## Operators with finite chain length and the ergodic theorem

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- by K. B. Laursen and M. Mbekhta
- Proc. Amer. Math. Soc.
**123**(1995), 3443-3448 - DOI: https://doi.org/10.1090/S0002-9939-1995-1277123-9
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## Abstract:

With a technical assumption (*E-k*), which is a relaxed version of the condition ${T^n}/n \to 0,n \to \infty$, where

*T*is a bounded linear operator on a Banach space, we prove a generalized uniform ergodic theorem which shows,

*inter alias*, the equivalence of the finite chain length condition $(X = {(I - T)^k}X \oplus \ker {(I - T)^k})$, of closedness of ${(I - T)^k}X$, and of quasi-Fredholmness of $I - T$. One consequence, still assuming (

*E-k*), is that $I - T$ is semi-Fredholm if and only if $I - T$ is Riesz-Schauder. Other consequences are: a uniform ergodic theorem and conditions for ergodicity for certain classes of multipliers on commutative semisimple Banach algebras.

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## Bibliographic Information

- © Copyright 1995 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**123**(1995), 3443-3448 - MSC: Primary 47A35; Secondary 46J20, 47A53, 47B06
- DOI: https://doi.org/10.1090/S0002-9939-1995-1277123-9
- MathSciNet review: 1277123