On the invariant faces associated with a cone-preserving map
| dc.creator | Tam, Bit-Shun | |
| dc.creator | Schneider, Hans | |
| dc.date | 1998-02-22 | |
| dc.date.accessioned | 2026-07-07T05:23:55Z | |
| dc.date.available | 2026-07-07T05:23:55Z | |
| dc.description | For an $n \times n$ 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 $\IR^{n}_{+}$. 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 $λ$ of $A$ for $K$, there is a chain of $m_λ$ 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 $m_λ$ is the maximal order of distinguished generalized eigenvectors of $A$ corresponding to $λ$, but there is no such chain with more than $m_λ$ 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. | |
| dc.identifier | https://arxiv.org/abs/math/9802107 | |
| dc.identifier | http://arxiv.org/abs/math/9802107 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/76638 | |
| dc.subject | Rings and Algebras | |
| dc.subject | Operator Algebras | |
| dc.subject | 15A48; 15A04; 47B65; 47B60 | |
| dc.title | On the invariant faces associated with a cone-preserving map | |
| dc.type | text |