  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'.
