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 Kreĭn-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 $A\in M_n(\mathbb{C})$ we have the following statement: Suppose that $0\lt c\lt +\infty$ is such that $ |\operatorname{Im}\, A_{ij}\bar{A}_{mn}| < c \leq \operatorname{Re}\, A_{ij}\bar{A}_{mn}$ for all indices. Then $A$ has a 'spectral gap'.