## Abstract

Suppose that K is a cone in a real Hilbert space script K with K^{1} = {0}, and that A : script K → script K is a self-adjoint operator which maps K into itself. If ∥A∥ is an eigenvalue of A, it is shown that it has an eigenvector in the cone. As a corollary, it follows that if ∥A∥^{n} is an eigenvalue of A^{n}, then ∥A∥ is an eigenvalue of A which has an eigenvector in K. The role of the support-boundary of K in the simplicity of the principal eigenvalue ∥A∥ is investigated. If H is a separable Hilbert space, it is shown that ∥A∥ ∈ σ(A); that is, the spectral radius of A lies in the spectrum of A. When A is compact, we obtain a very elementary proof of the Krein-Rutman Theorem in the self-adjoint case without assuming that K^{1} = {0}.

Original language | English |
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Pages (from-to) | 167-183 |

Number of pages | 17 |

Journal | Journal of the London Mathematical Society |

Volume | 53 |

Issue number | 1 |

DOIs | |

Publication status | Published - Feb 1996 |

## ASJC Scopus subject areas

- General Mathematics