The Helton-Howe measure of almost normal Toeplitz operators

TL;DR

This paper investigates the Helton-Howe measure for almost normal Toeplitz operators, providing a trace formula that relates to harmonic extensions of symbols, generalizing the Fredholm index winding number formula.

Contribution

It introduces a method to compute Helton-Howe measures for almost normal Toeplitz operators using harmonic extensions, extending previous index formulas.

Findings

Helton-Howe measure for these operators is expressed via harmonic extensions.

A generalization of the winding number formula for Fredholm index is established.

Trace formulas for commutators of almost normal Toeplitz operators are derived.

Abstract

The Helton-Howe measure associated with an almost normal operator was constructed by Helton and Howe. It provides a trace formula that allows us to calculate the trace of commutators that would otherwise be incalculable. We will investigate almost normal Toeplitz operators and determine their Helton-Howe measures in terms of the harmonic extensions of the symbols. Our result may be thought of as a generalization of the winding number formula of the Fredholm index of a Toeplitz operator.