Optimal Extension of Positive Order Continuous Operators with Values in Quasi-Banach Lattices
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MAL EXTENSION OF POSITIVE ORDER CONTINUOUS OPERATORS WITH VALUES IN QUASI-BANACH LATTICES B. B. Tasoev
UDC 517
Abstract: The goal of this article is to present some method of optimal extension of positive order continuous and σ-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. DOI: 10.1134/S0037446620050122 Keywords: quasi-Banach lattice, optimal extension, optimal domain, Bartle–Dunford–Schwartz integration, weakly integrable functions, Banach function space
1. Introduction Let (Ω, Σ, μ) be a measure space, let X(μ) be a Banach function space in L0 (μ) := L0 (Ω, Σ, μ), and let T : X(μ) → E be a linear operator with values in the Banach space E that possesses some property P . The optimal extension problem can be formulated as follows: Does T admit some extension to a wider domain with values in E that preserves P ? The optimal extension problem was studied in numerous articles (see, for example, [1–4]). In particular it was shown in [1] that if X(μ) is an order continuous Banach function space; then, given a continuous operator T , we can define the vector measure νT on a suitable δ-subring of the σ-algebra Σ by the formula νT (A) := T (χA ). Then the space L1 (νT ) of integrable functions in the sense of Bartle, Dunford, and Schwarz with respect to the vector measure νT is the greatest Banach ideal space to which the operator T extends with the preservation of continuity. The optimal extension problem for positive order continuous operators with values in the Banach lattices with the Fatou and Levi properties was investigated in [2]. It was shown that the space L1w (νT ) of weakly integrable functions with respect to the vector measure νT is the greatest Banach function space to which the operator T extends with the preservation of order continuity. The present article deals with the optimal extension problem for positive order continuous and σ-order continuous operators on quasi-Banach function spaces with values in arbitrary Dedekind complete quasiBanach lattices. It should be noted that, in the case of quasi-Banach lattices, the definition of L1w (νT ) is inapplicable since the latter is based on duality theory. The main tool for our constructions is the smallest extension of the Bartle–Dunford–Schwarz type integral which is proposed in [5, 6]. The article is organized as follows: In Section 2, we give preliminary information about quasiBanach lattices. In Section 3, we give the definition of Bartle–Dunford–Schwarz integral and some of the important properties of the integral. In Section 4, using the concept of the smallest extension of a positive operator, we define the space of weakly integrable functions and provide necessary information on these spac
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