Weak spectral mapping theorems for functional differential equations (Q1181680)

The authors start with a very interesting Banach space version of Gearhart's theorem [see \textit{I. Herbst}, J. Oper. Theory 10, 87-94 (1983; Zbl 0535.47024)] characterizing those spectral values \(\lambda\) of semigroup operators \(T(t)\) obtained as \(\lambda=e^{\mu t}\) for \(\mu\) in the spectrum \(\sigma(A)\) of the operator \(A\). For the solution semigroups of neutral differential equations and difference equations a weak spectral mapping \(\overline {e^{t\sigma(A)}}\backslash\{0\}=\sigma(T(t))\backslash\{0\}\) is deduced from this characterization.