scholarly journals On Bilinear Narrow Operators

Mathematics ◽  
2021 ◽  
Vol 9 (22) ◽  
pp. 2892
Author(s):  
Marat Pliev ◽  
Nonna Dzhusoeva ◽  
Ruslan Kulaev
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Vector Lattice ◽  
Cartesian Product ◽  
Bilinear Operator ◽  
Projection Property ◽  
New Class ◽  
Bilinear Operators ◽  
Vector Lattices ◽  
Order Continuous

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.

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.

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 Lebesgue's Differentiation Theorems in R.I. Quasi-Banach Spaces and Lorentz SpacesΓp,w

2012 ◽  
Vol 2012 ◽  
pp. 1-28 ◽  
Author(s):  
Maciej Ciesielski ◽  
Anna Kamińska
Keyword(s):  
Banach Space ◽  
Weight Function ◽  
Banach Spaces ◽  
Maximal Function ◽  
Measurable Function ◽  
Pointwise Convergence ◽  
Lorentz Spaces ◽  
Best Constant ◽  
Decreasing Rearrangement ◽  
Order Continuous

The paper is devoted to investigation of new Lebesgue's type differentiation theorems (LDT) in rearrangement invariant (r.i.) quasi-Banach spacesEand in particular on Lorentz spacesΓp,w={f:∫(f**)pw<∞}for any0<p<∞and a nonnegative locally integrable weight functionw, wheref**is a maximal function of the decreasing rearrangementf*for any measurable functionfon(0,α), with0<α≤∞. The first type of LDT in the spirit of Stein (1970), characterizes the convergence of quasinorm averages off∈E, whereEis an order continuous r.i. quasi-Banach space. The second type of LDT establishes conditions for pointwise convergence of the best or extended best constant approximantsfϵoff∈Γp,worf∈Γp-1,w,1<p<∞, respectively. In the last section it is shown that the extended best constant approximant operator assumes a unique constant value for any functionf∈Γp-1,w,1<p<∞.

scholarly journals Some stability properties of Arens regular bilinear operators

1991 ◽  
Vol 34 (3) ◽  
pp. 443-454
Author(s):  
A. Ülger
Keyword(s):  
Banach Spaces ◽  
Compact Hausdorff Space ◽  
Borel Measure ◽  
Hausdorff Space ◽  
Bilinear Operator ◽  
Stability Properties ◽  
Bilinear Operators

In this paper we present three results about Arens regular bilinear operators. These are: (a). Let X, Y be two Banach spaces, K a compact Hausdorff space, µ a Borel measure on K and m: X × Y →ℂ a bounded bilinear operator. Then the bilinear operator defined by is regular iff m is regular, (b) Let (Xα), (Xα),(Zα) be three families of Banach spaces and let mα:Xα ×Yα→Zα, be a family of bilinear operators with supα∥mα∥<∞. Then the bilinear operator defined by is regular iff each mα, is regular, (c) Let X, Y have the Dieudonné property and let m:X × Y→Z be a bounded bilinear operator with m(X×Y) separable and such that, for each z′ in ext Z′1, z′∘m is regular. Then m is regular. Several applications of these results are also given.

A new class of Banach space with the drop property

2012 ◽  
Vol 142 (1) ◽  
pp. 215-224 ◽  
Author(s):  
Suyalatu Wulede ◽  
Wudunqiqige Ha
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Uniformly Convex ◽  
Strongly Convex ◽  
New Class ◽  
Uniformly Convex Spaces ◽  
Convex Spaces

We discuss a new class of Banach spaces which are wider than the strongly convex spaces introduced by Congxin Wu and Yongjin Li. We prove that the new class of Banach spaces lies strictly between either the class of uniformly convex spaces and strongly convex spaces or the class of fully k-convex spaces and strongly convex spaces. The new class of Banach spaces has inclusive relations with neither the class of locally uniformly convex spaces nor the class of nearly uniformly convex spaces. We obtain in addition some characterizations of this new class of Banach spaces.

scholarly journals Components in vector lattices and extreme extensions of quasi-measures and measures

1993 ◽  
Vol 35 (2) ◽  
pp. 153-162 ◽  
Author(s):  
Z. Lipecki
Keyword(s):  
Convex Set ◽  
Vector Lattice ◽  
Convex Combination ◽  
Extreme Points ◽  
Main Body ◽  
Projection Property ◽  
Vector Lattices ◽  
Definition Of ◽  
Image Position ◽  
Theoretical Results

