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Closed semistable operators and singular differential equations. (English) Zbl 1080.47500
Summary: 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 H. R. Thieme, G. F. Webb and J. 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.

47A10 Spectrum, resolvent
47A60 Functional calculus for linear operators
34G10 Linear differential equations in abstract spaces
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