On Quasi-Compact Markov Operators

Abstract

Let $P$ be a conservative Markov operator on $L_\infty(X, \sum, m)$. The following conditions are proved to be equivalent: (i) $P$ is ergodic and quasi-compact. (ii) $P$ is ergodic and $(I - P)L_\infty$ is closed. (iii) For every $u \in L_1$ with $\int u dm = 0$ the sequence $\{\sum^N_{n=0} uP^n\}$ is weakly sequentially compact in $L_1$. (iv) There exists an invariant probability measure $\lambda \sim m$, and for every $u \in L_1$ with $\int u dm = 0, \sup_{N\geqq 0} \|\sum^N_{n=0} uP^n\|_1 < \infty$. These conditions are used to study the quasi-compactness of the induced operators $P_f = \sum^\infty_{n=0} (PT_{1-f})^nPT_f$ in the case that $P$ is Harris-recurrent: The following conditions are equivalent: (i) $P_f$ is quasi-compact. (ii) $f$ is "special" in the (modified) sense of Neveu. (iii) There exists a $\sigma$-finite measure $\lambda$, equivalent to $m$ on $A = \{f > 0\}$, with $\int f d\lambda < \infty$, and $\int fg d\lambda = 0$ implies $\sup_{N\geqq 0} \|\sum^N_{n=0} P^n(fg)\|_\infty < \infty$. Using this characterization certain limit theorems for $P$ are obtained.