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