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Title

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

Authors 1, 1, 1

Affiliations

  1. Department of Mathematics and Computer Science, Ben-Gurion University of the Negev, Beer-Sheva, Israel

Abstract

Let T be a power-bounded linear operator in a real Banach space X. We study the equality (*) (I-T)X={z∈X:⊃n∥∑k=0nTkz∥<∞}. 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 on the (positive) integers are obtained.

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Studia Mathematica
1996/121/1
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Pages:
67-85
Main language of publication
English
Received
1996-02-26
Accepted
1996-04-26
Published
1996
Exact and natural sciences