$L^2$-Quasi-compact and hyperbounded Markov operators

Abstract

A Markov operator $P$ on a probability space $(S,\Sigma,\mu)$, with $\mu$ invariant, is called {\it hyperbounded} if for some $1 \le p<q \le \infty$ it maps (continuously) $L^p$ into $L^q$. We deduce from a recent result of Gl\"uck that a hyperbounded $P$ is quasi-compact, hence uniformly ergodic, in all $L^r(S,\mu)$, $1<r< \infty$. We prove, using a method similar to Foguel's, that a hyperbounded Markov operator has periodic behavior similar to that of Harris recurrent operators, and for the ergodic case obtain conditions for aperiodicity. Given a probability $\nu$ on the unit circle, we prove that if the convolution operator $P_\nu f:=\nu*f$ is hyperbounded, then $\nu$ is atomless. We show that there is $\nu $ absolutely continuous such that $P_\nu$ is not hyperbounded, and there is $\nu$ with all powers singular such that $P_\nu$ is hyperbounded. As an application, we prove that if $P_\nu$ is hyperbounded, then for any sequence $(n_k)$ of distinct positive integers with bounded gaps, $(n_kx)$ is uniformly distributed mod 1 for $\nu$ almost every $x$ (even when $\nu$ is singular).