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SPECTRAL CONDITIONS FOR UNIFORM P-ERGODICITIES OF MARKOV OPERATORS ON ABSTRACT STATES SPACES

Published online by Cambridge University Press:  23 September 2020

NAZIFE ERKURŞUN-ÖZCAN
Affiliation:
Department of Mathematics, Faculty of Science, Hacettepe University, Ankara06800, Turkey, e-mail: [email protected]
FARRUKH MUKHAMEDOV
Affiliation:
Department of Mathematical Sciences, College of Science, United Arab Emirates University, Al Ain, Abu Dhabi15551, United Arab Emirates, e-mails: [email protected]; [email protected]

Abstract

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 TP. Moreover, we give Deoblin’s kind of conditions for the uniform P-ergodicity of Markov operators.

Type
Research Article
Copyright
© The Author(s), 2020. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

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