# 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).

**DOI: **http://dx.doi.org/10.7900/jot.2021apr04.2334

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

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