Asymptotic periodicity of the iterates of Markov operators

Abstract

We say P : L 1 → L 1 P:{L^1} \to {L^1} is a Markov operator if (i) P f ≥ 0 Pf \geq 0 for f ≥ 0 f \geq 0 and (ii) ‖ P f ‖ = ‖ f ‖ \| Pf\| = \| f\| if f ≥ 0 f \geq 0 . It is shown that any Markov operator P P has certain spectral decomposition if, for any f ∈ L 1 f \in {L^1} with f ≥ 0 f \geq 0 and ‖ f ‖ = 1 \| f\| = 1 , P n f → F {P^n}f \to \mathcal {F} when n → ∞ n \to \infty , where F \mathcal {F} is a strongly compact subset of L 1 {L^1} . It follows from this decomposition that P n f {P^n}f is asymptotically periodic for any f ∈ L 1 f \in {L^1} .