Cones and gauges in complex spaces: Spectral gaps and complex Perron-Frobenius theory | Annals of Mathematics

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∈Mn(ℂ) we have the following statement: Suppose that 0

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数学年刊(Annals of Mathematics)

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