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Weak \(L^ p\)-stability of a linear semigroup on a Hilbert space implies exponential stability. (English) Zbl 0675.47031
Summary: We prove that a strongly continuous semigroup of linear operators on a Hilbert space is weakly \(L^ p\)-stable for some \(p\in [1,\infty)\) if and only if the semigroup is exponentially stable. As an application, we prove that the Cauchy problems associated with the semigroup are well posed on the infinite time interval \((-\infty,0]\) if and only if the semigroup is exponentially stable.

MSC:
47D03 Groups and semigroups of linear operators
34G10 Linear differential equations in abstract spaces
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