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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.