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Proceedings of the American Mathematical Society

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

 

 

Two ergodic theorems for convex combinations of commuting isometries


Author: S. A. McGrath
Journal: Proc. Amer. Math. Soc. 40 (1973), 229-234
MSC: Primary 47A35; Secondary 28A65
DOI: https://doi.org/10.1090/S0002-9939-1973-0318927-1
MathSciNet review: 0318927
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Abstract: Let $ (X,\mathcal{F},\mu )$ be a measure space. In this paper we obtain $ {L^p}$ estimates for the supremum of the Cesàro averages of combinations of commuting isometries of $ {L^p}(X,\mathcal{F},\mu )$. In particular, we show that a convex combination of two invertible commuting isometries of $ {L^p}(X,\mathcal{F},\mu ),p$ fixed, $ 1 < p < \infty ,p \ne 2$, admits of adominated estimate with constant $ P/(p - 1)$. We also show that a convex combination of an arbitrary number of commuting positive invertible isometries of $ {L^2}(X,\mathcal{F},\mu )$ admits of a dominated estimate with constant 2.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1973-0318927-1
Keywords: Dominated estimates of $ {L^p}$ contractions, positive contractions, convex combinations of commuting isometries, positive fixed functions, normal positive contractions, periodic isometries
Article copyright: © Copyright 1973 American Mathematical Society