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On the spectrum of \(C_ 0\)-semigroups. (English) Zbl 0572.47030
Let X be a Banach space, A the generator of the \(C_ 0\)-semigroup \(E^{At}\) in X, and consider the inhomogeneous differential equation \[ (*)\quad u'=Au+f. \] We show that (*) has a unique 1-periodic mild solution for each continuous 1-periodic f iff \(1\in \rho (e^ A)\), the resolvent set of \(e^ A\), and that in case X is a Hilbert space the latter is equivalent to \(\{\) \(2\pi\) in:\(n\in {\mathbb{Z}}\}\subset \rho (A)and\sup \{| (2\pi in-A)^{-1}|:n\in {\mathbb{Z}}\}<\infty\). Then we give several applications of these results, including a characterization of the spectrum of \(e^{At}\) in Hilbert spaces, some remarks on the so called ”spectrum determined growth property” of semigroups and uniform asymptotic stability of (*), and a study of the existence of dichotomic projections for (*).

47D03 Groups and semigroups of linear operators
34G10 Linear differential equations in abstract spaces
47A10 Spectrum, resolvent
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