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 s et in the operator norm topology. This occurs if and only if the seque nce {n(-1)T(n)}(n greater than or equal to 1) is precompact and the po int 1 is at most a simple pole of the resolvent of T. Equivalent geome tric conditions are also obtained.