The critical spectrum of a strongly continuous semigroup.
\textit{A. Lyapunov} showed in [``General problem of the stability of motion'' (Russian), Charkov (1892; JFM 24.0876.02); French translation in Ann. Fac. Sci. Toulouse Sci. Math. Sci. Phys. (2) 9, 203-474 (1907; JFM 38.0738.07)] that the asymptotic behavior as \(t \to \infty\) of the exponential function \(\exp[tA]\) for a matrix \(A\) can be described by the location of the eigenvalues of \(A\). Later, \textit{E. Hille} and \textit{R. S. Phillips} [``Functional analysis and semi-groups'' (Am. Math. Soc. Colloq. Publ. 31, AMS, Providence/RI) (1957; Zbl 0078.10004)] discovered that an analogous statement does not hold for a strongly continuous semigroup \((T(t))_{t\geq0}\) with unbounded generator \(A\) on a Banach space \(X\). The reason is the failure of the spectral mapping theorem \[ \sigma(T(t)) = e^{t\sigma(A)}. \] In this paper, the authors introduce the critical spectrum \(\sigma_{\text{crit}}(T(t))\), which yields in an elegant and optimal way a spectral mapping theorem of the form \[ \sigma(T(t)) = e^{t\sigma(A)} \cup \sigma_{\text{crit}}(T(t)). \] As an immediate consequence of this theorem, they prove the spectral mapping theorem for eventually norm continuous semigroups. Moreover, the authors make use of the critical spectrum to describe the asymptotic behavior of the semigroup \({\mathcal T}= (T(t))_{t\geq0}\), improving classical stability results. To do this, they introduce the notion of the critical growth bound \(\omega_{\text{crit}}({\mathcal T})\) of \({\mathcal T}\) as the usual growth bound of a semigroup \(\widehat{T}(t)\) defined on a quotient space \(\widehat{X}\) involving \(\ell^{\infty}(X)\). For example, they give a characterization of the growth bound \(\omega_0({\mathcal T})\) \[ \omega_0({\mathcal T}) = \max\{s(A), \omega_{\text{crit}} ({\mathcal T})\}, \] where \(s(A)\) is the spectral bound of the generator \(A\).
scientific article; zbMATH DE number 1496433
scientific article