A handy formula for the Fredholm index of Toeplitz plus Hankel operators

TL;DR

This paper develops a clear symbol calculus and provides a simple formula for calculating the Fredholm index of Toeplitz plus Hankel operators with piecewise continuous symbols on l^p-spaces, enhancing understanding of their invertibility.

Contribution

It introduces a transparent symbol calculus and a practical formula for the Fredholm index of Toeplitz plus Hankel operators with piecewise continuous generating functions.

Findings

Derived a formula for the Fredholm index of these operators.

Established a symbol calculus for the associated Banach algebra.

Enhanced the understanding of the Fredholm property in this context.

Abstract

We consider Toeplitz and Hankel operators with piecewise continuous generating functions on $l^p$-spaces and the Banach algebra generated by them. The goal of this paper is to provide a transparent symbol calculus for the Fredholm property and a handy formula for the Fredholm index for operators in this algebra.