Classical Perron–Frobenius theory

In this appendix we provide proofs of most of the results from Section 1.1 concerning classical linear Perron–Frobenius theory. We begin (see Theorem B.1.1) by proving a generalization, valid for general cones, of Perron's theorem which is stated in Theorem 1.1.1. From this result we then derive the finite-dimensional Kreĭn–Rutman theorem (Theorem 1.1.6). We also show that many of the results in the general version of Perron's theorem remain valid for irreducible linear maps, and this yields Theorem 1.1.7. We subsequently give a complete proof of the third assertion in the classical Perron–Frobenius Theorem 1.1.2, which depends on special properties of the cone and is of a qualitatively different nature from the other two assertions (see Proposition B.4.3). We next use this part of the Perron–Frobenius theorem to prove Theorems 1.1.8 and 1.1.9 concerning the peripheral spectrum and iterative behavior of linear maps on polyhedral cones. Our treatment here is concise and meant only as an introduction to the linear theory. The reader should consult the books by Bapat and Raghavan [15], Berman and Plemmons [22], Minc [148], and Seneta [202], or the survey paper by Tam [214], for a more thorough discussion of linear Perron–Frobenius theory.