Complex symmetric weighted composition operators on Bergman spaces and Lebesgue spaces

TL;DR

This paper characterizes complex symmetric weighted composition operators on Bergman spaces of a half-plane, identifying conditions for hermitian and unitary operators, and explores their relation to Lebesgue spaces.

Contribution

It provides a comprehensive classification of complex symmetric weighted composition operators on Bergman spaces, including conditions for hermitian and unitary cases, and links to Lebesgue spaces.

Findings

Boundedness of symmetric weighted composition operators

Classification of hermitian and unitary operators

Connection to complex symmetry in Lebesgue spaces

Abstract

In the paper, we investigate weighted composition operators on Bergman spaces of a half-plane. We characterize weighted composition operators which are hermitian and those which are complex symmetric with respect to a family of conjugations. As it turns out, weighted composition operators enhanced by a symmetry must be bounded. Hermitian, and unitary weighted composition operators are proven to be complex symmetric with respect to an adapted and highly relevant conjugation. We classify which the linear fractional functions give rise to the complex symmetry of bounded composition operators. We end the paper with a natural link to complex symmetry in Lebesgue space.