Hankel operators and the Dixmier trace on the Hardy space

TL;DR

This paper characterizes Hankel operators on the Hardy space that belong to the Dixmier class, providing estimates for their Dixmier trace and revealing differences from the Bergman space setting, including non-measurable operators.

Contribution

It establishes criteria for Dixmier class membership of Hankel operators on the Hardy space and explores their trace properties, highlighting novel distinctions from Bergman space operators.

Findings

Existence of non-measurable Dixmier-class Hankel operators.

Dixmier-class Hankel operators not always in the $(1, abla)$ Schatten-Lorentz ideal.

Criteria for membership in the Dixmier class on Hardy space.

Abstract

We give criteria for the membership of Hankel operators on the Hardy space on the disc in the Dixmier class, and establish estimates for their Dixmier trace. In contrast to the situation in the Bergman space setting, it turns out that there exist Dixmier-class Hankel operators which are not measurable (i.e. their Dixmier trace depends on the choice of the underlying Banach limit), as well as Dixmier-class Hankel operators which do not belong to the $(1,\infty)$ Schatten-Lorentz ideal. A related question concerning logarithmic interpolation of Besov spaces is also discussed.