L2-Quasi-compact and hyperbounded Markov operators

Abstract A 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) L p into L q . We deduce from a recent result of Glück that a hyperbounded P is quasi-compact, hence uniformly ergodic, in all L r ( 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 P ν f := ν ⋇ f is hyperbounded, then ν is atomless. We show that there is ν absolutely continuous such that P ν is not hyperbounded, and there is ν with all powers singular such that P ν is hyperbounded. As an application, we prove that if P ν is hyperbounded, then for any sequence ( n k ) of distinct positive integers with bounded gaps, ( n k x ) is uniformly distributed mod 1 for ν almost every x (even when ν is singular).