Eigenvectors of order-preserving linear operators. (English) Zbl 0968.47010
Summary: Suppose that is a closed, total cone in a real Banach space , that is a bounded linear operator which maps into itself, and that denotes the Banach space adjoint of . Assume that , the spectral radius of , is positive, and that there exist and with (or, more generally, that there exist and with ). If, in addition, satisfies some hypotheses of a type used in mean ergodic theorems, it is proved that there exist and with , and . The support boundary of is used to discuss the algebraic simplicity of the eigenvalue . The relation of the support boundary to H. Schaefer’s ideas of quasi-interior elements of and irreducible operators is treated, and it is noted that, if , then there exists an which is not a quasi-interior point. The motivation for the results is recent work of Toland, who considered the case in which is a Hilbert space and is selfadjoint; the theorems in the paper generalize several of Toland’s propositions.
|47B65||Positive and order bounded operators|
|47A75||Eigenvalue problems (linear operators)|
|46B40||Ordered normed spaces|