PRECOMPACTNESS IN THE UNIFORM ERGODIC-THEORY

We characterize the Banach space operators T whose arithmetic means {n(-1)(I+T+...+T-n-1)}(n greater than or equal to 1) form a precompact set in the operator norm topology. This occurs if and only if the sequence {n(-1)T(n)}(n greater than or equal to 1) is precompact and the point 1 is at most a simple pole of the resolvent of T. Equivalent geometric conditions are also obtained.