Remarks on resolvent positive operators and their perturbation

Horst R. Thieme

doi:10.3934/dcds.1998.4.73

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1998, Volume 4, Issue 1: 73-90. Doi: 10.3934/dcds.1998.4.73

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Remarks on resolvent positive operators and their perturbation

Horst R. Thieme^1,

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

Received: December 31, 1995

Revised: December 31, 1996

Published: December 1997

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Abstract

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.

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Mathematics Subject Classification: 34G10, 47D06, 58F11.

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