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12 pages, 279 KB

Open AccessArticle

Construction of ε-Nets for the Space of Planar Convex Bodies Endowed with the Banach–Mazur Metric

by Yanmei Chen, Yunfang Lyu, Shenghua Gao and Senlin Wu

Mathematics 2025, 13(8), 1358; https://doi.org/10.3390/math13081358 - 21 Apr 2025

Viewed by 409

In Chuanming Zong’s program to attack Hadwiger’s covering conjecture, which is a long-standing open problem from convex and discrete geometry, the construction of

ε

-nets for the space of convex bodies endowed with the Banach–Mazur metric plays a crucial role. Recently, Gao et [...] Read more.

In Chuanming Zong’s program to attack Hadwiger’s covering conjecture, which is a long-standing open problem from convex and discrete geometry, the construction of

ε

-nets for the space of convex bodies endowed with the Banach–Mazur metric plays a crucial role. Recently, Gao et al. provided a possible way of constructing

ε

-nets for

(K^{n}, d_{B M})

based on finite subsets of

Z^{n}

theoretically. In this work, we present an algorithm to construct

ε

-nets for

(K^{2}, d_{B M})

and a

(1 / 4)

-net for

(C^{2}, d_{B M})

is constructed. To the best of our knowledge, this is the first concrete

ε

-net for

(C^{2}, d_{B M})

for such a small

ε

. Full article

(This article belongs to the Section B: Geometry and Topology)

13 pages, 277 KB

Open AccessArticle

Growth Spaces on Circular Domains Taking Values in a Banach Lattice, Embeddings and Composition Operators

by Nihat Gökhan Göğüş

Mathematics 2024, 12(16), 2554; https://doi.org/10.3390/math12162554 - 19 Aug 2024

Viewed by 1074

Abstract

We introduce the space of holomorphic growth spaces with values in a Banach lattice. We provide norm and essential norm estimates of the embedding operator, and we completely characterize the bounded and compact embeddings of the growth spaces using vector-valued Carleson measures. As [...] Read more.

We introduce the space of holomorphic growth spaces with values in a Banach lattice. We provide norm and essential norm estimates of the embedding operator, and we completely characterize the bounded and compact embeddings of the growth spaces using vector-valued Carleson measures. As an application, we prove a characterization of weighted composition operators. Full article

(This article belongs to the Special Issue Complex Analysis and Geometric Function Theory, 2nd Edition)

13 pages, 304 KB

Open AccessArticle

Lipschitz Transformations and Maurey-Type Non-Homogeneous Integral Inequalities for Operators on Banach Function Spaces

by Roger Arnau and Enrique A. Sánchez-Pérez

Mathematics 2023, 11(22), 4599; https://doi.org/10.3390/math11224599 - 9 Nov 2023

Viewed by 1203

Abstract

We introduce a method based on Lipschitz pointwise transformations to define a distance on a Banach function space from its norm. We show how some specific lattice geometric properties (p-convexity, p-concavity, p-regularity) or, equivalently, some types of summability conditions [...] Read more.

We introduce a method based on Lipschitz pointwise transformations to define a distance on a Banach function space from its norm. We show how some specific lattice geometric properties (p-convexity, p-concavity, p-regularity) or, equivalently, some types of summability conditions (for example, when the terms of the terms in the sums in the range of the operator are restricted to the interval

[- 1, 1]

) can be studied by adapting the classical analytical techniques of the summability of operators on Banach lattices, which recalls the work of Maurey. We show a technique to prove new integral dominations (equivalently, operator factorizations), which involve non-homogeneous expressions constructed by pointwise composition with Lipschitz maps. As an example, we prove a new family of integral bounds for certain operators on Lorentz spaces. Full article

(This article belongs to the Special Issue Ordered Topological Vector Spaces, Operators Acting on Them and Related Applications)

16 pages, 345 KB

Open AccessArticle

Property (h) of Banach Lattice and Order-to-Norm Continuous Operators

by Fu Zhang, Hanhan Shen and Zili Chen

Mathematics 2023, 11(12), 2747; https://doi.org/10.3390/math11122747 - 17 Jun 2023

Cited by 1 | Viewed by 2120

Abstract

In this paper, we introduce the property

(h)

on Banach lattices and present its characterization in terms of disjoint sequences. Then, an example is given to show that an order-to-norm continuous operator may not be

σ

-order continuous. Suppose [...] Read more.

