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Every generating isotone projection cone is latticial and correct. (English) Zbl 0725.46002
The subset K of the Hilbert space H is called a (pointed convex positive) cone provided that \(K+K\subseteq K\), \(\lambda\) \(K\subseteq K\) for all real numbers \(\lambda\geq 0\), and \(K\cap (-K)=\{0\}\). The cone K is said to be generating if \(K-K=H\). Assuming that K is closed, then, for each \(x\in H\), there is a unique element \(P_ K(x)\in K\), called the projection of x onto K, for which \(\| x-P_ K(x)\| \leq \| x- y\|\) for every y in K. The cone K also defines a partial order on H by the relation \(x\leq y\Leftrightarrow y-x\in K\). In the paper under review, the autors’ aim is to characterize those cones K for which the projection mapping \(P_ K\) is monotone increasing; that is, for which \(x\leq y\) implies that \(P_ K(x)\leq P_ K(y)\). Such a cone is called an isotone projection cone and the main result of the paper is the necessary condition for a generating cone to be an isotone projection cone announced in the paper’s title. (See Theorem 1 and Propositions 3 and 6.)
Two additional terms must be defined: latticial and correct. With the ordering defined by the cone K, the space H is a vector lattice and the cone K is called latticial if every pair of elements in H has a least upper bound in this lattice. In the proof of Proposition 3, showing that a generating isotone projection cone is latticial, the authors essentially construct the least upper bound of an arbitrary pair of elements. A face of the cone K is a convex cone \(F\subseteq K\) such that, if \(x\in K\), \(y\in F\) and \(x\leq y\), then \(x\in F\) and we say that K is correct if, for each face F of K, we have \(P_{spF}(K)\subseteq K\), where spF denotes the closed linear span of F.
In the finite-dimensional setting, the authors show (Theorem 9) that the necessary condition of the title is also sufficient; that is, a generating cone in \({\mathbb{R}}^ n\) is an isotone projection cone if and only if it is latticial and correct. In the infinite-dimensional case, it remains an open question whether or not a generating, latticial, and correct cone must be an isotone projection cone.

MSC:
46A40 Ordered topological linear spaces, vector lattices
46C05 Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product)
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