Some algebraic properties of Toeplitz and Hankel operators on polydisk

In this paper we characterize when the product of two Toeplitz operators on the Hardy space of polydisk is also a Toeplitz operator. We show that if the product of two Toeplitz operators is of finite rank, then one of them has to be zero. There are two natural ways of denfining Hankel operators on polydisk. Big Hankel operators are closely related to Toeplitz operators. However small Hankel operators on polydisk inherit many interesting properties of Hankel operators on the unit disk. We discuss when the product of two Hankel operators is of finite rank for both big and small Hankel operators. We also answer the question when the product of two small Hankel operators is also a Hankel operator. These results extend several results on unit disk obtained over the last four decades.