Mean ergodic semigroups of operators (Q1935064)

The article under review deals with mean ergodicity of semigroups of operators. It is related to recent work on operator ergodic theory, especially to the papers [Ann. Acad. Sci. Fenn., Math. 34, No. 2, 401--436 (2009; Zbl 1194.47012)] and [J. Funct. Anal. 187, No. 1, 146--162 (2001; Zbl 1017.46007)]. The setting is rather general; the authors deal with locally convex Hausdorff spaces and ask which of the mean ergodic results on semigroups of operators in Banach spaces extend to their general setting. It turns out that, under the additional assumption that the space be sequentially complete, a fruitful theory can also be developed. The paper starts with an introduction followed by a section treating preliminaries, where the crucial definitions are given. In the third section, a~sequence of results on mean ergodic \(C_0\)-semigroups is presented. In particular, Theorem~5 characterizes the reflexivity of a complete, barrelled lcHs with a Schauder basis in terms of mean ergodicity of equicontinuous \(C_0\)-semigroups. In the last section, the authors assume additionally that the space under consideration is the so-called GDP-space, that is, Grothendieck space with the Dunford-Pettis property. In such a setting, they show when mean ergodicity of a \(C_0\)-semigroup implies mean ergodicity of a dual one. In several proofs, the authors use Riemann integrability of vector-valued functions, the basics of which have been collected in a separate appendix.