Cones and gauges in complex spaces: Spectral gaps and complex Perron-Frobenius theory
成果类型：
Article
署名作者：
Rugh, Hans Henrik
刊物名称：
ANNALS OF MATHEMATICS
ISSN/ISSBN：
0003-486X
发表日期：
2010
页码：
1707-1752
关键词：
摘要：
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 is an element of M-n(sic) we have the following statement: Suppose that 0 < c < +infinity is such that vertical bar Im A(ij) (A) over bar (mn)vertical bar < c <= Re A(ij) <(A)over bar>(mn) for all indices. Then A has a 'spectral gap'.