A resolvent approach to the stability of essential and critical spectra of perturbed \(C_0\)-semigroups on Hilbert spaces with applications to transport theory
type: Item
instance of: scholarly article
title: A resolvent approach to the stability of essential and critical spectra of perturbed \(C_0\)-semigroups on Hilbert spaces with applications to transport theory
zbMATH Open document ID: 1117.35056
author: Mohammed Sbihi
published in: Journal of Evolution Equations
publication date: 2007-05-24T00:00:00Z
review text: Let \(T\) be the generator of a \(C_{0}\)-semigroup \((U(t))_{t \geq 0}\) on a Hilbert space \(H\). Let \(K\) be a bounded operator in \(H\) and \((V(t))_{t \geq 0}\) the \(C_{0}\)-semigroup generated by \(T+K\). \(V(t)\) is given by the Dyson-Phillips expansion \[ V(t)= \sum_{j=0}^{\infty} U_{j}(t). \] The stability of the essential and essential type spectrum is implied by the compactness of \(U_{1}(t)\) and \(U_{k}(t)\) for \(k \geq 1\), respectively. The stability of the critical and critical type spectrum is implied by the norm continuity of \(U_{1}(t)\) and \(U_{k}(t)\) for \(k \geq 1\), respectively. The author proves conditions for these properties to hold involving the resolvent of \(T\). Applications are given to transport theory, where \(T\) is the streaming operator \[ T: L^\infty(\Omega \times V) \supset D(T) \ni \varphi \mapsto -v\frac{\partial \varphi}{\partial x} - \sigma(x,v)\varphi(x,v), \] where \(\Omega\) is a domain in \({\mathbb R}^N\), \(V\) the space of velocities and \(D(T)\) involves some boundary conditions. The operator \(K\) is taken to be a collision operator of the form \[ K(x): L^2(V) \ni \varphi \mapsto \int_{V} k(x,v,v') \varphi(v') \,d\mu(v') \] which satisfies the assumption that \(\{K(x); x \in \Omega\}\) is collectively compact in \(L^2(V)\).
reviewed by: Rolf Dieter Grigorieff
Mathematics Subject Classification ID: 35P05
Mathematics Subject Classification ID: 47A10
Mathematics Subject Classification ID: 47D06
Mathematics Subject Classification ID: 82D75
Mathematics Subject Classification ID: 35Q72
Mathematics Subject Classification ID: 47A55
Mathematics Subject Classification ID: 82C70
zbMATH DE Number: 5156763
zbMATH Keywords: perturbed \(C_{0}\)-semigroup
zbMATH Keywords: essential spectrum
zbMATH Keywords: critical spectrum
zbMATH Keywords: neutron transport theory
MaRDI profile type: Publication
full work available at URL: s00028-006-0226-2
OpenAlex ID: W2022458591
DOI: 10.1007/S00028-006-0226-2
description: scientific article; zbMATH DE number 5156763