On the invariant faces associated with a cone-preserving map

Abstract

For an n×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 R+. 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 cone-preserving 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 λ ofA forK, there is a chain of mλ distinct A-invariant join-irreducible faces of K, each containing in its relative interior a generalized eigenvector of A corresponding to λ (referred to as semi-distinguished A-invariant faces associated with λ), where mλ is the maximal order of distinguished generalized eigenvectors of A corresponding to λ, but there is no such chain with more than mλ 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. 1991 Mathematics Subject Classification. Primary 15A48; Secondary 47B65, 47A25, 46B42.