In this paper, we introduce the property

(h)

on Banach lattices and present its characterization in terms of disjoint sequences. Then, an example is given to show that an order-to-norm continuous operator may not be

σ

-order continuous. Suppose

T : E \to F

is an order-bounded operator from Dedekind

σ

-complete Banach lattice E into Dedekind complete Banach lattice F. We prove that T is

σ

-order-to-norm continuous if and only if T is both order weakly compact and

σ

-order continuous. In addition, if E can be represented as an ideal of

L_{0} (μ)

, where

(Ω, Σ, μ)

is a

σ

-finite measure space, then T is

σ

-order-to-norm continuous if and only if T is order-to-norm continuous. As applications, we extend Wickstead’s results on the order continuity of norms on E and

E^{'}

. Full article

(This article belongs to the Special Issue Ordered Topological Vector Spaces, Operators Acting on Them and Related Applications)

16 pages, 359 KB

Open AccessReview

Markov Moment Problem and Sandwich Conditions on Bounded Linear Operators in Terms of Quadratic Forms

by Octav Olteanu

Mathematics 2022, 10(18), 3288; https://doi.org/10.3390/math10183288 - 10 Sep 2022

Cited by 2 | Viewed by 1544

Abstract

As is well-known, unlike the one-dimensional case, there exist nonnegative polynomials in several real variables that are not sums of squares. First, we briefly review a method of approximating any real-valued nonnegative continuous compactly supported function defined on a closed unbounded subset by [...] Read more.

As is well-known, unlike the one-dimensional case, there exist nonnegative polynomials in several real variables that are not sums of squares. First, we briefly review a method of approximating any real-valued nonnegative continuous compactly supported function defined on a closed unbounded subset by dominating special polynomials that are sums of squares. This also works in several-dimensional cases. To perform this, a Hahn–Banach-type theorem (Kantorovich theorem on an extension of positive linear operators), a Haviland theorem, and the notion of a moment-determinate measure are applied. Second, completions and other results on solving full Markov moment problems in terms of quadratic forms are proposed based on polynomial approximation. The existence and uniqueness of the solution are discussed. Third, the characterization of the constraints

T_{1} \leq T \leq T_{2}

for the linear operator

T,

only in terms of quadratic forms, is deduced. Here,

T_{1}, T, and T_{2}

are bounded linear operators. Concrete spaces, operators, and functionals are involved in our corollaries or examples. Full article

(This article belongs to the Special Issue Variational Problems and Applications)

12 pages, 305 KB

Open AccessArticle

Applications for Unbounded Convergences in Banach Lattices

by Zhangjun Wang and Zili Chen

Fractal Fract. 2022, 6(4), 199; https://doi.org/10.3390/fractalfract6040199 - 1 Apr 2022

Cited by 4 | Viewed by 2639

Abstract

Several recent papers investigated unbounded convergences in Banach lattices. The focus of this paper is to apply the results of unbounded convergence to the classical Banach lattice theory from a new perspective. Combining all unbounded convergences, including unbounded order (norm, absolute weak, absolute [...] Read more.

Several recent papers investigated unbounded convergences in Banach lattices. The focus of this paper is to apply the results of unbounded convergence to the classical Banach lattice theory from a new perspective. Combining all unbounded convergences, including unbounded order (norm, absolute weak, absolute weak*) convergence, we characterize L-weakly compact sets, L-weakly compact operators and M-weakly compact operators on Banach lattices. For applications, we introduce so-called statistical-unbounded convergence and use these convergences to describe KB-spaces and reflexive Banach lattices. Full article

7 pages, 243 KB

Open AccessArticle

Continuous Operators for Unbounded Convergence in Banach Lattices

by Zhangjun Wang and Zili Chen

Mathematics 2022, 10(6), 966; https://doi.org/10.3390/math10060966 - 17 Mar 2022

Cited by 26 | Viewed by 1995

Abstract

Recently, continuous functionals for unbounded order (norm, weak and weak*) in Banach lattices were studied. In this paper, we study the continuous operators with respect to unbounded convergences. We first investigate the approximation property of continuous operators for unbounded convergence. Then we show [...] Read more.

