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Reflexive cones

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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.

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Acknowledgments

The authors thank very much an anonymous referee for the bibliographical and structural remarks which helped us to improve our article.

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Authors and Affiliations

  1. Dipartimento di Scienza e Alta Tecnologia, Università dell’Insubria, via Valleggio, 11, 22100 , Como, Italy

    E. Casini

  2. Dipartimento di Discipline Matematiche, Finanza Matematica ed Econometria, Università Cattolica del Sacro Cuore, via Necchi 9, 20123 , Milan, Italy

    E. Miglierina

  3. Department of Mathematics, National Technical University of Athens, Zografou, 15780 , Athens, Greece

    I. A. Polyrakis & F. Xanthos

Authors
  1. E. Casini
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  2. E. Miglierina
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  3. I. A. Polyrakis
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  4. F. Xanthos
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Corresponding author

Correspondence to I. A. Polyrakis.

Additional information

I.A. Polyrakis and F. Xanthos have been co-financed by the European Union (European Social Fund, ESF) and Greek national funds through the Operational Program “Education and Lifelong Learning” of the National Strategic Reference Framework (NSRF)—Research Funding Program: Heracleitus II. Investing in Knowledge society through the European Social Fund.

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Casini, E., Miglierina, E., Polyrakis, I.A. et al. Reflexive cones. Positivity 17, 911–933 (2013). https://doi.org/10.1007/s11117-012-0212-6

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