Asymptotic periodicity of the iterates of Markov operators

Abstract:

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