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Journal of the London Mathematical Society 1996 53(1):167-183; doi:10.1112/jlms/53.1.167
© 1996 by London Mathematical Society
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© The London Mathematical Society

Self-Adjoint Operators and Cones

J. F. Toland

School of Mathematical Sciences, University of Bath Bath BA2 7AY

Received 8 June 1994.

Suppose that K is a cone in a real Hilbert space H with K{perp} = {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 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|| isin{sigma}(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{perp} = {0}.


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