Abstract
We prove that a Markov operatorT onL1 has an invariant density if and only if there exists a densityf that satisfies lim supn→∞‖Tnf − f‖ < 2. Using this result, we show that a Frobenius-Perron operatorP is mean ergodic if and only if there exists a densityw such that lim supn→∞ ‖Pnf − w‖<2 for every densityf. Corresponding results hold for strongly continuous semigroups.
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