The semi-commutator of Toeplitz operators on the bidisc

In this paper we characterize when the semi-commutator $T_fT_g-T_{fg}$ of two Toeplitz operators $T_f$ and $T_g$ on the Hardy space of the bidisc is zero. We also show that there is no nonzero finite rank semi-commutator on the bidisc. Furthermore explicit examples of compact semi-commutators with symbols continuous on the bitorus $T^2$ are given.