Eigenvectors of Order-Preserving Linear Operators |
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Authors: | Nussbaum Roger D |
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Institution: | Mathematics Department, Rutgers University New Brunswick, NJ 08903, USA. E-mail: nussbaum{at}math.rutgers.edu |
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Abstract: | Suppose that K is a closed, total cone in a real Banach spaceX, that A:XX is a bounded linear operator which maps K intoitself, and that A' denotes the Banach space adjoint of A. Assumethat r, the spectral radius of A, is positive, and that thereexist x00 and m1 with Am(x0)=rmx0 (or, more generally, thatthere 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 uK{0} and K'{0}with A(u)=ru, A'()=r and (u)>0. The support boundary of Kis used to discuss the algebraic simplicity of the eigenvaluer. The relation of the support boundary to H. Schaefer's ideasof quasi-interior elements of K and irreducible operators Ais treated, and it is noted that, if dim(X)>1, then thereexists an xK{0} which is not a quasi-interior point.The motivation for the results is recent work of Toland, whoconsidered 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. |
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