On the Mean Ergodic Theorem of Sine

Abstract

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