On the structure of the critical spectrum of strongly continuous semigroups (Q2708553)

In this paper the authors establish some properties of the critical spectrum \(\sigma_{\text{crit}} (T_t)\) where \((T_t)_{t \geq 0} \) is a strongly continuous semigroup of linear operators \(T_t\) on the Banach space \(X\). Let \(A\) be the generator of the semigroup. The main result of the paper is the following one on the rotational invariance of \(\sigma_{\text{crit}} (T_t)\):NEWLINENEWLINENEWLINELet \((\lambda_n)_n\) be a sequence in \(\mathbb{C}\) with \(\lim_{n \to \infty} \Re (\lambda_n) = w\) and \(\lim_{n \to \infty} \mid \Im (\lambda_n) \mid = \infty\). Assume that \(\lim_{n \to \infty} \inf \{ \|( A - \lambda_n) x \|: \|x \|= 1, x \in {\mathcal D} (A) \} = 0 \). Then for almost all \(t \geq 0 \qquad \{\exp (w t + i s) : |s |\leq \pi \} \subset \sigma (T_t) \) holds.NEWLINENEWLINENEWLINEUnformately up to now there exists only an extrinsic definition of \(\sigma_{c ri t} (T_t)\) using the Fréchet power or ultrapowers of \(X\). In so far also the following result of the paper under review is important.NEWLINENEWLINENEWLINELet \(\rho (t) = \lim_{\lambda \to \infty} \|A R (\lambda, A) T_t \|\). Then \(\inf \{ w \in \mathbb R : \sup \{ e^{-w t} \rho (t) : t \geq 0 \} \) is the critical growth bound of \((T_t)_t\) defined in an earlier paper by the extension of \((T_t)\) to some ultraproduct.NEWLINENEWLINEFor the entire collection see [Zbl 0957.00037].