The critical spectrum of a strongly continuous semigroup.
type: Item
instance of: scholarly article
title: The critical spectrum of a strongly continuous semigroup.
zbMATH Open document ID: 1054.47034
author: Rainer Nagel
author: Jan Poland
published in: Advances in Mathematics
publication date: 2000-01-01T00:00:00Z
review text: \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 \(A\).
reviewed by: Kazuaki Taira
Mathematics Subject Classification ID: 47D06
Mathematics Subject Classification ID: 47A10
zbMATH DE Number: 1496433
zbMATH Keywords: strongly continuous semigroup
zbMATH Keywords: generator
zbMATH Keywords: critical spectrum
zbMATH Keywords: spectral mapping theorem
zbMATH Keywords: critical growth bound
MaRDI profile type: Publication
OpenAlex ID: W2069433685
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DOI: 10.1006/AIMA.1998.1893
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description: scientific article; zbMATH DE number 1496433