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Journal of the London Mathematical Society Advance Access published online on April 20, 2009
Journal of the London Mathematical Society, doi:10.1112/jlms/jdp008
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© 2009 London Mathematical Society

Projective metrics and contraction principles for complex cones

Loïc Dubois

Department of Mathematics
University of Cergy-Pontoise
2 avenue Adolphe Chauvin
95302 Cergy-Pontoise cedex
France

In this article, we consider linearly convex complex cones in complex Banach spaces and we define a new projective metric on these cones. Compared to the hyperbolic gauge of Rugh, it has the advantages of being explicit and easier to estimate. We prove that this metric also satisfies a contraction principle similar to Birkhoff's theorem for the Hilbert metric. We are thus able to improve existing results on spectral gaps for complex matrices. Finally, we compare the contraction principles for the hyperbolic gauge and our metric on particular cones, including complexification of Birkhoff cones. It appears that the contraction principles for our metric and the hyperbolic gauge occur simultaneously on these cones. However, we get better contraction rates with our metric.

2000 Mathematics Subject Classification 15A48, 47A75, 47B65.

Received December 11, 2007; revised November 26, 2008;
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