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The resolvent growth assumption for semigroups on Hilbert spaces. (English) Zbl 0693.47034
The relation between the large t behaviour of a $$C_ 0$$ semigroup $$T_ t$$ on a Hilbert space and the growth of the resolvent of its generator A along vertical lines in the complex plane is considered. The Nth asymptotic type of $$T_ t$$, $$\omega^ a_ N$$ is defined as the inf of those $$\omega\in R$$ such that $$\exp (-\omega t)\cdot T_ tx$$ is norm- bounded for all x in the domain of $$A^ N$$. The Nth spectral type of $$T_ t$$, $$\omega^ s_ N$$ is defined as the inf of those $$\omega\in R$$, such that Re z$$>\omega$$ is in the resolvent set of A and $$(1+| Im z|)^{-N}\| R(z,A)\|$$ is bounded on vertical strips: $$\omega <a\leq Re z\leq b<\infty$$. The main result is: $$\omega^ a_ N=\omega^ s_ N$$. On a Banach space, only $$\omega^ a_ N\geq \omega^ s_ N$$ is proved.
Reviewer: N.Angelescu

##### MSC:
 47D03 Groups and semigroups of linear operators
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