Compact composition operators on the Dirichlet space and capacity of sets of contact points

TL;DR

This paper investigates the properties of composition operators on the Dirichlet space, focusing on their compactness, Schatten class membership, and the capacity of contact point sets, revealing nuanced distinctions between different function spaces.

Contribution

It introduces new constructions of composition operators with specific contact point sets and explores their membership in Schatten classes, advancing understanding of operator theory on the Dirichlet space.

Findings

Existence of Schur functions with contact sets of logarithmic capacity zero

Every bounded composition operator on the Dirichlet space has contact sets of capacity zero

Compact operators on the Dirichlet space are also compact on the Gaussian Hardy-Orlicz space

Abstract

In this paper, we prove that for every compact set of the unit disk of logarithmic capacity 0, there exists a Schur function both in the disk algebra and in the Dirichlet space such that the associated composition operator is in all Schatten classes (of the Dirichlet space), and for which the set of points whose image touches the unit circle is equal to this compact set. We show that for every bounded composition operator on the Dirichlet space and for every point of the unit circle, the logarithmic capacity of the set of point having this point as image is 0. We show that every compact composition operator on the Dirichlet space is compact on the gaussian Hardy-Orlicz space; in particular, it is in every Schatten class on the usual Hilbertian Hardy space. On the other hand, there exists a Schur function such that the associated composition operator is compact on the gaussian Hardy-Orlicz space, but which is not even bounded on the Dirichlet space. We prove that the Schatten classes on the Dirichlet space can be separated by composition operators. Also, there exists a Schur function such that the associated composition operator is compact on the Dirichlet space, but in no Schatten class.