The resolvent growth assumption for semigroups on Hilbert spaces (Q908501)

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.