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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 $A: X\to X$ 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 $x_0\ne 0$ and $m\ge 1$ with $A^m(x_0)= r^m x_0$ (or, more generally, that there exist $x_0\not\in (-K)$ and $m\ge 1$ with $A^m(x_0)\ge r^m x_0$). If, in addition, $A$ satisfies some hypotheses of a type used in mean ergodic theorems, it is proved that there exist $u\in K- \{0\}$ and $\theta\in K'- \{0\}$ with $A(u)= ru$, $A'(\theta)= r\theta$ and $\theta(u)> 0$. 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 $x\in K-\{0\}$ 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.

47B65Positive and order bounded operators
47A75Eigenvalue problems (linear operators)
46B40Ordered normed spaces
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