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Eigenvectors of order-preserving linear operators. (English) Zbl 0968.47010
Summary: Suppose that K is a closed, total cone in a real Banach space X, that A:XX 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 0 0 and m1 with A m (x 0 )=r m x 0 (or, more generally, that there exist x 0 (-K) and m1 with A m (x 0 )r m x 0 ). If, in addition, A satisfies some hypotheses of a type used in mean ergodic theorems, it is proved that there exist uK-{0} and θK ' -{0} 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 xK-{0} 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 selfadjoint; the theorems in the paper generalize several of Toland’s propositions.
MSC:
47B65Positive and order bounded operators
47A75Eigenvalue problems (linear operators)
46B40Ordered normed spaces