Narrow and C-compact Orthogonally Additive Operators in Lattice-Normed Spaces

Pliev, Marat; Polat, Faruk; Weber, Martin R.

doi:10.1007/s00025-019-1075-y

Cited by 23 publications

(4 citation statements)

References 26 publications

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“…showed that narrow operators actually have a vector lattice nature and extended the theory of these operators to the setting of vector lattices. Later some deep results on narrow operators were obtained for the case of nonlinear operators [1, 11, 18, 32, 34]. We note that authors of all above‐mentioned articles studied operators defined on real vector lattices.…”

Section: Introductionmentioning

confidence: 94%

Lateral order on complex vector lattices and narrow operators

Dzhusoeva

Huang

Pliev

et al. 2023

Mathematische Nachrichten

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In this paper, we continue investigation of the lateral order on vector lattices started in [25]. We consider the complexification 𝐸 ℂ of a real vector lattice 𝐸 and introduce the lateral order on 𝐸 ℂ . Our first main result asserts that the set of all fragments 𝔉 𝑣 of an element 𝑣 ∈ 𝐸 ℂ of the complexification of an uniformly complete vector lattice 𝐸 is a Boolean algebra. Then, we study narrow operators defined on the complexification 𝐸 ℂ of a vector lattice 𝐸, extending the results of articles [22,27,28] to the setting of operators defined on complex vector lattices.We prove that every order-to-norm continuous linear operator  ∶ 𝐸 ℂ → 𝑋 from the complexification 𝐸 ℂ of an atomless Dedekind complete vector lattice 𝐸 to a finite-dimensional Banach space 𝑋 is strictly narrow. Then, we prove that every 𝐶-compact order-to-norm continuous linear operator  from 𝐸 ℂ to a Banach space 𝑋 is narrow. We also show that every regular order-no-norm continuous linear operator from 𝐸 ℂ to a complex Banach lattice (𝓁 𝑝 () ℂ is narrow. Finally, in the last part of the paper we investigate narrow operators taking values in symmetric ideals of compact operators.

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Section: Introductionmentioning

confidence: 94%

Lateral order on complex vector lattices and narrow operators

Dzhusoeva

Huang

Pliev

et al. 2023

Mathematische Nachrichten

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“…In this section, we consider C-compact bilinear operators and show that C-compactness of a bilinear operator implies its narrowness. We note that C-compact operators on vector lattices and lattice-normed spaces were investigated recently in [5][6][7]. Definition 6.…”

Section: C-compact and Narrow Operatorsmentioning

confidence: 99%

On Bilinear Narrow Operators

2021

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In this article, we introduce a new class of operators on the Cartesian product of vector lattices. We say that a bilinear operator T:E×F→W defined on the Cartesian product of vector lattices E and F and taking values in a vector lattice W is narrow if the partial operators Tx and Ty are narrow for all x∈E,y∈F. We prove that, for order-continuous Köthe–Banach spaces E and F and a Banach space X, the classes of narrow and weakly function narrow bilinear operators from E×F to X are coincident. Then, we prove that every order-to-norm continuous C-compact bilinear regular operator T is narrow. Finally, we show that a regular bilinear operator T from the Cartesian product E×F of vector lattices E and F with the principal projection property to an order continuous Banach lattice G is narrow if and only if |T| is.

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“…In the context of vector lattices, orthogonally additive operators were originally considered by Mazón and Segura de León in the 1990s (see [28] and [45]). In the recent years the general theory of OAOs in ordered spaces was actively developed both in Russia and other countries (see [3], [13], [23]- [25], [36] and [43]).…”

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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Narrow and C-compact Orthogonally Additive Operators in Lattice-Normed Spaces

Cited by 23 publications

References 26 publications

Lateral order on complex vector lattices and narrow operators

Lateral order on complex vector lattices and narrow operators

On Bilinear Narrow Operators

Diffuse orthogonally additive operators

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