Mean ergodic operators and reflexive Fréchet lattices

Journal article (2011)
Department
Delft Institute of Applied Mathematics (EEMCS) (TU Delft)
Copyright: © 2011 Royal Society of Edinburgh
DOI: https://doi.org/doi:10.1017/S0308210510000314
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Published Date

26-09-2011

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Source:

Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 141, 2011

Faculty

Electrical Engineering, Mathematics and Computer Science

Department

Delft Institute of Applied Mathematics

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 Fréchet lattices (or more general locally convex solid Riesz spaces) there is virtually no information available. For a Fréchet 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 ?-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?? (no longer a Fréchet lattice) is mean ergodic. An important technique is to develop criteria that detect when E admits a (positively) complemented lattice copy of c0, l1 or l?.

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