On the uniform ergodic theorem

Abstract

We give an elementary proof of the uniform ergodic theorem: “Let T T be a linear operator on a Banach space with | | T n / n | | → 0 ||{T^n}/n|| \to 0 . The following are equivalent: (1) N − 1 ∑ n = 0 N − 1 T n {N^{ - 1}}\sum \nolimits _{n = 0}^{N - 1} {{T^n}} converges uniformly. (2) ( I − T ) 2 X {(I - T)^2}X is closed. (3) ( I − T ) X (I - T)X is closed."