Order Continuity of Orthogonally Additive Operators

Fotiy, O.; Krasikova, I.; Pliev, Marat; Popov, Mikhail

doi:10.1007/s00025-021-01543-x

Cited by 12 publications

(2 citation statements)

References 9 publications

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“…and disjointly continuous in terminology of [10]) on Similarly we define the horizontal σ-order continuity and σ-order continuity by replacing nets with sequences. Obviously, the order continuity implies the rest of continuities but not converse (see [4] for details).…”

Section: Proposition 1 ([21]mentioning

confidence: 99%

On linear sections of orthogonally additive operators

Gumenchuk¹,

Krasikova²,

Popov³

2022

Mat. Stud.

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Our first result asserts that, for linear regular operators acting from a Riesz space with the principal projection property to a Banach lattice with an order continuous norm, the $C$-compactness is equivalent to the $AM$-compactness. Next we prove that, under mild assumptions, every linear section of a $C$-compact orthogonally additive operator is $AM$-compact, and every linear section of a narrow orthogonally additive operator is narrow.

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Section: Proposition 1 ([21]mentioning

confidence: 99%

On linear sections of orthogonally additive operators

Gumenchuk¹,

Krasikova²,

Popov³

2022

Mat. Stud.

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“…The solution of the von Neumann problem depends substantially on the presence of 'rich' order structure on the set of regular linear integral operators on the L 2 [0, 1]. Various order properties of the space OA r (E, F ) were considered in [10]- [12], [26] and [32].…”

Section: § 1 Introduction and Preliminariesmentioning

confidence: 99%

Diffuse orthogonally additive operators

Abasov,

Dzhusoeva,

Pliev

2024

Sb. Math.

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A regular orthogonally additive operator is a diffuse operator if it is disjoint from all operators in the band generated by the disjointness preserving operators. We present a criterion for principal lateral projections in an order complete vector lattice $E$ to be disjoint. We also state a criterion for a regular orthogonally additive operator to be diffuse. A criterion for the regularity of an integral Urysohn operator acting on ideal spaces of measurable functions is also presented. This criterion is used to show that an integral operator is diffuse. Examples of vector lattices are considered in which the sets of diffuse operators consist only of the zero element. The general form of an order projection operator onto the band generated by the disjointness preserving operators is found. Bibliography: 47 titles.

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Orthogonally Additive Operators on Vector Lattices

Pliev

Popov

2022

Trends in Mathematics

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Order Continuity of Orthogonally Additive Operators

Cited by 12 publications

References 9 publications

On linear sections of orthogonally additive operators

On linear sections of orthogonally additive operators

Diffuse orthogonally additive operators

Orthogonally Additive Operators on Vector Lattices

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