# zbMATH — the first resource for mathematics

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 (*).

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