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Proceedings of the American Mathematical Society

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

 

 

Ergodic theorems of weak mixing type


Authors: Lee K. Jones and Michael Lin
Journal: Proc. Amer. Math. Soc. 57 (1976), 50-52
MSC: Primary 47A35
DOI: https://doi.org/10.1090/S0002-9939-1976-0405134-X
MathSciNet review: 0405134
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Abstract: Given a linear contraction $ T$ on a Banach space $ X$ and $ x \in X$, the convergence

$\displaystyle \forall {x^ \ast } \in {X^ \ast }{N^{ - 1}}\sum\limits_{i = 1}^N {\vert\langle {x^ \ast },{T^i}x\rangle \vert} \xrightarrow[{N \to \infty }]{}0$

is shown to be equivalent to the convergence

$\displaystyle \mathop {\sup }\limits_{\vert\vert{x^ \ast }\vert\vert \leqslant ... ...sum\limits_{j = 1}^N {\vert\langle {x^ \ast },{T^{{k_j}}}x\rangle \vert \to 0} $

for every subsequence with $ {k_j}/j$ bounded. A sufficient condition is that, for some $ \{ {n_i}\} ,{T^{{n_i}}}x \to 0$ weakly.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1976-0405134-X
Keywords: Ergodic theorems, weak mixing
Article copyright: © Copyright 1976 American Mathematical Society