Factorization theory for a class of Toeplitz + Hankel operators

TL;DR

This paper investigates the invertibility of operators formed by the sum of Toeplitz and Hankel operators on Hardy spaces, establishing that invertibility depends on a specific generalized factorization of the generating function.

Contribution

It introduces a new factorization criterion that characterizes the invertibility of Toeplitz plus Hankel operators on Hardy spaces.

Findings

Invertibility of $M()$ characterized by generalized factorization of $$

Provides a necessary and sufficient condition for invertibility

Extends classical results on Toeplitz operators to combined Toeplitz + Hankel operators.

Abstract

In this paper we study operators of the form $M(\phi)=T(\phi)+H(\phi)$ where $T(\phi)$ and $H(\phi)$ are the Toeplitz and Hankel operators acting on $H^p(\T)$ with generating function $\phi\in L^\iy(\T)$. It turns out that $M(\phi)$ is invertible if and only if the function $\phi$ admits a certain kind of generalized factorization.