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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2024 MCQ for Transactions of the American Mathematical Society is 1.48 .

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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

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.
References
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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