Valuations on Banach Lattices

Tradacete, Pedro; Villanueva, Ignacio

doi:10.1093/imrn/rny129

Cited by 19 publications

(15 citation statements)

References 27 publications

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“…Classification results for valuations on Lipschitz functions on S n−1 were obtained in [42,43] and on Banach lattices in [127]. A Hadwiger theorem for valuations on definable functions was established in [18].…”

Section: Theorem 72 ([40]mentioning

confidence: 99%

Geometric valuation theory

Ludwig¹

2021

Preprint

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Section: Theorem 72 ([40]mentioning

confidence: 99%

Geometric valuation theory

Ludwig¹

2021

Preprint

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“…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]. In applications, it is often necessary to study integral equations depending on several variables.…”

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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“…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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Valuations on Banach Lattices

Cited by 19 publications

References 27 publications

Geometric valuation theory

Geometric valuation theory

Orthogonally Biadditive Operators

Atomic Operators in Vector Lattices

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