The canonical half-norm, dual half-norms, and monotonic norms
01 Jan 1983-Tohoku Mathematical Journal (Mathematical Institute, Tohoku University)-Vol. 35, Iss: 3, pp 375-386
About: This article is published in Tohoku Mathematical Journal.The article was published on 1983-01-01 and is currently open access. It has received 12 citations till now. The article focuses on the topics: Norm (group).
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TL;DR: In this paper, the basic structure of positive continuous one-parameter semigroups acting on ordered Banach spaces is described, and the review is in two parts, in two stages.
Abstract: In this review we describe the basic structure of positive continuous one-parameter semigroups acting on ordered Banach spaces. The review is in two parts.
99 citations
TL;DR: In this article, the authors studied the metric projection onto a nonempty closed convex subset of a general Banach space and applied it to decompositions of Banach spaces along convex cones.
Abstract: This paper is devoted to the study of the metric projection onto a nonempty closed convex subset of a general Banach space. Thanks to a systematic use of semi-inner products and duality mappings, characterizations of the metric projection are given. Applications to decompositions of Banach spaces along convex cones and variational inequalities are presented.
22 citations
TL;DR: In this paper, the authors give an overview of normality and conormality properties of pre-ordered Banach spaces and define a class of ordered spaces called quasi-lattices which strictly contain the Banach lattices, and prove that every strictly convex reflexive ordered Banach space with a closed proper generating cone is a quasi lattice.
Abstract: We give an overview of normality and conormality properties of pre-ordered Banach spaces. For pre-ordered Banach spaces \(X\) and \(Y\) with closed cones we investigate normality of \(B(X,Y)\) in terms of normality and conormality of the underlying spaces \(X\) and \(Y\). Furthermore, we define a class of ordered Banach spaces called quasi-lattices which strictly contains the Banach lattices, and we prove that every strictly convex reflexive ordered Banach space with a closed proper generating cone is a quasi-lattice. These spaces provide a large class of examples \(X\) and \(Y\) that are not Banach lattices, but for which \(B(X,Y)\) is normal. In particular, we show that a Hilbert space \(\mathcal {H}\) endowed with a Lorentz cone is a quasi-lattice (that is not a Banach lattice if \(\dim \mathcal {H}\ge 3\)), and satisfies an identity analogous to the elementary Banach lattice identity \(\Vert |x|\Vert =\Vert x\Vert \) which holds for all elements \(x\) of a Banach lattice. This is used to show that spaces of operators between such ordered Hilbert spaces are always absolutely monotone and that the operator norm is positively attained, as is also always the case for spaces of operators between Banach lattices.
17 citations
Cites background from "The canonical half-norm, dual half-..."
...However, the concept of conormality occurs scattered under many names throughout the literature (chronologically, [13, 6, 3, 12, 9, 16, 21, 17, 22, 20, 19, 23, 5, 8, 18])....
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TL;DR: In this article, the authors define the canonical half-norm N associated with a convex cone B+ with a Riesz norm ∥ · ∥, and define N-dissipativity.
15 citations
TL;DR: In this paper, the order structure of the space of continuous linear operators on an ordered Banach space is studied and the main topic is the Robinson property, that is, the norm of a positive linear operator is attained on the positive unit cone.
Abstract: The order structure of the space of all continuous linear operators on an ordered Banach space is studied. The main topic is the Robinson property, that is, the norm of a positive linear operator is attained on the positive unit cone.
10 citations
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TL;DR: In this article, the authors define the canonical half-norm N associated with a convex cone B+ with a Riesz norm ∥ · ∥, and define N-dissipativity.
15 citations