Stability criteria for positive semigroups on ordered Banach spaces

Abstract

We consider generators of positive $C_0$-semigroups and, more generally, resolvent positive operators $A$ on ordered Banach spaces and seek for conditions ensuring the negativity of their spectral bound $s(A)$. Our main result characterizes $s(A)<0$ in terms of so-called \emph{small-gain conditions} that describe the behaviour of $Ax$ for positive vectors $x$. This is new even in case that the underlying space is an $L^p$-space or a space of continuous functions. We also demonstrate that it becomes considerably easier to characterize the property $s(A)<0$ if the cone of the underlying Banach space has non-empty interior or if the essential spectral bound of $A$ is negative. To treat the latter case, we discuss a counterpart of a Krein-Rutman theorem for resolvent positive operators. When $A$ is the generator of a positive $C_0$-semigroup, our results can be interpreted as stability results for the semigroup, and as such, they complement similar results recently proved for the discrete-time case. In the same vein, we prove a Collatz--Wielandt type formula and a logarithmic formula for the spectral bound of generators of positive semigroups.

This is version 2. Several results, examples, and reference were added compared to v1. The title was slightly changed. 39 pages