A Sufficient condition for compactness of Hankel operators

Abstract

Let $\Omega$ be a bounded convex domain in ${\mathbb{C}}^n$. We show that if $\phi \in C^1(\overline{\Omega })$ is holomorphic along analytic varieties in $b\Omega$, then $H_{\phi }^q$, the Hankel operator with symbol $\phi$, is compact. We have shown the converse earlier, 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 symbols are Fredholm (of index zero).

10 pages, to appear in J. Operator Theory, minor changes