Abstract
In this paper we study complex symmetric weighted composition operators on the Hardy space. We provide some characterizations of ψ and φ when a weighted composition operator Wψ,φ is complex symmetric. We investigate which combinations of weights ψ and maps of the open unit disk φ give rise to complex symmetric weighted composition operators with a special conjugation. As some applications, we obtain several examples for nonnormal complex symmetric operators. In addition, we give spectral properties of complex symmetric weighted composition operators. We examine eigenvalues and eigenvectors of such operators and find some conditions for which a complex symmetric weighted composition operator is Hilbert–Schmidt. Finally, we consider cyclicity, hypercyclicity, and the single-valued extension property for complex symmetric weighted composition operators.
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