Compact differences of composition operators on Holomorphic Function Spaces in the unit ball

We find a lower bound for the essential norm of the difference of two composition operators acting on H2(BN) or A2s(BN) (s > −1). This result plays an important role in proving a necessary and sufficient condition for the difference of linear fractional composition operators to be compact, which answers a question posed by MacCluer and Weir in 2005.