Characterizing Fréchet-Schwartz spaces via power bounded operators

[EN] We characterize Köthe echelon spaces (and, more generally, those Fréchet spaces with an unconditional basis) which are Schwartz, in terms of the convergence of the Cesàro means of power bounded operators defined on them. This complements similar known characterizations of reflexive and of Fréchet–Montel spaces with a basis. Every strongly convergent sequence of continuous linear operators on a FréchetSchwartz space does so in a special way. We single out this type of “rapid convergence” for a sequence of operators and study its relationship to the structure of the underlying space. Its relevance for Schauder decompositions and the connection to mean ergodic operators on Fréchet–Schwartz spaces is also investigated.The research of the first two authors was partially supported by the projects MTM2010-15200 and GVA Prometeo II/2013/013 (Spain). The second author gratefully acknowledges the support of the Alexander von Humboldt Foundation.Albanese, AA.; Bonet Solves, JA.; Ricker, WJ. (2014). Characterizing Fréchet-Schwartz spaces via power bounded operators. Studia Mathematica. 224(1):25-45. https://doi.org/10.4064/sm224-1-22545224