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Journal of Operator Theory

Volume 71, Issue 2,  Spring  2014  pp. 381-410.

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

Authors:  Joshua Isralowitz
Author institution:SUNY Albany, 1400 Washington Ave. Albany, NY, 12222, U.S.A.

Summary: In this paper, we characterize invertible Toeplitz products on a number of Banach spaces of analytic functions, including the weighted\break Bergman space $L^p _\mathrm a (\mathbb{B}_n, \mathrm dv_\gamma)$, the Hardy space $H^p(\partial \mathbb{D})$, and the standard 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. Furthermore, we prove weighted norm inequalities for the Fock projection, and compare the various A${}_p$ type conditions that arise in our results. Finally, we extend the "reverse H\"older inequality" of Zheng and Stroethoff (\textit{J. Funct. Anal.} \textbf{195}(2002), 48-70 and \textit{J. Operator Theory} \textbf{59}(2008), 277-308) for $p = 2$ to the general case of $p > 1$.

Keywords: Toeplitz operator, weighted norm inequalities, products of Toeplitz operators

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