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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
Keywords:
Hilbert space
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