Cones and gauges in complex spaces: Spectral gaps and complex Perron-Frobenius theory

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 $A\in M_n(\mathbb{C})$ 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'.

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      MRCLASS = {46.3X},
      MRNUMBER = {2,104b},
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      }
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    @article {Ves76, MRKEY = {0433228},
      AUTHOR = {Vesentini, Edoardo},
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      JOURNAL = {Ann. Scuola Norm. Sup. Pisa Cl. Sci.},
      VOLUME = {3},
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      NUMBER = {4},
      PAGES = {671--696},
      MRCLASS = {46N05 (54E35 32H15)},
      MRNUMBER = {55 \#6206},
      MRREVIEWER = {Alexander M. Rubinov},
      URL = {http://www.numdam.org/item?id=ASNSP_1976_4_3_4_671_0},
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      }
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      SERIES = {SMF/AMS Texts and Monographs},
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      PAGES = {x+82},
      ISBN = {0-8218-1948-8},
      MRCLASS = {37Fxx (30D05 37D35)},
      MRNUMBER = {2000h:37063},
      ZBLNUMBER = {0959.37003},
      }

Authors

Hans Henrik Rugh

Université de Cergy-Pontoise
Département de Mathématiques
UMR CNRS 8088
F-95800 Cergy-Pontoise
France