On the invariant faces associated with a cone-preserving map

doi:10.1090/S0002-9947-00-02597-6

機構典藏 > College of Sciences > Graduate Institute & Department of Mathematics > Journal Article > Item 987654321/41206

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Title:	On the invariant faces associated with a cone-preserving map
Authors:	譚必信;Tam, Bit-shun;Schneider, Hans
Contributors:	淡江大學數學學系
Date:	2001-01
Issue Date:	2010-01-28 06:57:13 (UTC+8)
Publisher:	Providence: American Mathematical Society (AMS)
Abstract:	For an n x n nonnegative matrix P, an isomorphism is obtained between the lattice of initial subsets (of {1, ,n}) for P and the lattice of P-invariant faces of the nonnegative orthant RI. Motivated by this isomorphism, we generalize some of the known combinatorial spectral results on a nonnegative matrix that are given in terms of its classes to results for a conepreserving map on a polyhedral cone, formulated in terms of its invariant faces. In particular, we obtain the following extension of the famous Rothblum index theorem for a nonnegative matrix: If A leaves invariant a polyhedral cone K, then for each distinguished eigenvalue A of A for K, there is a chain of m\ distinct A-invariant join-irreducible faces of K, each containing in its relative interior a generalized eigenvector of A corresponding to A (referred to as semidistinguished A-invariant faces associated with A), where mx is the maximal order of distinguished generalized eigenvectors of A corresponding to A, but there is no such chain with more than mx members. We introduce the important new concepts of semi-distinguished A-invariant faces, and of spectral pairs of faces associated with a cone-preserving map, and obtain several properties of a cone-preserving map that mostly involve these two concepts, when the underlying cone is polyhedral, perfect, or strictly convex and/or smooth, or is the cone of all real polynomials of degree not exceeding n that are nonnegative on a closed interval. Plentiful illustrative examples are provided. Some open problems are posed at the end.
Relation:	Transactions of the American Mathematical Society 353(1), pp.209-245
DOI:	10.1090/S0002-9947-00-02597-6
Appears in Collections:	[Graduate Institute & Department of Mathematics] Journal Article

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