Journal of Operator Theory

Volume 55, Issue 2, Spring 2006 pp. 269-283.

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

Authors: Estelle L. Basor (1) and Torsten Ehrhardt (2)
Author institution: (1) Department of Mathematics, California Polytechnic State University, San Luis Obispo, CA 93407, USA
(2) Mathematics Department, University of California, Santa Cruz, CA 95064, USA

Summary: 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 function $\phi$ is proved directly. Fredholm criteria are obtained as well.

Keywords: Toeplitz operator, Hankel operator, factorization, $A_{p}$-condition.

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