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1996 | 121 | 1 | 67-85
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On the uniform ergodic theorem in Banach spaces that do not contain duals

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Let T be a power-bounded linear operator in a real Banach space X. We study the equality (*) $(I-T)X = {z ∈ X: sup_{n} ∥∑_{k=0}^{n} T^{k}z∥ < ∞}$. For X separable, we show that if T satisfies and is not uniformly ergodic, then $\overline{(I-T)X}$ 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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  • Department of Mathematics and Computer Science, Ben-Gurion University of the Negev, Beer-Sheva, Israel
  • Department of Mathematics and Computer Science, Ben-Gurion University of the Negev, Beer-Sheva, Israel
  • Department of Mathematics and Computer Science, Ben-Gurion University of the Negev, Beer-Sheva, Israel
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