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) → X and in particular study their spectralproperties. We characterize the spectral bound of A, $s(A)$, interms of the resolvent outputs F(λ) = C (λ - B)⁻¹and 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.