Operators with finite chain length and the ergodic theorem

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.