## On the core of a cone-preserving map

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- by Bit Shun Tam and Hans Schneider
- Trans. Amer. Math. Soc.
**343**(1994), 479-524 - DOI: https://doi.org/10.1090/S0002-9947-1994-1242787-6
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## 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.

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## Bibliographic Information

- © Copyright 1994 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**343**(1994), 479-524 - MSC: Primary 15A48; Secondary 47B65
- DOI: https://doi.org/10.1090/S0002-9947-1994-1242787-6
- MathSciNet review: 1242787