Asymptotic Periodicity of the Iterates of Markov Operators

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