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A problem of exponential stability for linear dynamical systems in Hilbert spaces. (English) Zbl 0679.34053
Let H be a Hilbert space, $$e^{tA}$$ a $$C_ 0$$-semigroup with infinitesimal generator A and $\omega (A)=\lim_{t\to +\infty}\ln | e^{tA}| /t=\inf_{t\in]0,+\infty[}\ln | e^{tA}| /t.$ Then $$\omega (A)<0$$ if and only if $$\int^{+\infty}_{0}| <e^{tA}x,y>|^ pdt<+\infty$$ for some $$p\in [1,+\infty[$$ and $$x,y\in H$$.
Reviewer: G.Bottaro

##### MSC:
 34D05 Asymptotic properties of solutions to ordinary differential equations 34G10 Linear differential equations in abstract spaces
Hilbert space