## Abstract

This paper contains a unified account of the applications of Hilbert's projective

metric to the spectral properties of a class of positive linear maps, 'positive' in this

case meaning order-preserving with respect to a cone ordering of the underlying

vector space. We give a general discussion of the definition of the metric and

some of its important properties, an elementary proof of the identity central to

the theory, and applications to linear operators on partially ordered spaces, with

specialisations to integral operators on real function spaces and to real matrices.

For a suitable operator A (such as a real matrix with positive entries or an

integral operator with a positive, continuous kernel), we show that the spectral

radius r(A) of A is an algebraically simple eigenvalue of A whose eigenspace is

spanned by a positive vector, that no other eigenvalue of A has a positive

eigenvector, and we give an estimate for the magnitudes of any other eigenvalues

of A. We also show that if x is any positive vector, then the sequence j4"jc/||j4";t||

converges in norm to the unique normalised positive eigenvector u of A; in fact,

A^nx

• — u Mqn

\\A^nx\

where M and q are explicitly computable constants with M > 0 and 0 ^ q < 1.

metric to the spectral properties of a class of positive linear maps, 'positive' in this

case meaning order-preserving with respect to a cone ordering of the underlying

vector space. We give a general discussion of the definition of the metric and

some of its important properties, an elementary proof of the identity central to

the theory, and applications to linear operators on partially ordered spaces, with

specialisations to integral operators on real function spaces and to real matrices.

For a suitable operator A (such as a real matrix with positive entries or an

integral operator with a positive, continuous kernel), we show that the spectral

radius r(A) of A is an algebraically simple eigenvalue of A whose eigenspace is

spanned by a positive vector, that no other eigenvalue of A has a positive

eigenvector, and we give an estimate for the magnitudes of any other eigenvalues

of A. We also show that if x is any positive vector, then the sequence j4"jc/||j4";t||

converges in norm to the unique normalised positive eigenvector u of A; in fact,

A^nx

• — u Mqn

\\A^nx\

where M and q are explicitly computable constants with M > 0 and 0 ^ q < 1.

Original language | English |
---|---|

Pages (from-to) | 411-440 |

Number of pages | 30 |

Journal | Proc London Math Soc |

Volume | 70 |

Issue number | 2 |

DOIs | |

Publication status | Published - Mar 1995 |

## Keywords

- Analysis