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On the uniform ergodic theorem of Lin


Author: Stuart P. Lloyd
Journal: Proc. Amer. Math. Soc. 83 (1981), 710-714
MSC: Primary 47A35
DOI: https://doi.org/10.1090/S0002-9939-1981-0630042-0
MathSciNet review: 630042
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Abstract: Lin has given necessary and sufficient conditions for convergence in the uniform operator topology of $ {A_n} = (1 + T + \cdots + {T^{n - 1}})/n$, $ T$ being a Banach space operator satisfying $ \left\Vert {{T^n}} \right\Vert/n \to 0$. We prove a generalization in which the Cesàro means are replaced by any bounded sequence in the affine hull converging uniformly to invariance. In the case where $ T:{C_0}(X) \to {C_0}(X)$ is a transient Feller operator for noncompact locally compact Hausdorff space $ X$, we show that $ \{ {A_n}\} $ converges strongly but never uniformly.


References [Enhancements On Off] (What's this?)

  • [1] Nelson Dunford and Jacob T. Schwartz, Linear operators. I, Interscience, New York, 1958.
  • [2] Michael Lin, On the uniform ergodic theorem, Proc. Amer. Math. Soc. 43 (1974), 337-340. MR 0417821 (54:5869)
  • [3] Stuart P. Lloyd, On the mean ergodic theorem of Sine, Proc. Amer. Math. Soc. 56 (1976), 121-126. MR 0451007 (56:9297)
  • [4] Ryotaro Sato, The Hahn-Banach theorem implies Sine's mean ergodic theorem, Proc. Amer. Math. Soc. 77 (1979), 426. MR 545609 (80m:47007)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1981-0630042-0
Keywords: Uniform ergodic theorem
Article copyright: © Copyright 1981 American Mathematical Society

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