zbMATH — the first resource for mathematics

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)
Full Text:
References:
  Barker, G.P., Perfect cones, Linear algebra appl., 22, 211-221, (1978) · Zbl 0396.15009  Borwein, J.M.; Yost, D.T., Absolute norms on vector lattices, (), 215-222 · Zbl 0589.46005  Iochum, B., Cônes autopolaires et algèbres de Jordan, () · Zbl 0556.46040  Isac, G.; Németh, A.B.; Isac, G.; Németh, A.B., Corrigendum, Arch. math., Arch. mat., 49, 367-368, (1987) · Zbl 0624.41040  Isac, G.; Németh, A.B., Isotone projection cones in Hilbert spaces and the complementarity problem, Boll. U.M.I. (7), 3-B, (1989) · Zbl 0719.46011  Isac, G.; Németh, A.B., Ordered Hilbert spaces, (1987), preprint · Zbl 0624.41040  McArthur, C.W., In what spaces is every closed normal cone regular?, (), 121-125 · Zbl 0221.46013  Moreau, J., Décomposition orthogonale d’un espace hilbertien selon deux cônes mutuellement pollaires, C. R. acad. sci. Paris Sér. I math., 225, 238-240, (1962) · Zbl 0109.08105  Riesz, F.; Riesz, F., Sur quelques notions fondamentales dans la théorie générale des opérations linéaires, Mat. természett értes., Ann. of math., 41, 174-206, (1940) · JFM 66.0553.01  Zarantonello, E.H., Projections on convex sets in Hilbert space and spectral theory, (), 237-424  Youdine, A., Solution de deux problémes de la théorie des espaces semi-ordonnés, C. R. acad. sci. U.R.S.S., 27, 418-422, (1939) · Zbl 0021.36004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.