On the uniform ergodic theorem

Lin, Michael

doi:10.1090/S0002-9939-1974-0417821-6

On the uniform ergodic theorem
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Proc. Amer. Math. Soc. 43 (1974), 337-340 Request permission

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."

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