Extremal norms for positive linear inclusions

TL;DR

This paper proves the existence of extremal norms for positive linear semigroups that leave a cone invariant, extending the understanding of spectral properties for various types of systems.

Contribution

It establishes that irreducibility implies extremal norms for positive linear semigroups, including discrete, continuous, and jump systems, with special results for simplicial cones.

Findings

Existence of extremal norms under irreducibility

Extension of Lipschitz continuity results for joint spectral radius

Applicability to various types of linear systems

Abstract

For finite-dimensional linear semigroups which leave a proper cone invariant it is shown that irreducibility with respect to the cone implies the existence of an extremal norm. In case the cone is simplicial a similar statement applies to absolute norms. The semigroups under consideration may be generated by discrete-time systems, continuous-time systems or continuous-time systems with jumps. The existence of extremal norms is used to extend results on the Lipschitz continuity of the joint spectral radius beyond the known case of semigroups that are irreducible in the representation theory interpretation of the word.