We develop some ideas contained in the author's paper [8] which was, in turn, inspired by Bierlein and Stich [5]. The main body of the present paper is divided into three sections. Section 2 is concerned with some vector-lattice-theoretical results. They are then applied to extensions of quasi-measures and measures in Sections 3 and 4, respectively.Let X be a vector lattice, let x ε X+ and let S be a non-empty set. Theorems 1 and 2 describe some properties of the convex set(see Section 2 for the definition of the sum above). The extreme points of Dx,s are characterized in terms of the components of x. It is also shown that if X has the principal projection property and S is countable, then extr Dx,s is, in some sense, large in Dx,s. Furthermore, for finite S, each point in Dx,s is then a sσ-convex combination of extreme ones.

scholarly journals On Reich type λ−α-nonexpansive mapping in Banach spaces with applications to L1([0,1])

2018 ◽  
Vol 19 (2) ◽  
pp. 291
Author(s):  
Rabah Belbaki ◽  
E. Karapinar ◽  
Amar Ould-Hammouda,
Keyword(s):  
Banach Space ◽  
Nonexpansive Mapping ◽  
Strong Convergence ◽  
Partial Order ◽  
Banach Spaces ◽  
Nonexpansive Mappings ◽  
Weak And Strong Convergence ◽  
Convergence Theorems ◽  
New Class ◽  
Krasnoselskii Iteration

<p>In this manuscript we introduce a new class of monotone generalized nonexpansive mappings and establish some weak and strong convergence theorems for Krasnoselskii iteration in the setting of a Banach space with partial order. We consider also an application to the space L<sub>1</sub>([0,1]). Our results generalize and unify the several related results in the literature.</p>

scholarly journals Entire Analytic Functions of Unbounded Type on Banach Spaces and Their Lineability

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 150
Author(s):  
Andriy Zagorodnyuk ◽  
Anna Hihliuk
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Analytic Functions ◽  
Entire Functions ◽  
Dimensional Algebra ◽  
Dimensional Banach Space ◽  
Infinite Dimensional ◽  
Linear Subspaces ◽  
Infinite Dimensional Banach Space

In the paper we establish some conditions under which a given sequence of polynomials on a Banach space X supports entire functions of unbounded type, and construct some counter examples. We show that if X is an infinite dimensional Banach space, then the set of entire functions of unbounded type can be represented as a union of infinite dimensional linear subspaces (without the origin). Moreover, we show that for some cases, the set of entire functions of unbounded type generated by a given sequence of polynomials contains an infinite dimensional algebra (without the origin). Some applications for symmetric analytic functions on Banach spaces are obtained.

scholarly journals The dual of the space of bounded operators on a Banach space

2021 ◽  
Vol 8 (1) ◽  
pp. 48-59
Author(s):  
Fernanda Botelho ◽  
Richard J. Fleming
Keyword(s):  
Banach Space ◽  
Tensor Product ◽  
Banach Spaces ◽  
Dual Space ◽  
Tensor Products ◽  
Bounded Operators ◽  
Projective Tensor Product ◽  
Projective Tensor

Abstract Given Banach spaces X and Y, we ask about the dual space of the 𝒧(X, Y). This paper surveys results on tensor products of Banach spaces with the main objective of describing the dual of spaces of bounded operators. In several cases and under a variety of assumptions on X and Y, the answer can best be given as the projective tensor product of X ** and Y *.

The wigner property for CL-spaces and finite-dimensional polyhedral banach spaces

2021 ◽  
pp. 1-17
Author(s):  
Dongni Tan ◽  
Xujian Huang
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Phase Function ◽  
Linear Isometry ◽  
Basic Properties ◽  
Finite Dimensional ◽  
Image Position

Abstract We say that a map $f$ from a Banach space $X$ to another Banach space $Y$ is a phase-isometry if the equality \[ \{\|f(x)+f(y)\|, \|f(x)-f(y)\|\}=\{\|x+y\|, \|x-y\|\} \] holds for all $x,\,y\in X$ . A Banach space $X$ is said to have the Wigner property if for any Banach space $Y$ and every surjective phase-isometry $f : X\rightarrow Y$ , there exists a phase function $\varepsilon : X \rightarrow \{-1,\,1\}$ such that $\varepsilon \cdot f$ is a linear isometry. We present some basic properties of phase-isometries between two real Banach spaces. These enable us to show that all finite-dimensional polyhedral Banach spaces and CL-spaces possess the Wigner property.

Sign in / Sign up

Export Citation Format

Share Document