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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Pseudospectral operators and the pointwise ergodic theorem
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by R. E. Bradley PDF
Proc. Amer. Math. Soc. 112 (1991), 863-870 Request permission

Abstract:

We show that for a class of operators which properly contains the normal operators on ${L_2}$, \[ \frac {1}{n}\sum \limits _{i = 0}^{n - 1} {{T^i}f \to a.e.} {\text {iff}}\frac {1}{{{2^n}}}\sum \limits _{i = 0}^{{2^n} - 1} {{T^i}f \to a.e.} \] This theorem is used to give an alternate form of a theorem of Gaposhkin concerning the pointwise ergodic theorem for normal operators.
References
  • G. Alexits, Convergence problems of orthogonal series, International Series of Monographs in Pure and Applied Mathematics, Vol. 20, Pergamon Press, New York-Oxford-Paris, 1961. Translated from the German by I. Földer. MR 0218827
  • D. L. Burkholder, Semi-Gaussian subspaces, Trans. Amer. Math. Soc. 104 (1962), 123–131. MR 138986, DOI 10.1090/S0002-9947-1962-0138986-6
  • V. F. Gaposhkin, A theorem on the convergence almost everywhere of a sequence of measurable functions, and its application to sequences of stochastic integrals, Math. USSR Sbornik 33 (1977), 1-19. (English transl.) —, Criteria for the strong law of large numbers for some classes of second-order stationary processes and homogeneous random fields, Theory Probab. and Appl. 22 (1977), 286-310. (English transl.) —, Individual ergodic theorem for normal operators in ${L_2}$, Functional Anal. Appl. 15 (1981), 14-18. (English transl.)
  • P. R. Halmos, What does the spectral theorem say?, Amer. Math. Monthly 70 (1963), 241–247. MR 150600, DOI 10.2307/2313117
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Additional Information
  • © Copyright 1991 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 112 (1991), 863-870
  • MSC: Primary 47A35; Secondary 28D05, 47B15
  • DOI: https://doi.org/10.1090/S0002-9939-1991-1050017-6
  • MathSciNet review: 1050017