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

Let D n be the polydisk in C n and the symbols ϕ , ψ ∈ C ( D n ¯ ) such that ϕ and ψ are pluriharmonic on any ( n − 1 ) -dimensional polydisk in the boundary of D n . Then H ψ ∗ H ϕ is compact on A 2 ( D n ) if and only if for every 1 ⩽ j , k ⩽ n such that j ≠ k and any ( n − 1 ) -dimensional polydisk D, orthogonal to the z j -axis in the boundary of D n , either ϕ or ψ 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.