Spectral mapping theorem for integrated semigroups
type: Item
instance of: scholarly article
title: Spectral mapping theorem for integrated semigroups
zbMATH Open document ID: 0812.47036
DOI: 10.1007/BF02573773
author: Colin R. Day
published in: Semigroup Forum
publication date: 1993-11-11T00:00:00Z
full work available at URL: https://eudml.org/doc/135286
review text: \(\alpha\)-times integrated semigroups were introduced by W. Arendt for \(\alpha\) an integer and by M. Hieber for general \(\alpha\). They govern the Cauchy problem \[ u'(t)= Au(t),\;u(0)= x,\tag{*} \] where \(x\) is in the domain of \(A^{\alpha+1}\). The case of \(\alpha= 0\) corresponds to the usual \((C_ 0)\) semigroups. Thus for \(\alpha> 0\) the Cauchy problems under consideration are well posed in a certain sense (depending on \(\alpha\)) but not necessarily well-posed in the usual sense (for \(\alpha> 0\)). The \(\alpha\)-times integrated semigroup \(S= \{S(t): t>0\}\) governing (*) is given by \[ (\lambda- A)^{-1} x= \lambda^ \alpha \int^ \alpha_ 0 e^{-\lambda t} S(t) x dt. \] The author's spectral mapping theorem for point spectrum is \[ \sigma_ p(S(t))\cup \{0\} = \int^ t_ 0 e^{\lambda s}(t- s)^{\alpha-1} \Gamma(\alpha)^{-1} ds: \lambda(\sigma_ p(A))\cup \{0\}. \] Similar results are obtained for \(\sigma_{ap}(S(t)), \sigma_ r(S(t))\) (approximate point spectrum and residual spectrum).
Mathematics Subject Classification ID: 47D06
zbMATH DE Number: 438894
zbMATH Keywords: approximate point spectrum
zbMATH Keywords: \(\alpha\)-times integrated semigroups
zbMATH Keywords: Cauchy problem
zbMATH Keywords: \((C_ 0)\) semigroups
zbMATH Keywords: spectral mapping theorem for point spectrum
zbMATH Keywords: residual spectrum
reviewed by: Jerome A. Goldstein
MaRDI profile type: Publication
cites work: Vector-valued Laplace transforms and Cauchy problems
cites work: Interpolation of semigroups and integrated semigroups
cites work: The spectral mapping theorem for integrated semigroups
cites work: Laplace Transforms and α-Times Integrated Semigroups
cites work: Integrated semigroups and differential operators on \(L^ p\) spaces
cites work: One-parameter semigroups of positive operators
OpenAlex ID: W2062853234
description: scientific article; zbMATH DE number 438894