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# Wydawnictwa / Czasopisma IMPAN / Studia Mathematica / Wszystkie zeszyty

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

### Tom 121 / 1996

Studia Mathematica 121 (1996), 67-85 DOI: 10.4064/sm-121-1-67-85

#### Streszczenie

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.