Optimal extension of positive order continuous operators with values in quasi-banach lattices

B. B. Tasoev

Southern Mathematical Institute of the Vladikavkaz Scientific Center of the Russian Academy of Sciences, Vladikavkaz

Abstract: The goal of this article is to present some method of optimal extension of positive order continuous and $\sigma$-order continuous operators on quasi-Banach function spaces with values in Dedekind complete quasi-Banach lattices. The optimal extension of such an operator is the smallest extension of the Bartle–Dunford–Schwartz type integral. It is also shown that if a positive operator sends order convergent sequences to quasinorm convergent sequences, then its optimal extension is the Bartle–Dunford–Schwartz type integral.

Keywords: quasi-Banach lattice, optimal extension, optimal domain, Bartle–Dunford–Schwartz integration, weakly integrable functions, Banach function space.

UDC: 517

Received: 15.01.2020
Revised: 15.01.2020
Accepted: 17.06.2020

DOI: 10.33048/smzh.2020.61.512

 English version: Siberian Mathematical Journal, 2020, 61:5, 884–894

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