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

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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.

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Author information

Correspondence to José Barbosa Gomes.

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Gomes, J.B., Alves, M.B. On linear operators preserving certain cones in \(\mathbb {R}^n\). Positivity 24, 229–239 (2020). https://doi.org/10.1007/s11117-019-00675-3

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Keywords

  • Invariant cone
  • Positive eigenvalue
  • Permutation
  • Cycle

Mathematics Subject Classification

  • 15A18
  • 15A39
  • 52A20