Narrow orthogonally additive operators in lattice-normed spaces

Pliev, Marat; Fang, Xiaochun

doi:10.1134/s0037446617010177

Cited by 20 publications

(5 citation statements)

References 8 publications

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“…Then the notion was generalized to linear operators on Riesz spaces in [9], and for orthogonally additive operators on Riesz spaces in [18]. Remark also that narrow operators were defined on a more wide class of lattice normed spaces in [15] for linear operators and in [17] for orthogonally additive operators. Under natural assumptions, AM -compact (even C-compact) operators are narrow (see [9] and [18] for linear operators and [16] for orthogonally additive operators).…”

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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“…Moreover, U (E, F) is Dedekind complete, once F is [4,Theorem 3.2]. Simple examples of nonlinear abstract Uryson operators are: T 1 (x) = x + , T 2 (x) = |x| from E to E; more general, any order bounded linear operator T : E → F defines a positive abstract Uryson operator by S(x) = T|x| for all x ∈ E. For further examples (e.g., integral Uryson operators) see [4,5,7].…”

Section: Orthogonally Additive Operatorsmentioning

confidence: 99%

Projection lateral bands and lateral retracts

Kamińska

Krasikova

Popov

2020

Carpathian Math. Publ.

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A projection lateral band $G$ in a Riesz space $E$ is defined to be a lateral band which is the image of an orthogonally additive projection $Q: E \to E$ possessing the property that $Q(x)$ is a fragment of $x$ for all $x \in E$, called a lateral retraction of $E$ onto $G$ (which is then proved to be unique). We investigate properties of lateral retracts, that are, images of lateral retractions, and describe lateral retractions onto principal projection lateral bands (i.e. lateral bands generated by single elements) in a Riesz space with the principal projection property. Moreover, we prove that every lateral retract is a lateral band, and every lateral band in a Dedekind complete Riesz space is a projection lateral band.

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“…In this article, we analyse the notion of an atomic operator in the framework of the theory of vector lattices and orthogonally additive operators. Today, the theory of orthogonally additive operators in vector lattices is an active area in functional analysis; see for instance [1,2,7,8,10,11,13,14,[16][17][18]25]. Abstract results of this theory can be applied to the theory of nonlinear integral operators [12,21], and there are connections with problems of convex geometry [24].…”

Section: Introductionmentioning

confidence: 99%

Atomic Operators in Vector Lattices

Chill

Pliev

2020

Mediterr. J. Math.

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In this paper, we introduce a new class of operators on vector lattices. We say that a linear or nonlinear operator T from a vector lattice E to a vector lattice F is atomic if there exists a Boolean homomorphism $$\Phi $$ Φ from the Boolean algebra $${\mathfrak {B}}(E)$$ B ( E ) of all order projections on E to $${\mathfrak {B}}(F)$$ B ( F ) such that $$T\pi =\Phi (\pi )T$$ T π = Φ ( π ) T for every order projection $$\pi \in {\mathfrak {B}}(E)$$ π ∈ B ( E ) . We show that the set of all atomic operators defined on a vector lattice E with the principal projection property and taking values in a Dedekind complete vector lattice F is a band in the vector lattice of all regular orthogonally additive operators from E to F. We give the formula for the order projection onto this band, and we obtain an analytic representation for atomic operators between spaces of measurable functions. Finally, we consider the procedure of the extension of an atomic map from a lateral ideal to the whole space.

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Narrow orthogonally additive operators in lattice-normed spaces

Cited by 20 publications

References 8 publications

On linear sections of orthogonally additive operators

On linear sections of orthogonally additive operators

Projection lateral bands and lateral retracts

Atomic Operators in Vector Lattices

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