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Proceedings of the American Mathematical Society

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

 

 

Infinite matrices and invariant means


Author: Paul Schaefer
Journal: Proc. Amer. Math. Soc. 36 (1972), 104-110
MSC: Primary 40C05
DOI: https://doi.org/10.1090/S0002-9939-1972-0306763-0
MathSciNet review: 0306763
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Abstract: Let $ \sigma $ be a one-to-one mapping of the set of positive integers into itself such that $ {\sigma ^p}(n) \ne n$ for all positive integers n and p, where $ {\sigma ^p}(n) = \sigma ({\sigma ^{p - 1}}(n)),p = 1,2, \cdots $. A continuous linear functional $ \varphi $ on the space of real bounded sequences is an invariant mean if $ \varphi (x) \geqq 0$ when the sequence $ x = \{ {x_n}\} $ has $ {x_n} \geqq 0$ for all n, $ \varphi (\{ 1,1,1, \cdots \} ) = + 1$, and $ \varphi (\{ {x_{\sigma (n)}}\} ) = \varphi (x)$ for all bounded sequences x. Let $ {V_\sigma }$ be the set of bounded sequences all of whose invariant means are equal. If $ A = ({a_{nk}})$ is a real infinite matrix, then A is said to be (1) $ \sigma $-conservative if $ Ax = \{ {\Sigma _k}{a_{nk}}{x_k}\} \in {V_\sigma }$ for all convergent sequences x, (2) $ \sigma $-regular if $ Ax \in {V_\sigma }$ and $ \varphi (Ax) = \lim x$ for all convergent sequences x and all invariant means $ \varphi $, and (3) $ \sigma $-coercive if $ Ax \in {V_\sigma }$ for all bounded sequences x. Necessary and sufficient conditions are obtained to characterize these classes of matrices.


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DOI: https://doi.org/10.1090/S0002-9939-1972-0306763-0
Keywords: $ \sigma $-conservative matrices, $ \sigma $-regular matrices, $ \sigma $-coercive matrices, invariant means, almost convergence
Article copyright: © Copyright 1972 American Mathematical Society