Convergence of arithmetic means of operators in Fréchet spaces

Abstract

Every Köthe echelon Fréchet space XX that is Montel and not isomorphic to a countable product of copies of the scalar field admits a power bounded continuous linear operator TT such that I−TI−T does not have closed range, but the sequence of arithmetic means of the iterates of TT converges to 0 uniformly on the bounded sets in XX. On the other hand, if XX is a Fréchet space which does not have a quotient isomorphic to a nuclear Köthe echelon space with a continuous norm, then the sequence of arithmetic means of the iterates of any continuous linear operator TT (for which (1/n)Tn(1/n)Tn converges to 0 on the bounded sets) converges uniformly on the bounded subsets of XX, i.e., TT is uniformly mean ergodic, if and only if the range of I−TI−T is closed. This result extends a theorem due to Lin for such operators on Banach spaces. The connection of Browder’s equality for power bounded operators on Fréchet spaces to their uniform mean ergodicity is exposed. An analysis of the mean ergodic properties of the classical Cesàro operator on Banach sequence spaces is also given. © 2012 Elsevier Ltd. All rights reserved.