In this paper the stability and the perturbation bounds of Markov operators acting on abstract state spaces are investigated. Here, an abstract state space is an ordered Banach space where the norm has an additivity property on the cone of positive elements. We basically study uniform ergodic properties of Markov operators by means of so-called a generalized Dobrushin’s ergodicity coefficient. This allows us to get several convergence results with rates. Some results on quasi-compactness of Markov operators are proved in terms of the ergodicity coefficient. Furthermore, a characterization of uniformly P-ergodic Markov operators is given which enable us to construct plenty examples of such types of operators. The uniform mean ergodicity of Markov operators is established in terms of the Dobrushin ergodicity coefficient. The obtained results are even new in the classical and quantum settings.

中文翻译：

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马尔可夫算子的广义Dobrushin遍历系数和均匀遍历

本文研究了作用在抽象状态空间上的马尔可夫算子的稳定性和摄动界。在这里，抽象状态空间是有序的Banach空间，其中范数在正元素的锥面上具有可加性。我们基本上通过所谓的广义Dobrushin遍历系数研究Markov算子的一致遍历性质。这使我们可以获得速率的几种收敛结果。根据遍历系数，证明了有关Markov算子的拟紧性的一些结果。此外，均匀P的特征给出了遍历马尔可夫算子，这使我们能够构造出这类算子的大量示例。根据Dobrushin遍历系数来建立Markov算子的均匀平均遍历度。所获得的结果甚至在经典和量子设置中都是新的。