Compactness of products of Hankel operators on the polydisk and some product domains in $\mathbb{C}^2$

TL;DR

This paper characterizes when products of Hankel operators are compact on the polydisk in complex space, providing necessary and sufficient conditions based on boundary behavior and extending results to certain product domains in ^2.

Contribution

It offers a new characterization of compactness for Hankel operator products on polydisks and extends the analysis to specific product domains in ^2.

Findings

Characterization of compactness for Hankel products on ^n.

Necessary conditions for Hankel product compactness on product domains in ^2.

A new sufficient condition for the compactness of Hankel operator products.

Abstract

Let $\mathbb{D}^n$ be the polydisk in $\mathbb{C}^n$ and the symbols $\phi,\psi\in C(\bar{\mathbb{D}^n})$ such that $\phi$ and $\psi$ are pluriharmonic on any $(n-1)$-dimensional polydisk in the boundary of $\mathbb{D}^{n}.$ Then $H^*_{\psi}H_{\phi}$ is compact on $A^2(\mathbb{D}^n)$ if and only if for every $1\leq j,k\leq n$ such that $j\neq k$ and any $(n-1)$-dimensional polydisk $D$, orthogonal to the $z_j$-axis in the boundary of $\mathbb{D}^n,$ either $\phi$ or $\psi$ is holomorphic in $z_k$ on $D.$ Furthermore, we prove a different sufficient condition for compactnes of the products of Hankel operators. In $\mathbb{C}^2,$ our techniques can be used to get a necessary condition on some product domains involving annuli.