Stability and spectra of \(C_ 0\)-semigroups
type: Item
instance of: scholarly article
title: Stability and spectra of \(C_ 0\)-semigroups
zbMATH Open document ID: 0698.47023
DOI: 10.1007/BF01443514
published in: Mathematische Annalen
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164595
review text: Since its publication \textit{L. Gearhart}'s [Trans. Am. Math. Soc. 236, 385-394 (1978; Zbl 0326.47038)] on the boundary of the spectrum of the generator of a contraction semigroup has not lost its fascination, perhaps because the deep reasons for its truth are not quite well understood. The present paper sheds new light on this topic. Let us describe at least one of the interesting results contained in it: Let A be the infinitesimal generator of the strongly continuous semigroup \(U=(U_ t)_{t\geq 0}\) on the Banach space X. Then the intersection \(D_{\infty}(A)=\cap_{n}D(A^ n)\) of the domains of \(A^ n\) is a Fréchet-space when equipped with its natural projective topology. Moreover \(D_{\infty}(A)\) is invariant under U. Denote by \(\sigma (U_ t,D_{\infty}(A))\) the spectrum in the sense of Allan, \(\rho (U_ t,D_{\infty}(A))={\mathbb{C}}\setminus \sigma (U_ t,D_{\infty}(A))\). Let \(\sigma\) (A) denote the ordinary spectrum of A, and set \(\rho (A)={\mathbb{C}}\setminus \sigma (A).\) Theorem: For fixed t the following assertions are equivalent: (1) \(\exp (tz)\in \rho (U_ t,D_{\infty}(A)).\) (2) \(z+2\pi i{\mathbb{Z}}/t\subset \rho (A)\) and \(k\to \| (z+2\pi ik/t- A)^{-1}\|\) is slowly increasing. (3) There exists m such that \(D(A^ m)\subset (\exp (tz)-U_ t)(X)\) and \(\{z+2\pi i{\mathbb{Z}}/t\}\cap (P\sigma (A))=0.\) (There is one further equivalence.) Various applications to stability theory are given and moreover interesting theorems e.g. of Slemrod or Schaefer are improved.
Mathematics Subject Classification ID: 47D03
Mathematics Subject Classification ID: 47A10
Mathematics Subject Classification ID: 47B60
zbMATH DE Number: 4144778
zbMATH Keywords: boundary of the spectrum of the generator of a contraction semigroup
zbMATH Keywords: stability theory
author: Volker Wrobel
reviewed by: Manfred P. H. Wolff
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4144778