Spectral mapping theorem for integrated semigroups (Q688983)

\(\alpha\)-times integrated semigroups were introduced by W. Arendt for \(\alpha\) an integer and by M. Hieber for general \(\alpha\). They govern the Cauchy problem \[ u'(t)= Au(t),\;u(0)= x,\tag{*} \] where \(x\) is in the domain of \(A^{\alpha+1}\). The case of \(\alpha= 0\) corresponds to the usual \((C_ 0)\) semigroups. Thus for \(\alpha> 0\) the Cauchy problems under consideration are well posed in a certain sense (depending on \(\alpha\)) but not necessarily well-posed in the usual sense (for \(\alpha> 0\)). The \(\alpha\)-times integrated semigroup \(S= \{S(t): t>0\}\) governing (*) is given by \[ (\lambda- A)^{-1} x= \lambda^ \alpha \int^ \alpha_ 0 e^{-\lambda t} S(t) x dt. \] The author's spectral mapping theorem for point spectrum is \[ \sigma_ p(S(t))\cup \{0\} = \int^ t_ 0 e^{\lambda s}(t- s)^{\alpha-1} \Gamma(\alpha)^{-1} ds: \lambda(\sigma_ p(A))\cup \{0\}. \] Similar results are obtained for \(\sigma_{ap}(S(t)), \sigma_ r(S(t))\) (approximate point spectrum and residual spectrum).