Extremality and factorizability of Markov operators

Abstract

© 2020 Elsevier Inc. We prove that the extreme Markov operators that are factorizable as a composition of left and right invertible Markov operators must be one-sided invertible. It clearly follows that Markov operators, i.e. their corresponding doubly stochastic measures, with hairpin support are either invertible or non-factorizable. An alternative and more direct proof of this fact for Markov operators with full hairpin support is also given. We also demonstrate that there exists extreme, hence singular, Markov operator that is totally atomic.