On Compactness of Products of Toeplitz Operators.

We study compactness of product of Toeplitz operators with symbols continuous on the closure of the polydisc in terms of behavior of the symbols on the boundary. For certain classes of symbols and , we show that is compact if and only if vanishes on the boundary. We provide examples to show that for more general symbols, the vanishing of on the whole polydisc might not imply the compactness of . On the other hand, the reverse direction is closely related to the zero product problem for Toeplitz operators on the unit disc, which is still open.