Mean lower bounds for Markov operators

Eduard Emel'yanov¹ ; Manfred Wolff²

¹ Sobolev Institute of Mathematics Akad. Koptyug pr. 4 630090 Novosibirsk, Russia
² Mathematisches Institut Universität Tübingen Auf der Morgenstelle 10 D-72076 Tübingen, Germany

Annales Polonici Mathematici, Tome 83 (2004) no. 1, pp. 11-19.

Résumé

Let $T$ be a Markov operator on an $L^1$-space. We study conditions under which $T$ is mean ergodic and satisfies $\mathop {\rm dim}\nolimits \mathop {\rm Fix}\nolimits (T)\infty $. Among other things we prove that the sequence $(n^{-1}\sum _{k=0}^{n-1}T^k)_n$ converges strongly to a rank-one projection if and only if there exists a function $0\not =h\in L^1_+$ which satisfies $\mathop {\rm lim}_{n\to \infty }\| (h-n^{-1}\sum _{k=0}^{n-1}T^kf)_+\| =0$ for every density $f$. Analogous results for strongly continuous semigroups are given.

Détail

DOI : 10.4064/ap83-1-2

Keywords: markov operator space study conditions under which mean ergodic satisfies mathop dim nolimits mathop fix nolimits infty among other things prove sequence sum n converges strongly rank one projection only there exists function which satisfies mathop lim infty h n sum n every density analogous results strongly continuous semigroups given