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Positivity

, 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
Article
  • 45 Downloads

Abstract

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.

Keywords

Invariant cone Positive eigenvalue Permutation Cycle 

Mathematics Subject Classification

15A18 15A39 52A20 

Notes

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

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

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