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On the uniform ergodic theorem


Author: Michael Lin
Journal: Proc. Amer. Math. Soc. 43 (1974), 337-340
MSC: Primary 47A35
DOI: https://doi.org/10.1090/S0002-9939-1974-0417821-6
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."


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

  • [1] N. Dunford, Spectral theory. I. Convergence to projections, Trans. Amer. Math. Soc. 54 (1943), 185-217. MR 5, 39. MR 0008642 (5:39c)
  • [2] -, Spectral theory, Bull. Amer. Math. Soc. 49 (1943), 637-651. MR 5, 39. MR 0008643 (5:39d)
  • [3] N. Dunford and J. T. Schwartz, Linear operators. I: General theory, Pure and Appl. Math., vol. 7, Interscience, New York, 1958. MR 22 #8302. MR 0117523 (22:8302)
  • [4] W. H. Gottschalk and G. A. Hedlund, Topological dynamics, Amer. Math. Colloq. Publ., vol. 36, Amer. Math. Soc., Providence, R.I., 1955. MR 17, 650. MR 0074810 (17:650e)
  • [5] S. Horowitz, Transition probabilities and contractions of $ {L_\infty }$, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 24 (1972), 263-274. MR 0331516 (48:9849)
  • [6] M. Loève, Probability theory, 3rd ed., Van-Nostrand, Princeton, N.J., 1963. MR 34 #3596. MR 0203748 (34:3596)
  • [7] J. Neveu, Mathematical foundations of the calculus of probability, Holden-Day, San Francisco, Calif., 1965. MR 33 #6660. MR 0198505 (33:6660)
  • [8] K. Yosida and S. Kakutani, Operator theoretical treatment of Markoff's process and mean ergodic theorem, Ann. of Math. (2) 42 (1941), 188-228. MR 2, 230. MR 0003512 (2:230e)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1974-0417821-6
Keywords: Ergodic theorem, quasi-compact operators, Markov operators
Article copyright: © Copyright 1974 American Mathematical Society

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