Schatten classes of integration operators on Dirichlet spaces

TL;DR

This paper characterizes when certain integral, multiplication, Hankel, and Toeplitz operators belong to Schatten classes on Dirichlet spaces, extending known results for Toeplitz operators.

Contribution

It provides a comprehensive description of Schatten class membership for various operators on Dirichlet spaces, including an extension of Luecking's theorem.

Findings

Characterization of Schatten class membership for integral operators on Dirichlet spaces

Extension of Luecking's result to Toeplitz operators

Analysis of multiplication, Hankel, and Toeplitz operators in this context

Abstract

We address the question of describing the membership to Schatten-Von Neumann ideals $\mathcal{S}_ p$ of integration operators $(T_ g f)(z)=\int_{0}^{z}f(\zeta)\,g'(\zeta)\,d\zeta$ acting on Dirichlet type spaces. We also study this problem for multiplication, Hankel and Toeplitz operators. In particular, we provide an extension of Luecking's result on Toeplitz operators.