Boundedness, Compactness and Schatten-class Membership of Weighted Composition Operators

Abstract

The boundedness and compactness of weighted composition operators on the Hardy space \({{\mathcal H}^2}\) of the unit disc is analysed. Particular reference is made to the case when the self-map of the disc is an inner function. Schatten-class membership is also considered; as a result, stronger forms of the two main results of a recent paper of Gunatillake are derived. Finally, weighted composition operators on weighted Bergman spaces \({\mathcal{A}^2_\alpha(\mathbb{D})}\) are considered, and the results of Harper and Smith, linking their properties to those of Carleson embeddings, are extended to this situation.