Boundedness, compactness and Schatten-class membership of weighted composition operators

TL;DR

This paper investigates the boundedness, compactness, and Schatten-class membership of weighted composition operators on Hardy and Bergman spaces, extending recent results and analyzing specific cases like inner functions.

Contribution

It provides new criteria for boundedness, compactness, and Schatten-class membership of weighted composition operators, including extensions to weighted Bergman spaces and inner functions.

Findings

Characterization of bounded and compact weighted composition operators on Hardy spaces.

Derivation of conditions for Schatten-class membership, strengthening previous results.

Extension of properties of weighted composition operators to weighted Bergman spaces.

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.