# American Institute of Mathematical Sciences

January  1998, 4(1): 73-90. doi: 10.3934/dcds.1998.4.73

## Remarks on resolvent positive operators and their perturbation

 1 Department of Mathematics, Arizona State University, Tempe, AZ 85287-1804, United States

Received  January 1996 Revised  January 1997 Published  October 1997

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.
Citation: Horst R. Thieme. Remarks on resolvent positive operators and their perturbation. Discrete & Continuous Dynamical Systems, 1998, 4 (1) : 73-90. doi: 10.3934/dcds.1998.4.73
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