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Abstract:

1. We study Markov operators $T$, $A$, and $T^*$ generated by the classical transition function of general Markov chains on an arbitrary measurable space. The operator $T$ is defined on the Banach space of all bounded measurable functions. The operator $A$ is defined on the Banach space of all bounded countably additive measures. We construct an operator $T^*$, topologically conjugate to the operator $T$, acting in the space of all bounded finitely additive measures.

2. Sequences of Cesaro means of powers of Markov operators on the set of finitely additive probability measures are studied.

3. Our basic Theorem proves that the set of all limit measures (points) of such sequences in the weak topology generated by the predual space is nonempty, weakly compact, and all of them are invariant for this operator.

4. The theorem is proved that the well-known Deblin condition $(D)$ for the ergodicity of a Markov chain is equivalent to the condition $(*)$: all invariant finitely additive measures of a Markov chain are countably additive, i.e., there are no invariant purely finitely additive measures.

5. The main result is proved that, in the general case, a Markov operator $T^*$ is quasi-compact if and only if the operator $T$ is quasi-compact.

6. From the above theorems we obtain that the conjugate operator $T^*$ is quasi-compact if and only if the Deblin condition $(D)$ is satisfied.

7. It is shown that the quasi-compactness conditions for all three Markov operators $T$, $A$, and $T^*$ are equivalent to each other (here, results of other authors are also used).

8. As a consequence, we obtain that, for an operator $T^*$ to be quasi-compact, it is necessary and sufficient that it does not have invariant purely finitely additive measures.

9. A strong uniform reversible ergodic theorem is proved for the quasi-compact Markov operator $T^*$ in the space of finitely additive measures.

10. All proofs are given for the most general case, and the general theorems obtained cannot be improved.

11. A detailed analysis of Michael Lin's counterexample is given.

Latest publications:

1. Zhdanok, A. Quasi-Compactness of Operators for General Markov Chains and Finitely Additive

Measures. Mathematics 2024, 12, 3155. https://doi.org/10.3390/math12193155

2. Zhdanok, A. Invariant Finitely Additive Measures for General Markov Chains and the Doeblin

Condition. Mathematics 2023, 11, 3388. https://doi.org/10.3390/math11153388