OPERATORS WITH FINITE CHAIN-LENGTH AND THE ERGODIC THEOREM

With a technical assumption (E-k), which is a relaxed version of the c ondition T-n/n --> 0, n --> infinity, where T is a bounded linear oper ator 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)), of closedness of (I - T)( k)X, and of quasi-Fredholmness of I - T. One consequence, still assumi ng (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 condi tions for ergodicity for certain classes of multipliers on commutative semisimple Banach algebras.