Recently, continuous functionals for unbounded order (norm, weak and weak*) in Banach lattices were studied. In this paper, we study the continuous operators with respect to unbounded convergences. We first investigate the approximation property of continuous operators for unbounded convergence. Then we show some characterizations of the continuity of the continuous operators for

u o

,

u n

,

u a w

and

u a w^{*}

-convergence. Based on these results, we discuss the order-weakly compact operators on Banach lattices. Full article

9 pages, 333 KB

Open AccessArticle

A Note of Jessen’s Inequality and Their Applications to Mean-Operators

by Gul I Hina Aslam, Amjad Ali and Khaled Mehrez

Mathematics 2022, 10(6), 879; https://doi.org/10.3390/math10060879 - 10 Mar 2022

Cited by 1 | Viewed by 2281

Abstract

A variant of Jessen’s type inequality for a semigroup of positive linear operators, defined on a Banach lattice algebra, is obtained. The corresponding mean value theorems lead to a new family of mean-operators. Full article

(This article belongs to the Special Issue Advances in Mathematical Inequalities and Applications)

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12 pages, 293 KB

Open AccessArticle

On Bilinear Narrow Operators

by Marat Pliev, Nonna Dzhusoeva and Ruslan Kulaev

Mathematics 2021, 9(22), 2892; https://doi.org/10.3390/math9222892 - 13 Nov 2021

Cited by 2 | Viewed by 1778

Abstract

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 \times F \to W

defined on the Cartesian product of vector lattices E and F and taking [...] Read more.

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 \times F \to 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

T_{x}

and

T_{y}

are narrow for all

x \in E, y \in 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 \times 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 \times 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. Full article

18 pages, 356 KB

Open AccessArticle

Spaces of Pointwise Multipliers on Morrey Spaces and Weak Morrey Spaces

by Eiichi Nakai and Yoshihiro Sawano

Mathematics 2021, 9(21), 2754; https://doi.org/10.3390/math9212754 - 29 Oct 2021

Cited by 2 | Viewed by 1909

Abstract

The spaces of pointwise multipliers on Morrey spaces are described in terms of Morrey spaces, their preduals, and vector-valued Morrey spaces introduced by Ho. This paper covers weak Morrey spaces as well. The result in the present paper completes the characterization of the [...] Read more.

The spaces of pointwise multipliers on Morrey spaces are described in terms of Morrey spaces, their preduals, and vector-valued Morrey spaces introduced by Ho. This paper covers weak Morrey spaces as well. The result in the present paper completes the characterization of the earlier works of the first author’s papers written in 1997 and 2000, as well as Lemarié-Rieusset’s 2013 paper. As a corollary, the main result in the present paper shows that different quasi-Banach lattices can create the same vector-valued Morrey spaces. The goal of the present paper is to provide a complete picture of the pointwise multiplier spaces. Full article

(This article belongs to the Special Issue Recent Developments of Function Spaces and Their Applications I)

24 pages, 406 KB

Open AccessReview

On Markov Moment Problem, Polynomial Approximation on Unbounded Subsets, and Mazur–Orlicz Theorem

by Octav Olteanu

Symmetry 2021, 13(10), 1967; https://doi.org/10.3390/sym13101967 - 18 Oct 2021

Cited by 1 | Viewed by 1911

Abstract

We review earlier and recent results on the Markov moment problem and related polynomial approximation on unbounded subsets. Such results allow proving the existence and uniqueness of the solutions for some Markov moment problems. This is the first aim of the paper. Our [...] Read more.

We review earlier and recent results on the Markov moment problem and related polynomial approximation on unbounded subsets. Such results allow proving the existence and uniqueness of the solutions for some Markov moment problems. This is the first aim of the paper. Our solutions have a codomain space a commutative algebra of (linear) symmetric operators acting from the entire real or complex Hilbert space

H

to

H

; this algebra of operators is also an order complete Banach lattice. In particular, Hahn–Banach type theorems for the extension of linear operators having a codomain such a space can be applied. The truncated moment problem is briefly discussed by means of reference citations. This is the second purpose of the paper. In the end, a general extension theorem for linear operators with two constraints is recalled and applied to concrete spaces. Here polynomial approximation plays no role. This is the third aim of this work. Full article

(This article belongs to the Special Issue Symmetry and Approximation Methods)

14 pages, 339 KB

Open AccessArticle

On Markov Moment Problem and Related Results

by Octav Olteanu

Symmetry 2021, 13(6), 986; https://doi.org/10.3390/sym13060986 - 1 Jun 2021

Cited by 13 | Viewed by 2872

Abstract

We prove new results and complete our recently published theorems on the vector-valued Markov moment problem, by means of polynomial approximation on unbounded subsets, also applying an extension of the positive linear operators’ result. The domain is the Banach lattice of continuous real-valued [...] Read more.

