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Vladikavkazskii Matematicheskii Zhurnal, 2024, Volume 26, Number 2, Pages 47–53
DOI: https://doi.org/10.46698/s3201-6067-0570-n
(Mi vmj909)
 

On extreme extension of positive operators

A. G. Kusraev

North-Caucasus Center for Mathematical Research VSC RAS, 1 Williams St., Mikhailovskoye village 363110, Russia
References:
Abstract: Given vector lattices $E$, $F$ and a positive operator $S$ from a majorzing subspace $D$ of $E$ to $F$, denote by $\mathcal{E}(S)$ the collection of all positive extensions of $S$ to all of $E$. This note aims to describe the collection of extreme points of the convex set $\mathcal{E}(T\circ S)$. It is proved, in particular, that $\mathcal{E}(T\circ S)$ and $T\circ\mathcal{E}(S)$ coincide and every extreme point of $\mathcal{E}(T\circ S)$ is an extreme point of $T\circ\mathcal{E}(S)$, whenever $T:F\to G$ is a Maharam operator between Dedekind complete vector lattices. The proofs of the main results are based on the three ingredients: a characterization of extreme points of subdifferentials, abstract disintegration in Kantorovich spaces, and an intrinsic characterization of subdifferentials.
Key words: vector lattice, positive operator, extreme extension, subdifferential, Maharam operator.
Funding agency Grant number
Ministry of Science and Higher Education of the Russian Federation 075-02-2024-1379
The research was executed at the Regional mathematical center of North-Caucasus Center for Mathematical Research of the Vladikavkaz Scientific Centre of the Russian Academy of Sciences, agreement № 075-02-2024-1379.
Received: 24.04.2024
Document Type: Article
UDC: 517.98
Language: English
Citation: A. G. Kusraev, “On extreme extension of positive operators”, Vladikavkaz. Mat. Zh., 26:2 (2024), 47–53
Citation in format AMSBIB
\Bibitem{Kus24}
\by A.~G.~Kusraev
\paper On extreme extension of positive operators
\jour Vladikavkaz. Mat. Zh.
\yr 2024
\vol 26
\issue 2
\pages 47--53
\mathnet{http://mi.mathnet.ru/vmj909}
\crossref{https://doi.org/10.46698/s3201-6067-0570-n}
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