  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. $\|T^n-P\|\to 0$, here $P$ is a projection. We
have showed that $T$ is uniformly $P$-ergodic if and only if $\|T^n-P\|\leq
C\beta^n$, $0<\beta<1$. In this paper, we prove that such a $\beta$ 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.
