Eigenvectors of Order-preserving Linear Operators

Suppose that K is a closed, total cone in a real Banach space X, that A :X!X is a bounded linear operator which maps K into itself, and that A« denotes the Banach space adjoint of A. Assume that r, the spectral radius of A, is positive, and that there exist x ! 1 0 and m& 1 with Am(x ! ) ̄ rmx ! (or, more generally, that there exist x ! a (®K ) and m& 1 with Am(x ! )& rmx ! ). If, in addition, A satisfies some hypotheses of a type used in mean ergodic theorems, it is proved that there exist u `K®20 ́ and θ `K «®20 ́ with A(u) ̄ ru, A«(θ) ̄ rθ and θ(u)" 0. The support boundary of K is used to discuss the algebraic simplicity of the eigenvalue r. The relation of the support boundary to H. Schaefer’s ideas of quasi-interior elements of K and irreducible operators A is treated, and it is noted that, if dim(X )" 1, then there exists an x `K®20 ́ which is not a quasi-interior point. The motivation for the results is recent work of Toland, who considered the case in which X is a Hilbert space and A is self-adjoint ; the theorems in the paper generalize several of Toland’s propositions.