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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)


On the uniform ergodic theorem

Author: Michael Lin
Journal: Proc. Amer. Math. Soc. 43 (1974), 337-340
MSC: Primary 47A35
MathSciNet review: 0417821
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Abstract: We give an elementary proof of the uniform ergodic theorem: ``Let $ T$ be a linear operator on a Banach space with $ \vert\vert{T^n}/n\vert\vert \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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PII: S 0002-9939(1974)0417821-6
Keywords: Ergodic theorem, quasi-compact operators, Markov operators
Article copyright: © Copyright 1974 American Mathematical Society