Complex symmetric generalized weighted composition operators on Hilbert spaces of analytic functions

Abstract

We investigate complex symmetry of a generalized weighted composition operator Dn,v,ψ on the reproducing kernel Hilbert space Hγ of analytic functions on the unit disk D. We first characterize a class of anti-linear weighted composition operators that are conjugations with a new approach. Then we obtain necessary and sufficient conditions for Dn,v,ψ to be complex symmetric with respect to these conjugations. Our results not only generalize and unify the ones in the literature, but also provide an affirmative answer to a question from a paper of Lim and Khoi (2018) [16]. The classes of self-adjoint, normal, unitary generalized weighted composition operators on Hγ and their relations with the property of complex symmetry are also studied.