Compact and Hilbert-Schmidt Differences of Weighted Composition Operators

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 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.