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

The boundedness and compactness of weighted composition operators on the Hardy space 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 are considered, and the results of Harper and Smith, linking their properties to those of Carleson embeddings, are extended to this situation.