Quasi-Compactness of Operators for General Markov Chains and Finitely Additive Measures

We study Markov operators T , A , and T * 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. We prove the main result of the paper that, in general, a Markov operator T * is quasi-compact if and only if T is quasi-compact. It is proved that the conjugate operator T * is quasi-compact if and only if the Doeblin condition ( D ) is satisfied. It is shown that the quasi-compactness conditions for all three Markov operators T , A , and T * are equivalent to each other. In addition, 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. A strong uniform reversible ergodic theorem is proved for the quasi-compact Markov operator T * in the space of finitely additive measures. We give all the proofs for the most general case. A detailed analysis of Lin’s counterexample is provided.