Weighted and Compact Composition Operators of Complex Symmetric with Hilbert-Schmidt Norm Topologies and Symbol of Universal Covering Map

Abstract

We study complex symmetric weighted composition operators on the Hardy space .We provide some characterizations when a weighted composition operators is complex symmetric. We investigate which combinations of weights and maps of the open unit disk give rise to complex symmetric weighted composition operators with a special conjugation .We also investigate the topological structure of the space of weighted composition operators boundely acting between two Hilbert spaces of analytic functions on the open unit disk, satisfying some natural hypotheses. We study composition operators, acting on the Hardy spaces that have symbol and a universal covering map of the disk onto a finitely connected domain with distinct points in the interior of simply connected.