Perturbation theorems for \(\alpha\)-times integrated semigroups
type: Item
instance of: scholarly article
title: Perturbation theorems for \(\alpha\)-times integrated semigroups
zbMATH Open document ID: 1065.47007
published in: Archiv der Mathematik
publication date: 2004-03-07T00:00:00Z
review text: Under relative smallness conditions for the perturbing operator \(B\), perturbation results are proved for the generator \((A,D(A))\) of an \(\alpha\)-times integrated semigroup on a Banach space \(X\). More precisely, it is shown that if the estimate \[ \| B(\lambda-A)^{-1}\| \leq M<1\quad\text{or}\quad\| (\lambda-A)^{-1}Bx\| \leq M\| x\| \hskip5pt\text{for \(x\in D(B)\)} \] holds for \(\lambda\in\mathbb{C}\) on a vertical line, then (an extension of) \(A+B\) generates a \(\beta\)-times integrated semigroup, where \(\beta>\alpha+1\) if the \(\alpha\)-times integrated semigroup generated by \(A\) is exponentially bounded, or \(\beta>\alpha+2\) in the general case. For exponentially bounded \(\alpha\)-times integrated semigroups, the result can be sharpened if the Banach space has Fourier type \(p\in[1,2]\). Namely, in this case it suffices to take \(\beta>1+{{1}\over{p}}\). In particular, when \(A\) generates a \(C_0\)-semigroup and \(X\) is \(B\)-convex, the results yield that \(A+B\) (again with appropriate domain, depending on which estimate above is satisfied) is a generator of a once-integrated semigroup. As an illustration, Schrödinger operators and higher order differential operators are considered.
reviewed by: Bálint Farkas
Mathematics Subject Classification ID: 47A55
Mathematics Subject Classification ID: 47D62
Mathematics Subject Classification ID: 47D06
Mathematics Subject Classification ID: 47E05
zbMATH DE Number: 2051594
zbMATH Keywords: \(\alpha\)-times integrated semigroups
zbMATH Keywords: generators
zbMATH Keywords: perturbation theory
zbMATH Keywords: \(C_0\)-semigroups
zbMATH Keywords: Fourier type
zbMATH Keywords: \(B\)-convex Banach spaces
author: Cornelia Kaiser
author: Lutz Weis
MaRDI profile type: Publication
full work available at URL: s00013-003-0540-7
OpenAlex ID: W2092470032
DOI: 10.1007/S00013-003-0540-7
Recommended article: Q3017609
Recommended article: Q3169766
Recommended article: Q5499222
Recommended article: On perturbation of \(\alpha\)-times integrated \(C\)-semigroups
Recommended article: Q3373447
Recommended article: Perturbation of non-exponentially-bounded \(\alpha\)-times integrated \(C\)-semigroups
Recommended article: Perturbations and approximations of integrated semigroups
Recommended article: Perturbation theorems for local integrated semigroups
Recommended article: Multiplicative perturbations of local \(\alpha \)-times integrated \(C\)-semigroups
Recommended article: Perturbation theorems for local integrated semigroups and their applications
description: scientific article; zbMATH DE number 2051594