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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), t0. 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 x ˙(t)=Ax(t), t0, is asymptotically stable.

MSC:
34D05Asymptotic stability of ODE
34A30Linear ODE and systems, general
34G10Linear ODE in abstract spaces