scholarly journals Каждая латеральная полоса является ядром положительного ортогонально аддитивного оператора

2021 ◽  
pp. 115-118
Author(s):  
M.A. Pliev
Keyword(s):  
Open Problem ◽  
Vector Lattice ◽  
Linear Operators ◽  
Lateral Band ◽  
Vector Lattices ◽  
Additive Operator ◽  
Orthogonally Additive Operator ◽  
Disjointness Preserving ◽  
Orthogonally Additive ◽  
Orthogonally Additive Operators

{In this paper we continue a study of relationships between the lateral partial order $\sqsubseteq$ in a vector lattice (the relation $x \sqsubseteq y$ means that $x$ is a fragment of $y$) and the theory of orthogonally additive operators on vector lattices. It was shown in~\cite{pMPP} that the concepts of lateral ideal and lateral band play the same important role in the theory of orthogonally additive operators as ideals and bands play in the theory for linear operators in vector lattices. We show that, for a vector lattice $E$ and a lateral band $G$ of~$E$, there exists a vector lattice~$F$ and a positive, disjointness preserving orthogonally additive operator $T \colon E \to F$ such that ${\rm ker} \, T = G$. As a consequence, we partially resolve the following open problem suggested in \cite{pMPP}: Are there a vector lattice~$E$ and a lateral ideal in $E$ which is not equal to the kernel of any positive orthogonally additive operator $T\colon E\to F$ for any vector lattice $F$?

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).

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 Lateral continuity and orthogonally additive operators

2015 ◽  
Vol 7 (1) ◽  
pp. 49-56 ◽  
Author(s):  
A.I. Gumenchuk
Keyword(s):  
Additive Operator ◽  
Orthogonally Additive Operator ◽  
Orthogonally Additive ◽  
Orthogonally Additive Operators

We generalize the notion of a laterally convergent net from increasing nets to general ones and study the corresponding lateral continuity of maps. The main result asserts that, the lateral continuity of an orthogonally additive operator is equivalent to its continuity at zero. This theorem holds for operators that send laterally convergent nets to any type convergent nets (laterally, order or norm convergent).

Orthomorphisms of archimedean vector lattices

1981 ◽  
Vol 89 (1) ◽  
pp. 119-128 ◽  
Author(s):  
S. J. Bernau
Keyword(s):  
Linear Operator ◽  
Vector Lattice ◽  
Jordan Decomposition ◽  
Duality Results ◽  
Vector Lattices ◽  
Elementary Methods ◽  
Disjointness Preserving ◽  
Order Boundedness

AbstractA linear operator T on a vector lattice L preserves disjointness if Tx ⊥ y whenever x ⊥ y. If such a T is positive it is automatically order bounded. An ortho-morphism is an order bounded disjointness preserving linear operator on L. In this note we show that the theory of orthomorphisms on archimedean vector lattices admits a totally elementary exposition. Elementary methods are also effective in duality considerations when the order dual separates points of L. For the Jordan decomposition T = T+ − T− with T+x = (Tx+)+ − (Tx−)+ we can dtrop the order boundedness assumption if we assume either that T preserves ideals or that L is normed and T is continuous. Alternatively we may keep order boundedness and assume only |Tx| ⊥ |Ty| whenever x ⊥ y. The main duality results show: T preserves ideals if and only if T** does; T is an orthomorphism if and only if T* is; T is central (|T| is bounded by a multiple of the identity) if and only if T* is central if and only if T and T* preserve ideals.

scholarly journals Points of narrowness and uniformly narrow operators

2017 ◽  
Vol 9 (1) ◽  
pp. 37-47 ◽  
Author(s):  
A.I. Gumenchuk ◽  
I.V. Krasikova ◽  
M.M. Popov
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Measure Space ◽  
Linear Operators ◽  
Continuous Linear ◽  
Standard Tool ◽  
Orthogonally Additive ◽  
Orthogonally Additive Operators ◽  
Continuous Linear Operators

It is known that the sum of every two narrow operators on $L_1$ is narrow, however the same is false for $L_p$ with $1 < p < \infty$. The present paper continues numerous investigations of the kind. Firstly, we study narrowness of a linear and orthogonally additive operators on Kothe function spaces and Riesz spaces at a fixed point. Theorem 1 asserts that, for every Kothe Banach space $E$ on a finite atomless measure space there exist continuous linear operators $S,T: E \to E$ which are narrow at some fixed point but the sum $S+T$ is not narrow at the same point. Secondly, we introduce and study uniformly narrow pairs of operators $S,T: E \to X$, that is, for every $e \in E$ and every $\varepsilon > 0$ there exists a decomposition $e = e' + e''$ to disjoint elements such that $\|S(e') - S(e'')\| < \varepsilon$ and $\|T(e') - T(e'')\| < \varepsilon$. The standard tool in the literature to prove the narrowness of the sum of two narrow operators $S+T$ is to show that the pair $S,T$ is uniformly narrow. We study the question of whether every pair of narrow operators with narrow sum is uniformly narrow. Having no counterexample, we prove several theorems showing that the answer is affirmative for some partial cases.

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.

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