Positivity

, Volume 16, Issue 3, pp 543–563

New results on q-positivity

  • Y. García Ramos
  • J. E. Martínez-Legaz
  • S. Simons
Article

Abstract

In this paper we discuss symmetrically self-dual spaces, which are simply real vector spaces with a symmetric bilinear form. Certain subsets of the space will be called q-positive, where q is the quadratic form induced by the original bilinear form. The notion of q-positivity generalizes the classical notion of the monotonicity of a subset of a product of a Banach space and its dual. Maximal q-positivity then generalizes maximal monotonicity. We discuss concepts generalizing the representations of monotone sets by convex functions, as well as the number of maximally q -positive extensions of a q-positive set. We also discuss symmetrically self-dual Banach spaces, in which we add a Banach space structure, giving new characterizations of maximal q-positivity. The paper finishes with two new examples.

Keywords

q-Positive sets Symmetrically self-dual spaces Monotonicity Symmetrically self-dual Banach spaces Lipschitz mappings 

Mathematics Subject Classification

47H05 47N10 46N10 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bauschke H.H., Wang X., Yao L.: Monotone linear relations: maximality and fitzpatrick functions. J. Convex Anal. 16(3–4), 673–686 (2009)MathSciNetMATHGoogle Scholar
  2. 2.
    Fitzpatrick, S.: Representing monotone operators by convex functions. In: Workshop/Miniconference on Functional Analysis and Optimization, Canberra 1988, Proceedings of the Centre for Mathematical Analysis, vol. 20, pp. 59–65, Australian National University, Canberra (1988)Google Scholar
  3. 3.
    Kirszbraun M.D.: Uber die zusammenziehende und Lipschitzsche Transformationen. Fund. Math. 22, 77–108 (1934)Google Scholar
  4. 4.
    Marques Alves M., Svaiter B.F.: Maximal monotone operators with a unique extension to the bidual. J. Convex Anal. 16(2), 409–421 (2009)MathSciNetMATHGoogle Scholar
  5. 5.
    Martínez-Legaz J.E.: On maximally q -positive sets. J. Convex Anal. 16(3–4), 891– (2009)MathSciNetMATHGoogle Scholar
  6. 6.
    Martínez-Legaz J.E., Svaiter B.F.: Minimal convex functions bounded below by the duality product. Proc. Am. Math. Soc. 136(3), 873– (2008)MATHCrossRefGoogle Scholar
  7. 7.
    Simons S.: The range of a monotone operator. J. Math. Anal. Appl. 199(1), 176–201 (1996)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Simons S.: Positive sets and monotone sets. J. Convex Anal. 14(2), 297–317 (2007)MathSciNetMATHGoogle Scholar
  9. 9.
    Simons S.: From Hahn–Banach to monotonicity, Lecture Notes in Mathematics, vol. 1693. Springer, New York (2008)Google Scholar
  10. 10.
    Simons S.: Banach SSD spaces and classes of monotone sets. J. Convex Anal. 18(1), 227–258 (2011)MathSciNetMATHGoogle Scholar
  11. 11.
    Valentine F.A.: A Lipschitz condition preserving extension for a vector function. Am. J. Math. 67(1), 83–93 (1945)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Basel AG 2012

Authors and Affiliations

  • Y. García Ramos
    • 1
  • J. E. Martínez-Legaz
    • 2
  • S. Simons
    • 3
  1. 1.Centro de Investigación de la Universidad del PacíficoLimaPeru
  2. 2.Universitat Autònoma de BarcelonaBarcelonaSpain
  3. 3.University of CaliforniaSanta BarbaraUSA

Personalised recommendations