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.
Categories: math.FA math.CV