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Proceedings of the American Mathematical Society

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On the dominated ergodic theorem in $ L\sb{2}$ space


Authors: M. A. Akcoglu and L. Sucheston
Journal: Proc. Amer. Math. Soc. 43 (1974), 379-382
MSC: Primary 47A35; Secondary 28A65
DOI: https://doi.org/10.1090/S0002-9939-1974-0333770-6
MathSciNet review: 0333770
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Abstract: Let $ T$ be a contraction on $ {L_2}$ of a $ \sigma $-finite measure space, $ {A_n}(T)$ the operator $ (1/n)({T^0} + \cdots + {T^n}),S(T)f$ the function $ {\sup _n}\vert{A_n}(T)f\vert$. Theorem 1. Assume that, whatever be the measure space, $ S(U)f \in {L_2}$ for each unitary operator $ U$ on $ {L_2}$ and each function $ f \in {L_2}$. Then there exists a universal constant $ K$ such that $ \vert\vert S(T)f\vert\vert \leqq K\vert\vert f\vert\vert$ for each contraction $ T$ on $ {L_2}$ and each $ f \in {L_2}$. Theorem 2. Let $ T$ be a contraction on $ {L_2}$ and let $ U$ be a unitary dilation of $ T$ acting on a Hilbert space $ H$ containing $ {L_2}$. If all expressions of the form $ \Sigma_{n = 1}^\infty {{P_n}{A_n}(U)} $, where $ {P_n}$ are mutually orthogonal projections, are bounded operators on $ H$, then for each $ f \in {L_2}, S(T)f \in {L_2}$ and $ {A_n}(T)f$ converges a.e.


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DOI: https://doi.org/10.1090/S0002-9939-1974-0333770-6
Keywords: Contraction, unitary operator, dilation, Hilbert space, dominated ergodic theorem, pointwise convergence
Article copyright: © Copyright 1974 American Mathematical Society

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