O ct 2 00 6 Cones and gauges in complex spaces : Spectral gaps and complex Perron - Frobenius theory

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 Krĕın-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 |ImAijAmn| < c ≤ ReAijAmn for all indices. Then A has a ‘spectral gap’.