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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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On the mean ergodic theorem of Sine
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by Stuart P. Lloyd PDF
Proc. Amer. Math. Soc. 56 (1976), 121-126 Request permission

Abstract:

Robert Sine has shown that $(1/n)(I + T + \cdots + {T^{n - 1}})$, the ergodic averages, converge in the strong operator topology iff the invariant vectors of $T$ separate the invariant vectors of the adjoint operator ${T^ \ast },T$ being any Banach space contraction. We prove a generalization in which (spectral radius of $T$) $\leqq 1$ replaces $||T|| \leqq 1$, and any bounded averaging sequence converging uniformly to invariance replaces the ergodic averages; it is necessary to assume that such sequences exist.
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Additional Information
  • © Copyright 1976 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 56 (1976), 121-126
  • MSC: Primary 47A35
  • DOI: https://doi.org/10.1090/S0002-9939-1976-0451007-6
  • MathSciNet review: 0451007