Remarks on resolvent positive operators and their perturbation

We consider positive perturbations $A = B+ C $of resolvent positive operators $B$ by positiveoperators $C: D(A) \to X$ and in particular study their spectralproperties. We characterize the spectral bound of $A$, $s(A)$, interms of the resolvent outputs $F(\lambda) = C (\lambda - B)^{-1}$and derive conditions for $s(A)$ to be an eigenvalue of $A$ and a(first order) pole of the resolvent of $A$.On our way we show that the spectral radii of a completely monotonicoperator family form a superconvex function. Our results will beused in forthcoming publications to study the spectral and large-timeproperties of positive operator semigroups.