On positive semigroups
type: Item
instance of: scholarly article
title: On positive semigroups
zbMATH Open document ID: 0568.47032
author: Derek W. Robinson
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 1984-01-01T00:00:00Z
review text: Let \((E,E_+,\|.\|)\) be an ordered Banach space. Assume that the given norm \(\|.\|\) as well as the norm on the dual space are monotonous. Define the associated half norm N on E by \(N(a):=\inf \{\| b\|: b\geq a,\quad b\geq 0\}.\) Under these assumptions the following characterization of the generator A of a positive semigroup is proved: Let A be a densely-defined, closed operator in E. The following assertions are equivalent: (i) A is the generator of a positive \(C_ 0\)-semigroup \((S_ t)\) with bounds \(\| S_ t\|_+(:=\sup \{\| S_ ta\|: a\geq 0,\quad \| a\| \leq 1\})\leq Me^{\omega t}\) for all \(t\geq 0.\) (ii) \(range(I+\alpha A)=E\) and \(N((I+\alpha A)^ na)\geq (1-\alpha \omega)^ nN(a)/M\) for all \(a\in {\mathcal D}(A^ n)\), \(n\geq 1\) and \(\alpha >0\) small enough.
Mathematics Subject Classification ID: 47D03
Mathematics Subject Classification ID: 47B60
zbMATH DE Number: 3907201
zbMATH Keywords: ordered Banach space
zbMATH Keywords: generator of a positive \(C_ 0\)-semigroup
reviewed by: Manfred P. H. Wolff
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/1195181837
OpenAlex ID: W2041681781
cites work: Continuous semigroups on ordered Banach spaces
cites work: The canonical half-norm, dual half-norms, and monotonic norms
cites work: The Structure and Ideal Theory of the Predual of a Banach Lattice
cites work: Q5776298
cites work: The Jordan decomposition and half-norms
DOI: 10.2977/PRIMS/1195181837
description: scientific article; zbMATH DE number 3907201