Products of Toeplitz Operators on the Polydisk

This paper studies products of Toeplitz operators on the Hardy space of the polydisk. We show that T f T g = 0 if and only if T f T g is a finite rank if and only if T f or T g is zero. The product T f T g is still a Toeplitz operator if and only if there is a h (element of) L ^(infinity) (T ^n ) such that T f T g - T h is a finite rank operator. We also show that there are no compact simicommutators with symbols pluriharmonic on the polydisk.