Mean ergodic operators and reflexive Fréchet lattices

Abstract

Connections between (positive) mean ergodic operators acting in Banach lattices and properties of the underlying lattice itself are well understood (see the works of Emel'yanov, Wolff and Zaharopol). For Frechet lattices (or more general locally convex solid Riesz spaces) there is virtually no information available. For a Frechet lattice E, it is shown here that (amongst other things) every power-bounded linear operator on E is mean ergodic if and only if E is reflexive if and only if E. is Dedekind sigma-complete and every positive power-bounded operator on E is mean ergodic if and only if every positive power-bounded operator in the strong dual E'(beta) (no longer a Frechet lattice) is mean ergodic. An important technique is to develop criteria that detect when E admits a (positively) complemented lattice copy of c(0) l(1) or l(infinity). © Copyright © Royal Society of Edinburgh 2011.