On the Uniform Ergodic Theorem

Abstract

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

Locations

• Proceedings of the American Mathematical Society - View - PDF