An ergodic theorem on Banach lattices
type: Item
instance of: scholarly article
title: An ergodic theorem on Banach lattices
zbMATH Open document ID: 0592.47005
DOI: 10.1007/BF02772665
published in: Israel Journal of Mathematics
publication date: 1985-01-01T00:00:00Z
review text: A well known theorem of \textit{U. Krengel} [Z. Wahrscheinlichkeitstheorie Verw. Gebiete 6, 302-316 (1966; Zbl 0218.60060)] states that for any positive linear contraction T in \(L_ 1\) and for each \(f\in L_ 1\) the averages \(A_ nf=n^{-1}\sum^{n-1}_{k=0}T^ kf\) are convergent in probability. In the present paper the above theorem is extended to positive linear contractions T acting in a Banach lattice E. Instead of the convergence in probability, the authors propose to consider the existence of the truncated limit \(TL(A_ nf)=\Phi\), which means that \(\lim_{n} (A_ nf\wedge ku)=\Phi_ k\) exists in norm for some weak unit u and for each k, and that \(\Phi_ k\uparrow \Phi\). If \(E=L_ 1\) then this notion is equivalent to the convergence in probability. It is shown that \(TL(A_ nf)\) exists for each \(f\in E_+\) if E has a weak unit, is weakly sequentially complete, and satisfies an additional condition (C). In case of Orlicz spaces on a finite non-atomic measure space, the condition (C) is equivalent to the condition \(\Delta_ 2\).
Mathematics Subject Classification ID: 47A35
Mathematics Subject Classification ID: 47B60
Mathematics Subject Classification ID: 46B42
Mathematics Subject Classification ID: 46A40
zbMATH DE Number: 3951341
zbMATH Keywords: averages
zbMATH Keywords: convergent in probability
zbMATH Keywords: positive linear contractions
zbMATH Keywords: Banach lattice
zbMATH Keywords: existence of the truncated limit
zbMATH Keywords: weak unit
zbMATH Keywords: weakly sequentially complete
zbMATH Keywords: Orlicz spaces
zbMATH Keywords: condition (C)
zbMATH Keywords: condition \(\Delta _ 2\)
author: Mustafa Akcoglu
author: Louis Sucheston
reviewed by: Janusz Woś
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3951341