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Generalized Dobrushin ergodicity coefficient and uniform ergodicities of Markov operators

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Abstract

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.

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Acknowledgements

The authors thanks Dr. Ho Hon Leung for his help in checking the text of this paper. The authors would like to thank an anonymous referee whose useful suggestions allowed us to improve the content of the paper.

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  1. Department of Mathematical Sciences, College of Science, United Arab Emirates University, Al-Ain, 15551, United Arab Emirates

    Farrukh Mukhamedov & Ahmed Al-Rawashdeh

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  1. Farrukh Mukhamedov
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  2. Ahmed Al-Rawashdeh
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Correspondence to Ahmed Al-Rawashdeh.

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Mukhamedov, F., Al-Rawashdeh, A. Generalized Dobrushin ergodicity coefficient and uniform ergodicities of Markov operators. Positivity 24, 855–890 (2020). https://doi.org/10.1007/s11117-019-00713-0

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