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Laplace transform and asymptotic behavior of strongly continuous semigroups. (English) Zbl 0624.47031
We give some necessary and some sufficient conditions for the strong stability of a strongly continuous semigroup (T(t)) on a Banach space E. It is shown that for a positive semigroup on a Banach space ordered by a normal, generating cone, the spectral bound of the generator (A,D(A)) coincides with the exponential growth bound of $$\{$$ T(t)x, $$x\in D(A)\}$$.

##### MSC:
 47D03 Groups and semigroups of linear operators 47B60 Linear operators on ordered spaces