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On asymptotic cyclicity of doubly stochastic operators. (English) Zbl 0957.47011

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.

MSC:

47A35 Ergodic theory of linear operators
47B38 Linear operators on function spaces (general)
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