On the uniform ergodic theorem of Lin

Author:
Stuart P. Lloyd

Journal:
Proc. Amer. Math. Soc. **83** (1981), 710-714

MSC:
Primary 47A35

MathSciNet review:
630042

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Abstract: Lin has given necessary and sufficient conditions for convergence in the uniform operator topology of , being a Banach space operator satisfying . 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 is a transient Feller operator for noncompact locally compact Hausdorff space , we show that converges strongly but never uniformly.

**[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**, 10.1090/S0002-9939-1974-0417821-6**[3]**Stuart P. Lloyd,*On the mean ergodic theorem of Sine*, Proc. Amer. Math. Soc.**56**(1976), 121–126. MR**0451007**, 10.1090/S0002-9939-1976-0451007-6**[4]**Ryōtarō Satō,*The Hahn-Banach theorem implies Sine’s mean ergodic theorem*, Proc. Amer. Math. Soc.**77**(1979), no. 3, 426. MR**545609**, 10.1090/S0002-9939-1979-0545609-9

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DOI:
https://doi.org/10.1090/S0002-9939-1981-0630042-0

Keywords:
Uniform ergodic theorem

Article copyright:
© Copyright 1981
American Mathematical Society