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 → 0 , n → ∞ {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 ⊕ ker ⁡ ( I − T ) k ) (X = {(I - T)^k}X \oplus \ker {(I - T)^k}) , of closedness of ( I − T ) k X {(I - T)^k}X , and of quasi-Fredholmness of I − T I - T . One consequence, still assuming (E-k), is that I − T I - T is semi-Fredholm if and only if I − T 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.