, Volume 24, Issue 1, pp 229–239 | Cite as

On linear operators preserving certain cones in \(\mathbb {R}^n\)

  • José Barbosa GomesEmail author
  • Magno Branco Alves


A distinguished class of polyhedral cones is considered. For a linear operator \(\mathcal {L}\) preserving a cone in this class, we prove, under some assumption on the number of edges of the cone, that its spectrum contains exactly one strictly positive real eigenvalue. As an application, we characterize those \(\mathcal {L}\) that acts as a transposition on the edges of this cone. As other application, we show that if \(\mathcal {L}\) fixes the edges of an \((n-1)\)-face of this cone then \(\mathcal {L}\) must be a positive multiple of the identity map.


Invariant cone Positive eigenvalue Permutation Cycle 

Mathematics Subject Classification

15A18 15A39 52A20 



  1. 1.
    Aliprantis, C.D., Tourky, R.: Cones and Duality. American Mathematical Society, Providence (2007)CrossRefGoogle Scholar
  2. 2.
    Berman, A., Plemmons, R.J.: Nonnegative Matrices in the Mathematical Sciences. Academic Press, London (1979)zbMATHGoogle Scholar
  3. 3.
    Frobenius, G.: Über Matrizen aus nicht negativen Elementen. Sitzungsberichte Preussische Akademie der Wissenschaft. Berlin, 456–477 (1912)Google Scholar
  4. 4.
    Greer, R.: Trees and Hills: Methodology for Maximizing Functions of Systems of Linear Relations. Elsevier, Amsterdam (1984)zbMATHGoogle Scholar
  5. 5.
    Kreĭn, M.G., Rutman, M.A.: Linear operators leaving invariant a cone in a Banach space. Amer. Math. Soc. Translation 26, (1950). Translated from Uspehi Matem. Nauk (N.S.) 3, 3–95 (1948)Google Scholar
  6. 6.
    Lemmens, B., Nussbaum, R.: Nonlinear Perron–Frobenius Theory. Cambridge University Press, Cambridge (2012)CrossRefGoogle Scholar
  7. 7.
    Perron, O.: Zur Theorie der Matrices. Math. Ann. 64, 248–263 (1907)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Rockafellar, R.T.: Princeton University Press. Convex Analysis, Princeton (1970)Google Scholar
  9. 9.
    Schneider, H., Tam, B.-S.: On the core of a cone-preserving map. Trans. Am. Math. Soc. 343, 479–524 (1994)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Schrijver, A.: Theory of Linear and Integer Programming. Wiley, West Sussex (1998)zbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsFederal University of Juiz de ForaJuiz de ForaBrazil

Personalised recommendations