Mean lower bounds for Markov operators

Volume 83 / 2004

Eduard Emel'yanov, Manfred Wolff Annales Polonici Mathematici 83 (2004), 11-19 MSC: 37A30, 47A35. DOI: 10.4064/ap83-1-2


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.


  • Eduard Emel'yanovSobolev Institute of Mathematics
    Akad. Koptyug pr. 4
    630090 Novosibirsk, Russia
  • Manfred WolffMathematisches Institut
    Universität Tübingen
    Auf der Morgenstelle 10
    D-72076 Tübingen, Germany

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