Complex symmetric weighted composition operators in several variables

Abstract

In this paper, we study the complex symmetric weighted composition operators on the Hilbert space Hs of analytic functions over the open unit ball with reproducing kernels Kws(z)=(1−〈z,w〉)−s, where s∈N+. We completely characterize complex symmetric weighted composition operators with respect to two important conjugations JV and Jσ, which include Hermitian weighted composition operators and normal weighted composition operators appeared in [17]. Surprisingly, they are essential representatives of general conjugations of the form Cψ,φJ, where Cψ,φ is unitary and J-symmetric. Indeed, these complex symmetric weighted composition operators can only be divided into two classes. The size estimates and some other properties of these complex symmetric weighted composition operators are obtained. Moreover, involutive weighted composition operators are completely characterized too. As an application, we construct concrete conjugations for the involutive composition operators, which solves a problem raised by Garcia and Hammond in a general case.