© 1998 by London Mathematical Society
© The London Mathematical Society
Eigenvectors of Order-Preserving Linear Operators
Mathematics Department, Rutgers University New Brunswick, NJ 08903, USA. E-mail: nussbaum{at}math.rutgers.edu
Received 17 October 1995. Revision received 10 March 1996.
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 x0
0 and m
1 with Am(x0)=rmx0 (or, more generally, that there exist x0
(–K) and m1 with Am(x0)rmx0). If, in addition, A satisfies some hypotheses of a type used in mean ergodic theorems, it is proved that there exist u
K–{0} and 
K'–{0} with A(u)=ru, A'()=r and (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 self-adjoint; the theorems in the paper generalize several of Toland's propositions.