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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 ${A}^{\text{'}}$ 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 ${A}^{m}\left({x}_{0}\right)={r}^{m}{x}_{0}$ (or, more generally, that there exist ${x}_{0}\notin \left(-K\right)$ and $m\ge 1$ with ${A}^{m}\left({x}_{0}\right)\ge {r}^{m}{x}_{0}$). If, in addition, $A$ satisfies some hypotheses of a type used in mean ergodic theorems, it is proved that there exist $u\in K-\left\{0\right\}$ and $\theta \in {K}^{\text{'}}-\left\{0\right\}$ with $A\left(u\right)=ru$, ${A}^{\text{'}}\left(\theta \right)=r\theta$ and $\theta \left(u\right)>0$. 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 $dim\left(X\right)>1$, then there exists an $x\in K-\left\{0\right\}$ 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:
 47B65 Positive and order bounded operators 47A75 Eigenvalue problems (linear operators) 46B40 Ordered normed spaces