Bartoszek, Wojciech On asymptotic cyclicity of doubly stochastic operators. (English) Zbl 0957.47011 Ann. Pol. Math. 72, No. 2, 145-152 (1999). Let \((X,{\mathcal A},\mu)\) be a complete \(\sigma\)-finite measure space. An operator \(P: L^1(\mu)\to L^1(\mu)\) is a Markov operator if \(Pf\geq 0\) and \(\|Pf\|_1=\|f\|_1\) for all \(f\in L^1(\mu)\) with \(f\geq 0\). If further \(\mu(x)= 1\) and \(P1_X= 1_X\), then \(P\) is called doubly stochastic. This paper deals with several notions of support overlapping, and of asymptotic cyclocity of doubly stochastic operators. The author obtains a result that states that a doubly stochastic operator with almost overlapping support is weakly asymptotically cyclic, and a result that gives necessary and sufficient conditions for a doubly stochastic operator with almost overlapping support to be strongly asymptotically cyclic. Reviewer: Karma Dajani (Utrecht) Cited in 1 Document MSC: 47A35 Ergodic theory of linear operators 47B38 Linear operators on function spaces (general) Keywords:Markov operator; asymptotic cyclocity; doubly stochastic operators; weakly asymptotically cyclic; almost overlapping support PDFBibTeX XMLCite \textit{W. Bartoszek}, Ann. Pol. Math. 72, No. 2, 145--152 (1999; Zbl 0957.47011) Full Text: DOI