Mean lower bounds for Markov operators
Volume 83 / 2004
Annales Polonici Mathematici 83 (2004), 11-19
MSC: 37A30, 47A35.
DOI: 10.4064/ap83-1-2
Abstract
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.
