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

Zeljko Cuckovic and Sonmez Sahutoglu

Arxiv ID: 1004.0720•Last updated: 3/8/2021

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.

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