Characterizing Fréchet–Schwartz spaces via power bounded operators

Angela A. Albanese ¹   ; José Bonet ²   ; Werner J. Ricker ³

¹ Dipartimento di Matematica e Fisica “E. De Giorgi” Università del Salento C.P. 193 I-73100 Lecce, Italy
² Instituto Universitario de Matemática Pura y Aplicada IUMPA Universidad Politécnica de Valencia E-46071 Valencia, Spain
³ Math.-Geogr. Fakultät Katholische Universität Eichstätt-Ingolstadt D-85072 Eichstätt, Germany

Studia Mathematica, Tome 224 (2014) no. 1, pp. 25-45

Résumé

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 Fréchet–Schwartz 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.

Détail

DOI : 10.4064/sm224-1-2

Keywords: characterize echelon spaces generally those chet spaces unconditional basis which schwartz terms convergence ces means power bounded operators defined complements similar known characterizations reflexive chet montel spaces basis every strongly convergent sequence continuous linear operators chet schwartz space does special single out type rapid convergence sequence operators study its relationship structure underlying space its relevance schauder decompositions connection mean ergodic operators chet schwartz spaces investigated