Suppose that K is a closed, total cone in a real Banach space X, that A[ratio ]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[ges ]1 with Am(x0) =rmx0 (or, more generally, that there exist x0∉(−K) and m[ges ]1 with Am(x0) [ges ]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 θ∈K′−{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 x∈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 self-adjoint; the theorems in the paper generalize several of Toland's propositions.