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Journal of Operator Theory

Volume 89, Issue 1, Winter 2023  pp. 75-85.

A sufficient condition for compactness of Hankel operators

Authors:  Mehmet Celik (1), Sonmez Sahutolu (2), Emil J. Straube (3)
Author institution:(1) Department of Mathematics, Texas A & M University-Commerce, Commerce, TX 75429, U.S.A.
(2) Department of Mathematics & Statistics, University of Toledo, Toledo, OH 43606, U.S.A.
(3) Department of Mathematics, Texas A & M University, College Station, TX, 77843, U.S.A.

Summary: Let $\Omega$ be a bounded convex domain in $\mathbb{C}^{n}$. We show that if $\varphi \in C^{1}(\overline{\Omega})$ is holomorphic along analytic varieties in $b\Omega$, then $H^{q}_{\varphi}$, the Hankel operator with symbol $\varphi$, is compact. We have shown the converse earlier (Compactness of Hankel operators with continuous symbols on convex domains, \textit{Houston J. Math.} \textbf{46}(2020), 1005--1016) so that we obtain a characterization of compactness of these operators in terms of the behavior of the symbol relative to analytic structure in the boundary. A corollary is that Toeplitz operators with these nonvanishing symbols are Fredholm (of index zero).

Keywords: Hankel operators, convex domains, compactness, Fredholm Toeplitz operators

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