Weighted composition operators between Hilbert spaces of analytic functions in the operator norm and Hilbert–Schmidt norm topologies

We here investigate the topological structure of the space of weighted composition operators boundedly acting between two Hilbert spaces of analytic functions on the open unit disk, satisfying some natural hypotheses. We will consider the operator norm topology and the Hilbert–Schmidt norm topology respectively. These results will be involved in the investigation for the explicit cases of the classical Hardy–Hilbert space, the weighted Bergman spaces and the Dirichlet space. Furthermore we will estimate the Hilbert–Schmidt norms of difference of two composition operators acting from the Dirichlet space to the Hardy and the weighted Bergman spaces.