Characterizing Fréchet–Schwartz spaces via power bounded operators

Tom 224 / 2014

Angela A. Albanese, José Bonet, Werner J. Ricker Studia Mathematica 224 (2014), 25-45 MSC: Primary 46A04; Secondary 46A11, 46A45, 47A35, 47B37. DOI: 10.4064/sm224-1-2


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.


  • Angela A. AlbaneseDipartimento di Matematica e Fisica “E. De Giorgi”
    Università del Salento
    C.P. 193
    I-73100 Lecce, Italy
  • José BonetInstituto Universitario de Matemática Pura y Aplicada IUMPA
    Universidad Politécnica de Valencia
    E-46071 Valencia, Spain
  • Werner J. RickerMath.-Geogr. Fakultät
    Katholische Universität Eichstätt-Ingolstadt
    D-85072 Eichstätt, Germany

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