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Self‐Adjoint Operators and Cones
Author(s) -
Toland J. F.
Publication year - 1996
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/53.1.167
Subject(s) - eigenvalues and eigenvectors , mathematics , hilbert space , self adjoint operator , cone (formal languages) , spectrum (functional analysis) , corollary , operator (biology) , mathematical analysis , boundary (topology) , pure mathematics , separable space , space (punctuation) , physics , quantum mechanics , biochemistry , chemistry , linguistics , philosophy , algorithm , repressor , transcription factor , gene
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}.