Fredholm and invertibility theory for a special class of Toeplitz + Hankel operators

TL;DR

This paper develops a comprehensive Fredholm and invertibility framework for a specific class of Toeplitz+Hankel operators on Hardy spaces, providing formulas for defect numbers and applying results to key examples.

Contribution

It introduces a complete Fredholm and invertibility theory for Toeplitz+Hankel operators with piecewise continuous symbols satisfying a symmetry condition, including formulas for defect numbers.

Findings

Established Fredholm criteria for the operators.

Derived formulas for defect numbers.

Applied theory to important operator examples.

Abstract

We develop a complete Fredholm and invertibility theory for Toeplitz+Hankel operators $T(a)+H(b)$ on the Hardy space $H^p$, $1<p<\infty$, with piecewise continuous functions $a,b$ defined on the unit circle which are subject to the condition $a(t)a(t^{-1})=b(t)b(t^{-1})$, $|t|=1$. In particular, in the case of Fredholmness, formulas for the defect numbers are established. The results are applied to several important examples.