Commutators of composition operators with adjoints of composition operators on the ball

In this paper, we investigate compactness of the commutator [ C*., C.] on the Hardy space H2( BN) or the weighted Bergman space A2s ( BN) ( s > - 1), when. and. are automorphisms of the unit ball BN. We obtain that [ C*., C.] is compact if and only if both. and. are unitary and they commute. This generalizes the corresponding result in one variable. Moreover, our technique is different and simple. In addition, we also discuss the commutator [ C*., C.] on the Dirichlet space D( BN), where. and. are linear fractional self- maps or both automorphisms of B-N.