Existence of spectral values for irreducible \(C_ 0\)-semigroups
type: Item
instance of: scholarly article
title: Existence of spectral values for irreducible \(C_ 0\)-semigroups
zbMATH Open document ID: 0639.47033
DOI: 10.1016/0022-1236(87)90042-5
published in: Journal of Functional Analysis
publication date: 1987-01-01T00:00:00Z
review text: A \(C_ 0\)-semigroup \((T_ t)\) of positive linear operators on a Banach lattice E is called irreducible if it leaves no closed lattice ideals \(\neq \{0\}\), E invariant. Let us denote by A the generator of \((T_ t)\) and s(A) the spectral bound sup\(\{\) Re \(\lambda\) : \(\lambda\in \sigma (A)\}\). It is proved that each of the following conditions on E and \(T_ t\) implies that \(s(A)>-\infty:\) (i) \(E=C_ 0(U)\), U locally compact. (ii) The positive cone \(E_+\) contains an extreme ray. (iii) For some \(t>0\), \(T_ t\) is compact. (iii') The generator A has compact resolvent. (iv) E is order complete and for some \(t>0\), \(T_ t\) is an abstract kernel operator. In particular, in cases (iii) and (iii') the generator A of \(T_ t\) has the eigenvalue s(A) with corresponding positive eigenvector. Analogous results are established for the band irreducible \(C_ 0\)- semigroups.
reviewed by: Y. Eidelman
Mathematics Subject Classification ID: 47D03
Mathematics Subject Classification ID: 47B60
zbMATH DE Number: 4042632
zbMATH Keywords: C\({}_ 0\)-semigroup of positive linear operators on a Banach lattice
zbMATH Keywords: spectral bound
zbMATH Keywords: abstract kernel operator
zbMATH Keywords: band irreducible \(C_ 0\)- semigroups
author: Helmut H. Schaefer
MaRDI profile type: Publication
full work available at URL: 0022-1236(87)90042-5
OpenAlex ID: W2068753677
cites work: Q3923136
cites work: Q4726921
cites work: One-parameter semigroups of positive operators
cites work: Irreducible compact operators
cites work: Topologische Nilpotenz irreduzibler Operatoren
cites work: Q4042131
cites work: On theorems of de Pagter and Ando-Krieger
cites work: Q3669960
description: scientific article; zbMATH DE number 4042632