By a News Reporter-Staff News Editor at Journal of Mathematics -- Data detailed on Positivity have been presented. According to news reporting out of Leiden, Netherlands, by VerticalNews editors, the research stated, “If L and M are partially ordered vector spaces, then one can consider regular linear maps from L to M, i.e. linear maps which can be written as the difference of two positive linear maps. If the space L is directed, then the space of all regular linear operators becomes a partially ordered vector space itself.”
Financial support for this research came from Leiden University.
Our news journalists obtained a quote from the research from Leiden University, “We will mainly concern ourselves with the questions when the space is itself a Riesz space and how, even if it is not a Riesz space, its lattice operations work. The so-called Riesz-Kantorovich theorem gives sufficient conditions for which is a Riesz space and it also specifies the lattice operations by means of the Riesz-Kantorovich formula: if and with then the supremum in the point x is given by (S boolean OR T) (x) = sup {S(y) + T(x - y): 0<= y<= x}. It is still an open problem if whenever in a more general setting the supremum of two regular operators exists in , it automatically is given by the Riesz-Kantorovich formula. Our main result concerns the special case where L is a partially ordered vector space with a strong order unit and M is a (possibly infinite) product of copies of the real line, equipped with the lexicographic ordering.”
According to the news editors, the research concluded: “It will turn out that under some mild continuity and regularity conditions the lattice operations on are indeed given by the Riesz-Kantorovich formula, even though the space is not necessarily a Riesz space.”
For more information on this research see: The Riesz-Kantorovich formula for lexicographically ordered spaces. Positivity , 2018;22(2):609-627. Positivity can be contacted at: Springer, Van Godewijckstraat 30, 3311 Gz Dordrecht, Netherlands. (Springer - www.springer.com; Positivity - http://www.springerlink.com/content/1385-1292/)
Our news journalists report that additional information may be obtained by contacting W.M. Schouten, Leiden University, Math Inst, NL-2300 RA Leiden, Netherlands.
The direct object identifier (DOI) for that additional information is: https://doi.org/10.1007/s11117-017-0531-8. This DOI is a link to an online electronic document that is either free or for purchase, and can be your direct source for a journal article and its citation.
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CITATION: (2018-04-24), Findings in Positivity Reported from Leiden University (The Riesz-Kantorovich formula for lexicographically ordered spaces), Journal of Mathematics, 264, ISSN: 1945-8746, BUTTER® ID: 015560693
From the newsletter Journal of Mathematics.
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