In the present paper, we deal 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 β 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.

中文翻译：

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抽象状态空间上马尔可夫算子一致 P-遍历的谱条件

在本文中，我们处理作用于抽象状态空间（即有序巴拿赫空间，其中范数在正元素锥上具有可加性）的马尔可夫算子的渐近稳定性。基本上，我们对马尔可夫算子的收敛速度感兴趣吨满足制服磷-遍历性，即$\|T^nP\|\到 0$， 这里磷是一个投影。我们已经证明吨是一致的磷-遍历当且仅当$\|T^nP\|\leq C\beta^n$,$0<\beta<1$. 在本文中，我们证明了这样一个β的特点是光谱半径吨-磷. 此外，我们给出了 Deoblin 的制服条件磷-马尔可夫算子的遍历性。