On the core of a cone-preserving map

Abstract:

This is the third of a sequence of papers in an attempt to study the Perron-Frobenius theory of a nonnegative matrix and its generalizations from the cone-theoretic viewpoint. Our main object of interest here is the core of a cone-preserving map. If A is an $n \times n$ real matrix which leaves invariant a proper cone K in ${\mathbb {R}^n}$, then by the core of A relative to K, denoted by ${\text {core}}_K(A)$, we mean the convex cone $\bigcap \nolimits _{i = 1}^\infty {{A^i}K}$. It is shown that when ${\text {core}}_K(A)$ is polyhedral, which is the case whenever K is, then ${\text {core}}_K(A)$ is generated by the distinguished eigenvectors of positive powers of A. The important concept of a distinguished A-invariant face is introduced, which corresponds to the concept of a distinguished class in the nonnegative matrix case. We prove a significant theorem which describes a one-to-one correspondence between the distinguished A-invariant faces of K and the cycles of the permutation induced by A on the extreme rays of ${\text {core}}_K(A)$, provided that the latter cone is nonzero, simplicial. By an interplay between cone-theoretic and graph-theoretic ideas, the extreme rays of the core of a nonnegative matrix are fully described. Characterizations of K-irreducibility or A-primitivity of A are also found in terms of ${\text {core}}_K(A)$. Several equivalent conditions are also given on a matrix with an invariant proper cone so that its spectral radius is an eigenvalue of index one. An equivalent condition in terms of the peripheral spectrum is also found on a real matrix A with the Perron-Schaefer condition for which there exists a proper invariant cone K suchthat ${\text {core}}_K(A)$ is polyhedral, simplicial, or a single ray. A method of producing a large class of invariant proper cones for a matrix with the Perron-Schaefer condition is also offered.