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Proceedings of the American Mathematical Society

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

 

 

Vector-valued Hausdorff summability methods and ergodic theorems


Author: Takeshi Yoshimoto
Journal: Proc. Amer. Math. Soc. 107 (1989), 915-926
MSC: Primary 47A35
MathSciNet review: 984826
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Abstract: Suppose $ X$ and $ Y$ are two general Banach spaces. Let $ H = ({\Lambda _{n,k}})$ be a general $ {\mathbf{B}}[X,Y]$-operator valued Hausdorff summability method: $ {\Lambda _{n,k}} = (_k^n){\Delta ^{n - k}}{U_k}$ for $ k \leq n$ and $ {\Lambda _{n,k}} = {\theta _{X,Y}}$ for $ k > n$, where $ \{ {U_k}\} _{k = 0}^\infty $ is a sequence of operators in $ {\mathbf{B}}[X,Y]$ and $ \Delta $ denotes the backward difference (operator) and $ {\theta _{X,Y}}(x) = {0_Y}$ (the zero element in $ Y$) for all $ x \in $. Then some necessary and sufficient conditions are given for the mean and uniform convergence of the averages

$\displaystyle \sum\limits_{k = 0}^n {(_k^n){\Delta ^{n - k}}{U_k}({T^k}x)} \quad (x \in X,T \in {\mathbf{B}}[X]).$


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DOI: https://doi.org/10.1090/S0002-9939-1989-0984826-7
Keywords: Hausdorff summability method, URS-method, moment sequence, quasiregularity, mean ergodic theorem, uniform ergodic theorem
Article copyright: © Copyright 1989 American Mathematical Society