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Operators with finite chain length and the ergodic theorem

Authors: K. B. Laursen and M. Mbekhta
Journal: Proc. Amer. Math. Soc. 123 (1995), 3443-3448
MSC: Primary 47A35; Secondary 46J20, 47A53, 47B06
MathSciNet review: 1277123
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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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