Uniform mean ergodicity of $C_0$-semigroups\newline in a class of Fréchet spaces

Dissipative operators and additive perturbations in locally convex spaces

“…So, there is an interest to understand what is possible to extend from the Banach space setting to this more general setting. This analysis has been carried out in recent years; see, for instance, –, , , –, and the references therein. For instance, Domański and Langenbruch introduce a general notion of resolvent for operators on locally convex spaces to provide a complete solution of the abstract Cauchy problem for operator valued Laplace distributions or hyperfunctions on complete ultrabornological locally convex spaces.…”

“…In a recent paper Jacob, Wegner and Wintermayr prove a Desch–Schappacher perturbation theorem for locally equicontinuous C 0 ‐semigroups of continuous linear operators acting on sequentially complete locally convex spaces. On the other hand, Albanese, Bonet and Ricker show that the uniform mean ergodicity of a C 0 ‐semigroup on a locally convex Hausdorff space X is equivalent to the closedness of the range of its infinitesimal generator whenever X belongs to a class of Fréchet spaces, the so‐called quojection Fréchet spaces, as it happens in Banach space case. They show also that this characterization fails in the setting of general Fréchet spaces; see also for more information.…”