Level sets and Composition operators on the Dirichlet space

TL;DR

This paper investigates the properties of composition operators on the Dirichlet space, establishing criteria for their boundedness, compactness, and Hilbert-Schmidt class membership, some of which are proven to be optimal.

Contribution

It provides new criteria for boundedness, compactness, and Hilbert-Schmidt membership of composition operators on the Dirichlet space, with some criteria proven to be optimal.

Findings

Criteria for boundedness of composition operators

Criteria for compactness of composition operators

Criteria for Hilbert-Schmidt class membership

Abstract

We consider composition operators in the Dirichlet space of the unit disc in the plane. Various criteria on boundedness, compactness and Hilbert-Schmidt class membership are established. Some of these criteria are shown to be optimal.