Spectral conditions for uniform $P$-ergodicities of Markov operators on abstract states spaces

In the present paper deals with asymptotical stability of Markov operators acting on abstract state spaces (i.e. an ordered Banach space, where the norm has an additivity property on the cone of positive elements). Basically, we are interested in the rate of convergence when a Markov operator $T$ satisfies the uniform $P$-ergodicity, i.e. $\mathrm{\parallel }{T}^n-P\mathrm{\parallel }\to 0$, here $P$ is a projection. We have showed that $T$ is uniformly $P$-ergodic if and only if $|T^n-P|\leq C\beta^n$,$0<\beta<1$\mathrm{.}Inthispaper,weprovethatsucha$\beta$ is characterized by the spectral radius of $T-P$. Moreover, we give Deoblin’s kind of conditions for the uniform $P$-ergodicity of Markov operators.