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Asymptotic stability of linear differential equations in Banach spaces. (English) Zbl 0639.34050
Let A be a generator of a strongly continuous bounded semigroup T(t), $$t\geq 0$$. We prove that if the intersection of the spectrum of A and the imaginary axis is at most countable and $$A^*$$ has no purely imaginary eigenvalues, then the Cauchy problem for the differential equation $$\dot x(t)=Ax(t)$$, $$t\geq 0$$, is asymptotically stable.

##### MSC:
 34D05 Asymptotic properties of solutions to ordinary differential equations 34A30 Linear ordinary differential equations and systems, general 34G10 Linear differential equations in abstract spaces
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