 pages