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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