On the invariant faces associated with a cone-preserving map

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 R-n(+). 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 lambda of A for K, there is a chain of m(lambda) distinct A-invariant join-irreducible faces of K, each containing in its relative interior a generalized eigenvector of A corresponding to lambda (referred to as semi-distinguished A-invariant faces associated with lambda), where m(lambda) is the maximal order of distinguished generalized eigenvectors of A corresponding to lambda, but there is no such chain with more than m(lambda) 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.