Characterization of latticial cones in Hilbert spaces by isotonicity and generalized infimum |
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Authors: | S Z Németh |
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Institution: | (1) Royal Military College of Canada, Kingston, Ontario, Canada |
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Abstract: | A mapping is called isotone if it is monotone increasing with respect to the order defined by a pointed closed convex cone.
Finding the pointed closed convex generating cones for which the projection mapping onto the cone is isotone is a difficult
problem which was analyzed in 1, 2, 3, 4, 5]. Such cones are called isotone projection cones. In particular it was shown
that any isotone projection cone is latticial 2]. This problem is extended by replacing the projection mapping with a continuous
isotone retraction onto the cone. By introducing the notion of sharp mappings, it is shown that a pointed closed convex generating
cone is latticial if and only if there is a continuous isotone retraction onto the cone whose complement is sharp. This result
is used for characterizing a subdual latticial cone by the isotonicity of a generalization of the positive part mapping x ↦ x
+. This generalization is achieved by generalizing the infimum for subdual cones. The theoretical results of this paper exhibit
fundamental properties of the lattice structure of the space which were not analysed before. |
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Keywords: | |
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