L2-Quasi-compact and hyperbounded Markov operators

Abstract

AbstractA Markov operator P on a probability space (S, Σ μ) with μ invariant, is called hyperbounded if for some 1 ≤ p≤ q ≤ ∞ it maps (continuously) Lp into Lq.We deduce from a recent result of Glück that a hyperbounded P is quasi-compact, hence uniformly ergodic, in all Lr(S, μ), 1 < r < ∞. 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 ν on the unit circle, we prove that if the convolution operator Pνf:= ν ⋇ f is hyperbounded, then ν is atomless. We show that there is ν absolutely continuous such that Pν is not hyperbounded, and there is ν with all powers singular such that Pν is hyperbounded. As an application, we prove that if Pν is hyperbounded, then for any sequence (nk) of distinct positive integers with bounded gaps, (nkx) is uniformly distributed mod 1 for ν almost every x (even when ν is singular).