Commutators of composition operators with adjoints of composition operators on weighted Bergman spaces

For linear-fractional self-maps phi and of the unit disc ?, where at least one of phi and is a non-automorphism, we show that the commutator [C-psi*, C-phi] is non-trivially compact on the weighted Bergman space A(alpha)(2)(D) if and only if either phi and are both parabolic or phi and are both hyperbolic, with associated conclusions about their fixed points in each case. In the automorphism case, we show that this commutator is compact if and only if both phi and are rotations.