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On the uniform ergodic theorem in Banach spaces that do not contain duals

Vladimir FonfMichael LinAlexander Rubinov — 1996

Studia Mathematica

Let T be a power-bounded linear operator in a real Banach space X. We study the equality (*) ( I - T ) X = z X : s u p n k = 0 n T k z < . For X separable, we show that if T satisfies and is not uniformly ergodic, then ( I - T ) X ¯ contains an isomorphic copy of an infinite-dimensional dual Banach space. Consequently, if X is separable and does not contain isomorphic copies of infinite-dimensional dual Banach spaces, then (*) is equivalent to uniform ergodicity. As an application, sufficient conditions for uniform ergodicity of irreducible Markov chains...

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