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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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