Compact and Hilbert–Schmidt Differences of Weighted Composition Operators

Abstract

In this paper, we first obtain a characterization of compact difference of two weighted composition operators acting between the standard weighted Bergman spaces, under certain restrictions on the weights. We also calculate (upto equivalence) the Hilbert–Schmidt norm of a difference of two weighted composition operators acting from a Bergman space or Hardy space to an \(L^{2}(\mu )\) space. This result is followed by a few corollaries involving certain particular types of weights. We also investigate conditions for two weighted composition operators to lie on the same path component under the Hilbert–Schmidt norm topology.