On the contraction properties of a pseudo-Hilbert projective metric

TL;DR

This paper introduces a bounded variant of the Hilbert projective metric in infinite-dimensional spaces and demonstrates that positive linear operators act as 1-Lipschitz maps, with strict contraction linked to uniform positivity.

Contribution

It defines a new bounded projective metric and establishes the contraction properties of positive linear operators relative to this metric in infinite-dimensional spaces.

Findings

Positive linear operators are 1-Lipschitz with respect to the new metric.

Strict contraction is equivalent to uniform positivity.

The new metric provides a bounded framework for analyzing projective contractions.

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.