The spectral mapping theorem for integrated semigroups (Q2367898)

The article is devoted to one of the general theorems in the theory of strongly continuous semigroups -- the spectral mapping theorem. We denote as \(\sigma \bigl( T(t)\bigr)\) the spectrum of the semigroup \(\bigl( T(t) \bigr)_{t\geq 0}\), \(T:[0,\infty] \to L(E)\). The following main result for \(n\)-times integrated semigroups is represented in the article as Theorem 2. Let \(t\to S(t)\) \((t\geq 0)\) be an exponentially bounded \(n\)- times integrated semigroup. Then the spectral mapping theorem holds, i.e. \[ \sigma(S(t)) \cup\{0\}=\biggl\{ {1\over \lambda^ n}e^{\lambda t}- \sum^{n-1}_{k=0}{t^ k \over k! \lambda^{n-k}}| \lambda \in \sigma(A) \biggr\} \cup\{0\}. \]