Remarks on resolvent positive operators and their perturbation

We consider positive perturbations $A = B+ C $ of resolvent positive operators $B$ by positive operators $C: D(A) \to X$ and in particular study their spectral properties. We characterize the spectral bound of $A$, $s(A)$, in terms 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 monotonic operator family form a superconvex function. Our results will be used in forthcoming publications to study the spectral and large-time properties of positive operator semigroups.