On the contraction properties of a pseudo-Hilbert projective metric
Maxime Ligonnière
TL;DR
The paper introduces a bounded pseudo-Hilbert distance $d$ on the projective cone $\Pi(C)$ and analyzes the projective action of positive linear operators $M$ on $E$. It proves every positive operator is $1$-Lipschitz with respect to $d$, and shows strict contraction (i.e., $c(M)<1$) is equivalent to uniform positivity, with $c(M)=\psi(A^*(M))$ where $\psi(s)=(1-s^{-2})/(1+s^{-2})$ and $A^*(M)$ is the infimum of uniform positivity constants. The authors provide explicit criteria and formulas: in finite dimensions, contraction is governed by the zero-pattern of $M$ and an exact coefficient-based expression for $c(M)$; in the density/kernel setting, contraction reduces to uniform positivity bounds on the kernel $K(x,y)$, yielding $c(M_K)=\psi^{-1}(A^*)$. These results yield practical contraction criteria and underpin fixed-point and ergodicity analyses for semigroups of uniformly positive operators in both finite and infinite-dimensional contexts.
Abstract
In this note, we define a bounded variant on the Hilbert projective metric on an infinite dimensional space $E$ and study the contraction properties of the projective maps associated with positive linear operators on $E$. More precisely, we prove that any positive linear operator acts projectively as a $1$-Lipschitz map relatively to this distance. We also show that for a positive linear operator, strict projective contraction is equivalent to a property called uniform positivity.