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Subadditive mean ergodic theorems

Published online by Cambridge University Press:  19 September 2008

Y. Derriennic  and
U. Krengel
Y. Derriennic
Affiliation:
Département de Mathématiques, Université de Bretagne Occidentale, Brest, France
U. Krengel*
Affiliation:
Institüt für Mathematische Statistik, Göttingen, West Germany
*
†Dr U. Krengel, Institüt für Mathematische Statistik, Lotzestrasse 13, 3400 Göttingen, West Germany.
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Abstract

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The authors investigate which results of the classical mean ergodic theory for bounded linear operators in Banach spaces have analogues for subadditive sequences (Fn) in a Banach lattice B. A sequence (Fn) is subadditive for a positive contraction T in B if Fn+k ≤ Fn + TnFk (n, k ≥ 1). For example, von Neumann's mean ergodic theorem fails to extend to the general subadditive case, but it extends to the non-negative subadditive case. It is shown that the existence of a weak cluster point f = Tf for (n−1Fn) implies In Lp (1 ≤ p < ∞) the existence of a weak cluster point for non-negative (n−1Fn) is equivalent with norm convergence. If T is an isometry in Lp (1 < p < ∞) and sup then n−1Fn converges weakly. If T in L1 has a strictly positive fixed point and sup then n−1Fn converges strongly. Most results are proved even in the d-parameter case.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 1 , Issue 1 , March 1981 , pp. 33 - 48
Copyright
Copyright © Cambridge University Press 1981

References

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