Continuous semigroups on ordered Banach spaces
type: Item
instance of: scholarly article
title: Continuous semigroups on ordered Banach spaces
zbMATH Open document ID: 0535.47021
DOI: 10.1016/0022-1236(83)90014-9
author: Derek W. Robinson
published in: Journal of Functional Analysis
publication date: 1983-01-01T00:00:00Z
review text: The Hille-Yosida theorem establishes that a linear operator H generates a \(C_ 0\)-contraction semigroup if and only if H is norm closed norm densely defined and satisfies both the range condition \(R(I+\alpha H)=\beta\) and the dissipativity condition, \(\|(I+\alpha H)a\| \geq \| a\|,a\in D(H)for\alpha>0.\) The author obtains a direct analogue for \(C_ 0\)-semigroup S on an ordered Banach space \((\beta,\beta_+,\|.\|)\) with the additional property \(S\beta_+\subseteq \beta_+\) in the following theorem. Let H be a linear operator on the ordered Banach space \((\beta,\beta_+,\|.\|)\) and let N denote the canonical half-norm associated with \(\beta_+\). Assume \(\|.\|\) is a Riesz norm. The following conditions are equivalent. (1) H generates a \(C_ 0\)-semigroup S of positive contractions (2) H is norm-densely defined, norm-closed, and satisfies both the range condition \(R(I+\alpha H)=\beta\) and the dissipativity condition \(N(I+\alpha H)a\geq N(a),a\in D(H),\) for all small \(\alpha>0.\)
Mathematics Subject Classification ID: 47B60
Mathematics Subject Classification ID: 47B44
Mathematics Subject Classification ID: 47D03
zbMATH DE Number: 3849917
zbMATH Keywords: \(C_ 0\)-semigroup on an ordered Banach space
zbMATH Keywords: Hille-Yosida theorem
zbMATH Keywords: dissipativity condition,
zbMATH Keywords: Riesz norm
reviewed by: D. Somasundaram
MaRDI profile type: Publication
full work available at URL: 0022-1236(83)90014-9
OpenAlex ID: W2044241303
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description: scientific article; zbMATH DE number 3849917