Self‐Adjoint Operators and Cones

Suppose that K is a cone in a real Hilbert space ℋ with K ⊥ = {0}, and that A : ℋ → ℋ 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 ⊥ = {0}.