A Sufficient condition for compactness of Hankel operators

TL;DR

This paper provides a boundary-analytic condition under which Hankel operators on convex domains in complex space are compact, linking boundary behavior of symbols to operator properties.

Contribution

It establishes a sufficient condition for the compactness of Hankel operators based on the holomorphicity of symbols along boundary varieties, completing a characterization.

Findings

Hankel operators are compact if symbols are holomorphic along boundary varieties.

The characterization of compactness is both necessary and sufficient.

Toeplitz operators with these symbols are Fredholm of index zero.

Abstract

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, 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).