Toeplitz operators and Wiener-Hopf factorisation: an introduction

TL;DR

This paper introduces the relationship between Toeplitz operators and Wiener-Hopf factorisation, detailing their Fredholm properties, invertibility conditions, and applications to singular integral equations in Hardy spaces.

Contribution

It provides a comprehensive review of how Wiener-Hopf factorisation characterizes Toeplitz operators' Fredholm properties and invertibility in Hardy spaces, with explicit formulas and applications.

Findings

Fredholm and invertibility conditions for Toeplitz operators derived

Explicit formulas for inverses and one-sided inverses provided

Application to singular integral equations in $L^p(\mathbb R)$

Abstract

Wiener-Hopf factorisation plays an important role in the theory of Toeplitz operators. We consider here Toeplitz operators in the Hardy spaces $H^p$ of the upper half-plane and we review how their Fredholm properties can be studied in terms of a Wiener-Hopf factorisation of their symbols, obtaining necessary and sufficient conditions for the operator to be Fredholm or invertible, as well as formulae for their inverses or one-sided inverses when these exist. The results are applied to a class of singular integral equations in $L^p(\mathbb R)$.