Geodesic
Closed semistable operators and singular differential equations
Czechoslovak Mathematical Journal, Tome 53 (2003) no. 3, pp. 605-620
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We study a class of closed linear operators on a Banach space whose nonzero spectrum lies in the open left half plane, and for which $0$ is at most a simple pole of the operator resolvent. Our spectral theory based methods enable us to give a simple proof of the characterization of $C_0$-semigroups of bounded linear operators with asynchronous exponential growth, and recover results of Thieme, Webb and van Neerven. The results are applied to the study of the asymptotic behavior of the solutions to a singularly perturbed differential equation in a Banach space.
We study a class of closed linear operators on a Banach space whose nonzero spectrum lies in the open left half plane, and for which $0$ is at most a simple pole of the operator resolvent. Our spectral theory based methods enable us to give a simple proof of the characterization of $C_0$-semigroups of bounded linear operators with asynchronous exponential growth, and recover results of Thieme, Webb and van Neerven. The results are applied to the study of the asymptotic behavior of the solutions to a singularly perturbed differential equation in a Banach space.
Classification : 34G10, 47A10, 47A60, 47D06
Keywords: closed linear operator; $C_0$-semigroup; infinitesimal generator; semistable operator; singular differential equation 
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Koliha, J. J.; Tran, Trung Dinh. Closed semistable operators and singular differential equations. Czechoslovak Mathematical Journal, Tome 53 (2003) no. 3, pp. 605-620. http://geodesic.mathdoc.fr/item/CMJ_2003_53_3_a8/

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