Factorization of a class of Toeplitz + hankel operators and the A_p-condition

Abstract

Let $M(\phi)=T(\phi)+H(\phi)$ be the Toeplitz plus Hankel operator acting on $H^p(\T)$ with generating function $\phi\in L^\iy(\T)$. In a previous paper we proved that $M(\phi)$ is invertible if and only if $\phi$ admits a factorization $\phi(t)=\phi_{-}(t)\phi_{0}(t)$ such that $\phi_{-}$ and $\phi_{0}$ and their inverses belong to certain function spaces and such that a further condition formulated in terms of $\phi_{-}$ and $\phi_{0}$ is satisfied. In this paper we prove that this additional condition is equivalent to the Hunt-Muckenhoupt-Wheeden condition (or, $A_{p}$-condition) for a certain function $\sigma$ defined on $[-1,1]$, which is given in terms of $\phi_{0}$. As an application, a necessary and sufficient criteria for the invertibility of $M(\phi)$ with piecewise continuous functions $\phi$ is proved directly. Fredholm criteria are obtained as well.