Article

Positivity

, Volume 17, Issue 3, pp 911-933

Reflexive cones

  • E. CasiniAffiliated withDipartimento di Scienza e Alta Tecnologia, Università dell’Insubria
  • , E. MiglierinaAffiliated withDipartimento di Discipline Matematiche, Finanza Matematica ed Econometria, Università Cattolica del Sacro Cuore
  • , I. A. PolyrakisAffiliated withDepartment of Mathematics, National Technical University of Athens Email author 
  • , F. XanthosAffiliated withDepartment of Mathematics, National Technical University of Athens

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Abstract

Reflexive cones in Banach spaces are cones with weakly compact intersection with the unit ball. In this paper we study the structure of this class of cones. We investigate the relations between the notion of reflexive cones and the properties of their bases. This allows us to prove a characterization of reflexive cones in term of the absence of a subcone isomorphic to the positive cone of \(\ell _{1}\). Moreover, the properties of some specific classes of reflexive cones are investigated. Namely, we consider the reflexive cones such that the intersection with the unit ball is norm compact, those generated by a Schauder basis and the reflexive cones regarded as ordering cones in Banach spaces. Finally, it is worth to point out that a characterization of reflexive spaces and also of the spaces with the Schur property by the properties of reflexive cones is given.

Keywords

Cones Base for a cone Vector lattices Ordered Banach spaces Geometry of cones Weakly compact sets Reflexivity Positive Schauder bases

Mathematics Subject Classification (2010)

46B10 46B20 46B40 46B42