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 $H^2(B_N)$ or $A_s^2(B_N)$ ($s>-1$). 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.