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

Studia Mathematica (1996)

• Volume: 121, Issue: 1, page 67-85
• ISSN: 0039-3223

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## Abstract

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Let T be a power-bounded linear operator in a real Banach space X. We study the equality (*) $\left(I-T\right)X=z\in X:su{p}_{n}\parallel {\sum }_{k=0}^{n}{T}^{k}z\parallel <\infty$. For X separable, we show that if T satisfies and is not uniformly ergodic, then $\overline{\left(I-T\right)X}$ 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.

## How to cite

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Fonf, Vladimir, Lin, Michael, and Rubinov, Alexander. "On the uniform ergodic theorem in Banach spaces that do not contain duals." Studia Mathematica 121.1 (1996): 67-85. <http://eudml.org/doc/216343>.

@article{Fonf1996,
abstract = {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.},
author = {Fonf, Vladimir, Lin, Michael, Rubinov, Alexander},
journal = {Studia Mathematica},
keywords = {power-bounded linear operator; uniformly ergodic; dual Banach spaces; irreducible Markov chains},
language = {eng},
number = {1},
pages = {67-85},
title = {On the uniform ergodic theorem in Banach spaces that do not contain duals},
url = {http://eudml.org/doc/216343},
volume = {121},
year = {1996},
}

TY - JOUR
AU - Lin, Michael
AU - Rubinov, Alexander
TI - On the uniform ergodic theorem in Banach spaces that do not contain duals
JO - Studia Mathematica
PY - 1996
VL - 121
IS - 1
SP - 67
EP - 85
AB - 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.
LA - eng
KW - power-bounded linear operator; uniformly ergodic; dual Banach spaces; irreducible Markov chains
UR - http://eudml.org/doc/216343
ER -

## References

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