ORTHOGONALLY ADDITIVE POLYNOMIALS ON DEDEKIND σ-COMPLETE VECTOR LATTICES

2010 ◽  
Vol 110A (1) ◽  
pp. 83-94
Author(s):  
Mohamed Ali Toumi
Keyword(s):  
Vector Lattices ◽  
Complete Vector ◽  
Orthogonally Additive

scholarly journals Atomic Operators in Vector Lattices

2020 ◽  
Vol 17 (5) ◽  
Author(s):  
Ralph Chill ◽  
Marat Pliev
Keyword(s):  
Nonlinear Operator ◽  
Vector Lattice ◽  
Projection Property ◽  
New Class ◽  
Vector Lattices ◽  
Complete Vector Lattice ◽  
Whole Space ◽  
Complete Vector ◽  
Orthogonally Additive ◽  
Orthogonally Additive Operators

Abstract 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.

scholarly journals Orthogonally Biadditive Operators

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Nonna Dzhusoeva ◽  
Ruslan Kulaev ◽  
Marat Pliev
Keyword(s):  
Vector Lattice ◽  
Cartesian Product ◽  
New Class ◽  
Mild Conditions ◽  
Vector Lattices ◽  
Complete Vector Lattice ◽  
Measurable Functions ◽  
Complete Vector ◽  
Orthogonally Additive ◽  
Order Continuous

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.

Orthogonality of polynomials and orthosymmetry

2015 ◽  
Vol 27 (3) ◽  
Author(s):  
Mohamed Ali Toumi
Keyword(s):  
Vector Space ◽  
Vector Lattices ◽  
Multilinear Maps ◽  
Orthogonally Additive

AbstractOrthogonality of homogeneous orthogonally additive polynomials and orthosymmetry of multilinear maps are in general independent properties. In this paper, we show, for some important vector lattices, that the vector space of all orthogonally additive scalar-valued polynomials and the vector space of orthosymmetric scalar-valued multilinear maps are isomorphic.

scholarly journals ON SEPARATE ORDER CONTINUITY OF ORTHOGONALLY ADDITIVE OPERATORS

2021 ◽  
Vol 9 (1) ◽  
pp. 200-209
Author(s):  
I. Krasikova ◽  
O. Fotiy ◽  
M. Pliev ◽  
M. Popov
Keyword(s):  
Order Continuity ◽  
Vector Lattices ◽  
Additive Operator ◽  
Uniformly Convergent ◽  
Orthogonally Additive Operator ◽  
Orthogonally Additive ◽  
Order Continuous ◽  
Orthogonally Additive Operators

Our main result asserts that, under some assumptions, the uniformly-to-order continuity of an order bounded orthogonally additive operator between vector lattices together with its horizontally-to-order continuity implies its order continuity (we say that a mapping f : E → F between vector lattices E and F is horizontally-to-order continuous provided f sends laterally increasing order convergent nets in E to order convergent nets in F, and f is uniformly-to-order continuous provided f sends uniformly convergent nets to order convergent nets).

Multiplication in Vector Lattices

1968 ◽  
Vol 20 ◽  
pp. 1136-1149 ◽  
Author(s):  
Norman M. Rice
Keyword(s):  
Vector Lattice ◽  
Weak Order ◽  
Order Unit ◽  
Dedekind Complete Vector Lattice ◽  
Vector Lattices ◽  
Complete Vector Lattice ◽  
Complete Vector

B. Z. Vulih has shown (13) how an essentially unique intrinsic multiplication can be defined in a Dedekind complete vector lattice L having a weak order unit. Since this work is available only in Russian, a brief outline is given in § 2 (cf. also the review by E. Hewitt (4), and for details, consult (13) or (11)).

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