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Proceedings of the American Mathematical Society

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

 

 

Local ergodicity of nonpositive contractions on $ C(X)$


Author: Robert E. Atalla
Journal: Proc. Amer. Math. Soc. 88 (1983), 419-425
MSC: Primary 47A35; Secondary 47B55
DOI: https://doi.org/10.1090/S0002-9939-1983-0699406-5
MathSciNet review: 699406
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Abstract: Let $ T$ be an operator on $ C(X)$, $ X$ compact, with $ \left\Vert T \right\Vert \leqslant 1$, and suppose $ T$ has a nowhere vanishing invariant function $ {\psi ^{ - 1}}$. The operator $ R$ defined by $ Rf = T(f{\psi ^{ - 1}})\psi $ is (a) "locally" a Markov operator, and (b) (locally) strongly ergodic iff $ T$ is. This is used to prove Sine's local strong ergodicity theorem without assuming that $ T$ is positive.


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DOI: https://doi.org/10.1090/S0002-9939-1983-0699406-5
Keywords: Markov operator, contraction, strongly ergodic, invariant measure
Article copyright: © Copyright 1983 American Mathematical Society