Uniform convergence of operators and Grothendieck spaces with the Dunford-Pettis property
A theorem of D. W. Dean states that all Grothendieck spaces with the Dunford-Pettis property fail to admit Schauder decompositions. Similar results of H. P. Lotz concern the uniform convergence of ergodic operators and semigroups of operators. This paper considers extensions of these results to wider classes of Banach spaces. We say that a Banach space E has the surjective Dunford-Pettis property if every operator from E onto a reflexive Banach space maps weakly compact sets onto norm compact sets. The main result can now be stated. Theorem. Let \(E\) be a Grothendieck space with the surjective Dunford- Pettis property, then (1) The space \(E\) admits no Schauder decompositions; (2) Every strongly ergodic operator \(T\) on \(E\) with \(\| T^ n/n\| \to 0\) is uniformly ergodic; and (3) Every strongly continuous semi-group of operators \((T_ t)_{t\geq 0}\) on E is uniformly continuous. Examples are presented to distinguish the surjective Dunford-Pettis property from the closely related Dunford- Pettis property.
Grothendieck spaces with the Dunford-Pettis property
uniform convergence of ergodic operators
scientific article; zbMATH DE number 4002803