Self-adjoint operators and cones

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}.