Essential normality of automorphic composition operators

We first characterize those composition operators that are essentially normal on the weighted Bergman space A_s^2(D) for any real s > −1, where induced symbols are automorphisms of the unit disk D . Using the same technique, we investigate automorphic composition operators on the Hardy space H 2 ( B N ) and the weighted Bergman spaces A_s^2(B_N) ( s > −1). Furthermore, we give some composition operators induced by linear fractional self-maps of the unit ball B N that are not essentially normal.