Stability criteria for positive semigroups on ordered Banach spaces

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.