We prove new results and complete our recently published theorems on the vector-valued Markov moment problem, by means of polynomial approximation on unbounded subsets, also applying an extension of the positive linear operators’ result. The domain is the Banach lattice of continuous real-valued functions on a compact subset or an

L_{ν}^{1}

space, where

ν

is a positive moment determinate measure on a closed unbounded set. The existence and uniqueness of the operator solution are proved. Our solutions satisfy the interpolation moment conditions and are between two given linear operators on the positive cone of the domain space. The norm controlling of the solution is emphasized. The most part of the results are stated and proved in terms of quadratic forms. This type of result represents the first aim of the paper. Secondly, we construct a polynomial solution for a truncated multidimensional moment problem. Full article

(This article belongs to the Special Issue Functional Equations and Analytic Inequalities)

18 pages, 402 KB

Open AccessArticle

Geometric Characterization of Injective Banach Lattices

by Anatoly Kusraev and Semën Kutateladze

Mathematics 2021, 9(3), 250; https://doi.org/10.3390/math9030250 - 27 Jan 2021

Cited by 1 | Viewed by 2190

Abstract

This is a continuation of the authors’ previous study of the geometric characterizations of the preduals of injective Banach lattices. We seek the properties of the unit ball of a Banach space which make the space isometric or isomorphic to an injective Banach [...] Read more.

This is a continuation of the authors’ previous study of the geometric characterizations of the preduals of injective Banach lattices. We seek the properties of the unit ball of a Banach space which make the space isometric or isomorphic to an injective Banach lattice. The study bases on the Boolean valued transfer principle for injective Banach lattices. The latter states that each such lattice serves as an interpretation of an

A L

-space in an appropriate Boolean valued model of set theory. External identification of the internal Boolean valued properties of the corresponding

A L

-spaces yields a characterization of injective Banach lattices among Banach spaces and ordered Banach spaces. We also describe the structure of the dual space and present some dual characterization of injective Banach lattices. Full article

(This article belongs to the Special Issue Boolean Valued Analysis with Applications)

14 pages, 329 KB

Open AccessArticle

Kuelbs–Steadman Spaces for Banach Space-Valued Measures

by Antonio Boccuto, Bipan Hazarika and Hemanta Kalita

Mathematics 2020, 8(6), 1005; https://doi.org/10.3390/math8061005 - 19 Jun 2020

Cited by 7 | Viewed by 2351

Abstract

We introduce Kuelbs–Steadman-type spaces (

K S^{p}

spaces) for real-valued functions, with respect to countably additive measures, taking values in Banach spaces. We investigate the main properties and embeddings of

L^{q}

-type spaces into

K S^{p}

spaces, considering both the [...] Read more.

We introduce Kuelbs–Steadman-type spaces (

K S^{p}

spaces) for real-valued functions, with respect to countably additive measures, taking values in Banach spaces. We investigate the main properties and embeddings of

L^{q}

-type spaces into

K S^{p}

spaces, considering both the norm associated with the norm convergence of the involved integrals and that related to the weak convergence of the integrals. Full article

(This article belongs to the Special Issue Set-Valued Analysis)

20 pages, 361 KB

Open AccessArticle

Banach Lattice Structures and Concavifications in Banach Spaces

by Lucia Agud, Jose Manuel Calabuig, Maria Aranzazu Juan and Enrique A. Sánchez Pérez

Mathematics 2020, 8(1), 127; https://doi.org/10.3390/math8010127 - 14 Jan 2020

Viewed by 3878

Abstract

Let

(Ω, Σ, μ)

be a finite measure space and consider a Banach function space

Y (μ)

. We say that a Banach space E is representable by

Y (μ)

if there is a continuous [...] Read more.

Let

(Ω, Σ, μ)

be a finite measure space and consider a Banach function space

Y (μ)

. We say that a Banach space E is representable by

Y (μ)

if there is a continuous bijection

I : Y (μ) \to E

. In this case, it is possible to define an order and, consequently, a lattice structure for E in such a way that we can identify it as a Banach function space, at least regarding some local properties. General and concrete applications are shown, including the study of the notion of the pth power of a Banach space, the characterization of spaces of operators that are isomorphic to Banach lattices of multiplication operators, and the representation of certain spaces of homogeneous polynomials on Banach spaces as operators acting in function spaces. Full article

(This article belongs to the Section E1: Mathematics and Computer Science)

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