Compactness of products of Hankel operators on the polydisk and some product domains in C2

Let D-n be the polydisk in C-n and the symbols phi, psi is an element of C((D-n) over bar) such that phi and psi are pluriharmonic on any (n - 1)-dimensional polydisk in the boundary of D-n. Then H-psi*H-phi is compact on A(2)(D-n) if and only if for every 1 <= j, k <= n such that j not equal k and any (n - 1)-dimensional polydisk D, orthogonal to the z(j)-axis in the boundary of D-n, either phi or psi is holomorphic in z(k) on D. Furthermore, we prove a different sufficient condition for compactness of the products of Hankel operators. In C-2, our techniques can be used to get a necessary condition on some product domains involving annuli. (C) 2010 Elsevier Inc. All rights reserved.