A Few Remarks on Asymptotic Stabilities of Markov Operators on L¹-Spaces

Abstract: In this paper, the investigation of a generalized Dobrushin ergodicity coefficient to obtain uniform ergodicity and uniform mean ergodicities of positive contractions of L¹-spaces is carried out. Through the introduction of notions such as mean P-completely mixing and P-completely mixing, the last one being an extension of the complete mixing, several analogues of the Akcoglu and Sucheston theorem are proved. As an applications of these results, we establish mean ergodicities of positive contractions of L¹-spaces. It is vital to note that P stands for a Markov projection of L¹.