Smooth perturbations in ordered Banach spaces and similarity for the linear transport operators
type: Item
instance of: scholarly article
title: Smooth perturbations in ordered Banach spaces and similarity for the linear transport operators
zbMATH Open document ID: 0601.47011
author: Tomio Umeda
published in: Journal of the Mathematical Society of Japan
publication date: 1986-01-01T00:00:00Z
review text: A new method for establishing a similarity for a pair of linear operators in an ordered Banach space is discussed. The author considers a pair of (in general unbounded) linear operators \(B_ 1\) and \(B_ 2=B_ 1+A\) with A a bounded operator, and assumes that both \(-B_ 1\) and \(-B_ 2\) generate bounded \(C_ 0\)-groups in the Banach space. He assumes, in addition, that \(e^{-tB_ 1}\) and -A are positivity preserving, and that \[ \int^{\infty}_{-\infty}\| Ae^{-tB_ 1}u\| dt<\infty \] for all u in the Banach space. He shows that \(B_ 2\) is similar to \(B_ 1\) i.e., \[ B_ 2=W_+(B_ 2,B_ 1)B_ 1W_+(B_ 2,B_ 1)^{-1} \] where \(W_+(B_ 2,B_ 1)=s-\lim_{t\to \infty}e^{tB_ 2}e^{-tB_ 1}\). The main idea depends on an analysis of positivity preserving \(C_ 0\)-groups. The method is applied to establish the similarity for the collisionless transport operator and the linear transport operator in multiple scattering problem in \(L^ 1\)- space.
Mathematics Subject Classification ID: 47A40
Mathematics Subject Classification ID: 47D03
Mathematics Subject Classification ID: 47A55
Mathematics Subject Classification ID: 82C70
zbMATH DE Number: 3969429
zbMATH Keywords: similarity for a pair of linear operators in an ordered Banach space
zbMATH Keywords: positivity preserving \(C_ 0\)-groups
zbMATH Keywords: collisionless transport operator
zbMATH Keywords: linear transport operator
zbMATH Keywords: multiple scattering problem in \(L^ 1\)-space
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/03840617
OpenAlex ID: W2144147111
DOI: 10.2969/JMSJ/03840617
description: scientific article; zbMATH DE number 3969429