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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'.
@article{10_4007_annals_2010_171_1707,
author = {Hans Henrik Rugh},
title = {Cones and gauges in complex spaces: {Spectral} gaps and complex {Perron-Frobenius} theory},
journal = {Annals of mathematics},
pages = {1707--1752},
publisher = {mathdoc},
volume = {171},
number = {3},
year = {2010},
doi = {10.4007/annals.2010.171.1707},
mrnumber = {2680397},
zbl = {1208.15026},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4007/annals.2010.171.1707/}
}
TY - JOUR AU - Hans Henrik Rugh TI - Cones and gauges in complex spaces: Spectral gaps and complex Perron-Frobenius theory JO - Annals of mathematics PY - 2010 SP - 1707 EP - 1752 VL - 171 IS - 3 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.4007/annals.2010.171.1707/ DO - 10.4007/annals.2010.171.1707 LA - en ID - 10_4007_annals_2010_171_1707 ER -
%0 Journal Article %A Hans Henrik Rugh %T Cones and gauges in complex spaces: Spectral gaps and complex Perron-Frobenius theory %J Annals of mathematics %D 2010 %P 1707-1752 %V 171 %N 3 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.4007/annals.2010.171.1707/ %R 10.4007/annals.2010.171.1707 %G en %F 10_4007_annals_2010_171_1707
Hans Henrik Rugh. Cones and gauges in complex spaces: Spectral gaps and complex Perron-Frobenius theory. Annals of mathematics, Tome 171 (2010) no. 3, pp. 1707-1752. doi: 10.4007/annals.2010.171.1707
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