Eventually Positive Semigroups

Abstract

We consider one-parameter semigroups of linear operators $e^{tA}$ on $C\left(K\right)$ such that for every $f>0$ there exists $t_0>0$ so that $e^{tA}f>0$ for all $t>t_0$. The purpose of the talk is to give a general theory of such eventually positive semigroups and characterise them in terms of positivity properties of the resolvent ${\left(\lambda I-A\right)}^{-1}$ and the spectral projection associated with the spectral bound.

Examples of eventually positive semigroups include the semigroup generated by the Dirichlet-to-Neumann operator, delay differential equations, higher order parabolic equations and some matrix semigroups.

This is joint work with Wolfgang Arendt, Jochen Glück and James Kennedy.