Invertible Toeplitz products, weighted norm inequalities, and A${}_p$ weights

Abstract

In this paper, we characterize invertible Toeplitz products on a number of Banach spaces of analytic functions, including weighted Bergman space $L_a^p({\mathbb{B}}_n,d{v}_{\gamma })$, the Hardy space $H^p(\partial \mathbb{D})$, and the weighted Fock space F${}_{\alpha }^p$ for $p>1$. The common tool in the proofs of our characterizations will be the theory of weighted norm inequalities and A${}_p$ type weights. Moreover, we analyze and compare the various A${}_p$ type conditions that arise in our characterizations. Finally, we extend the "reverse Hölder inequality" of Zheng and Stroethoff \cite{SZ1, SZ2} for $p=2$ to the general case of $p>1$.

arXiv commentv1: 31 pages, no figures, submitted. v2: 34 pages, no figures, minor changes and additions made, submitted. v3: 35 pages, no figures, Propositions 4.1 and 4.4 and their proofs are corrected, Theorem 3.7 improved, minor changes made, to appear in the Journal of Operator Theory