On the mean ergodic theorem of Sine

Abstract

Robert Sine has shown that ( 1 / n ) ( I + T + ⋯ + T n − 1 ) (1/n)(I + T + \cdots + {T^{n - 1}}) , the ergodic averages, converge in the strong operator topology iff the invariant vectors of T T separate the invariant vectors of the adjoint operator T ∗ , T {T^ \ast },T being any Banach space contraction. We prove a generalization in which (spectral radius of T T ) ≦ 1 \leqq 1 replaces | | T | | ≦ 1 ||T|| \leqq 1 , and any bounded averaging sequence converging uniformly to invariance replaces the ergodic averages; it is necessary to assume that such sequences exist.