Cones and gauges in complex spaces: Spectral gaps and complex Perron-Frobenius theory

Abstract

We introduce complex cones and associated projective gauges, generalizing a real Birkhoff cone and its Hilbert metric to complex vector spaces. We deduce a variety of spectral gap theorems in complex Banach spaces. We prove a dominated complex cone contraction theorem and use it to extend the classical Perron-Frobenius Theorem to complex matrices, Jentzsch's Theorem to complex integral operators, a Kreǐn-Rutman Theorem to compact and quasi-compact complex operators and a Ruelle-Perron-Frobenius Theorem to complex transfer operators in dynamical systems. In the simplest case of a complex n by n matrix A ∈ Mn.(C) we have the following statement: Suppose that 0 < c < + ∞ is such that ?|ImAijĀmn| < c ≤ ReAijĀxmn for all indices. Then A has a 'spectral gap'. © 2010 Annals of Mathematics.