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On extreme extension of positive operators
A. G. Kusraev North-Caucasus Center for Mathematical Research VSC RAS, 1 Williams St., Mikhailovskoye village 363110, Russia
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.
Received: 24.04.2024
Citation:
A. G. Kusraev, “On extreme extension of positive operators”, Vladikavkaz. Mat. Zh., 26:2 (2024), 47–53
Linking options:
https://www.mathnet.ru/eng/vmj909 https://www.mathnet.ru/eng/vmj/v26/i2/p47
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Abstract page: | 31 | Full-text PDF : | 10 | References: | 7 |
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