On sums of narrow and compact operators

Fotiy, O.; Gumenchuk, A. I.; Krasikova, I.; Popov, Mikhail

doi:10.1007/s11117-019-00666-4

Cited by 25 publications

(8 citation statements)

References 13 publications

(29 reference statements)

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“…A number of results concerning linear narrow operators in ordered spaces were obtained in [4,[11][12][13]. The state-of-the-art theory of narrow operators is presented in [14][15][16][17].…”

Section: Discussionmentioning

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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Section: Discussionmentioning

confidence: 99%

On Bilinear Narrow Operators

2021

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“…|x − x n | ≤ u n for some sequence u n ↓ 0 is impossible. 9 The element z is maximal in D x,T,ε , if ∄ u ∈ D x,T,ε such that z ⊑ u and z = u.…”

Section: C-compact and Narrow Orthogonally Additive Operatorsmentioning

confidence: 99%

“…Note that linear narrow operators in function spaces first appeared in [17]. Nowadays the theory of narrow operators is a well-studied object of functional analysis and is presented in many research articles ( [9,11,14,15,18,23]) and in the monograph [21].…”

Section: Introductionmentioning

confidence: 99%

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

2019

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We consider C-compact orthogonally additive operators in vector lattices. After providing some examples of C-compact orthogonally additive operators on a vector lattice with values in a Banach space we show that the set of those operators is a projection band in the Dedekind complete vector lattice of all regular orthogonally additive operators. In the second part of the article we introduce a new class of vector lattices, called C-complete, andshow that any laterally-to-norm continuous C-compact orthogonally additive operator from a C-complete vector lattice to a Banach space is narrow, which generalizes a result of Pliev and Popov.

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“…Orthogonally additive operators in vector lattices first were introduced by Mazón and Segura de León in [1]. Today, the theory of these operators is an active field of the modern analysis (see [2][3][4][5][6][7][8][9]). We note that the study of orthogonally additive operators has useful applications in different areas of modern mathematics, e.g., convex geometry [10,11], dynamical systems [12], and nonlinear integral equations [13,14].…”

Section: Introductionmentioning

confidence: 99%

Orthogonally Biadditive Operators

Dzhusoeva

Kulaev

Pliev

2021

Journal of Function Spaces

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In this article, we introduce and study a new class of operators defined on a Cartesian product of ideal spaces of measurable functions. We use the general approach of the theory of vector lattices. We say that an operator T : E × F ⟶ W defined on a Cartesian product of vector lattices E and F and taking values in a vector lattice W is orthogonally biadditive if all partial operators T y : E ⟶ W and T x : F ⟶ W are orthogonally additive. In the first part of the article, we prove that, under some mild conditions, a vector space of all regular orthogonally biadditive operators O B A r E , F ; W is a Dedekind complete vector lattice. We show that the set of all horizontally-to-order continuous regular orthogonally biadditive operators is a projection band in O B A r E , F ; W . In the last section of the paper, we investigate orthogonally biadditive operators on a Cartesian product of ideal spaces of measurable functions. We show that an integral Uryson operator which depends on two functional variables is orthogonally biadditive and obtain a criterion of the regularity of an orthogonally biadditive Uryson operator.

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On sums of narrow and compact operators

Cited by 25 publications

References 13 publications

On Bilinear Narrow Operators

On Bilinear Narrow Operators

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

Orthogonally Biadditive Operators

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