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Proceedings of the American Mathematical Society

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

 

 

On the mean ergodic theorem of Sine


Author: Stuart P. Lloyd
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
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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 $ \vert\vert T\vert\vert \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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DOI: https://doi.org/10.1090/S0002-9939-1976-0451007-6
Keywords: Mean ergodic theorem
Article copyright: © Copyright 1976 American Mathematical Society