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 Krein-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 we have the following statement: Suppose that 0 < c < +infinity is such that |Im A_{ij} \bar{A}_{mn}| < c < Re A_{ij} \bar{A}_{mn} for all indices. Then A has a `spectral gap'.