Self-adjoint operators and cones

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15 Citations (Scopus)

Abstract

Suppose that K is a cone in a real Hilbert space script K with K1 = {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 An, 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 K1 = {0}.

Original languageEnglish
Pages (from-to)167-183
Number of pages17
JournalJournal of the London Mathematical Society
Volume53
Issue number1
DOIs
Publication statusPublished - Feb 1996

ASJC Scopus subject areas

  • Mathematics(all)

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