Semi-commutators of Toeplitz operators on the Bergman space

In this paper several necessary and sufficient conditions are obtained for the semi-commutator (T-f over bar T-g)-(T-f over bar g) of Toeplitz operators (T-f over bar) and T-g with bounded pluriharmonic symbols on the unit ball to be compact on the Bergman space. Using M-harmonic function theory on the unit ball we show that (T-f over bar T-g) - (T-f over bar g) with bounded pluriharmonic symbols f and g is zero on the Bergman space of the unit ball or the Hardy space of the unit sphere if and only if either f or g is holomorphic.