Ergodic theorem for positive operators on \(L_p\)-spaces \((1<p<\infty)\) revisited
type: Item
instance of: scholarly article
title: Ergodic theorem for positive operators on \(L_p\)-spaces \((1<p<\infty)\) revisited
zbMATH Open document ID: 1010.47008
DOI: 10.1016/S1631-073X(02)02246-X
author: Antoine Brunel
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2002-08-04T00:00:00Z
review text: Let \((\Omega, \Sigma, \mu)\) be a probability space and let \(1<p<\infty.\) A positive linear operator \(A\) on \(L^p(\Omega,\Sigma, \mu)\) is said to be subadditive with respect to powers (s.r.p.) if \(A^{m+n} <A^m +A^n\) for all \((m,n)\in {\mathbb N}^2\), where \(A^0=I.\) The author proves the following structure theorem for these operators. Let \(A\) be a positive linear s.r.p. operator on \(L^p(\Omega, \Sigma, \mu)\), then there exists a partition \(\Omega_1,\Omega_2,\Omega_3\) of \(\Omega\) into disjoint subsets such that \(A\) can be written as \[ A=(E_1+E_2+E_1AE_2)+(E_3AE_3), \] where \(E_if=f\cdot I_{\Omega_i}.\) As an application of this structure theorem, the author proves the following result. Let \(A\) be a positive linear s.r.p. operator on \(L^p(\Omega, \Sigma, \mu)\). If \(A^n/n\) converges to \(0\), then \(A^n\) is bounded for all \(n\), and for all \(f\in L^p(\Omega, \Sigma, \mu)\) the sequence \(A^nf\) converges almost everywhere.
reviewed by: K. Dajani
Mathematics Subject Classification ID: 47A35
Mathematics Subject Classification ID: 47A60
zbMATH DE Number: 1744379
zbMATH Keywords: subadditive with respect to powers
MaRDI profile type: Publication
cites work: Théorème ergodique pour les opérateurs positifs à moyennes bornées sur les espaces Lp(1 < p < ∞)
cites work: Q3704477
cites work: Mean-bounded operators and mean ergodic theorems
cites work: Ergodic theorems. With a supplement by Antoine Brunel
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description: scientific article; zbMATH DE number 1744379