Research

Jump to: Review

13 pages, 304 KiB

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 - 09 Nov 2023

Viewed by 593

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)

13 pages, 281 KiB

Open AccessArticle

Fejér-Type Inequalities for Some Classes of Differentiable Functions

by Bessem Samet

Mathematics 2023, 11(17), 3764; https://doi.org/10.3390/math11173764 - 01 Sep 2023

Viewed by 551

Abstract

We let

υ

be a convex function on an interval

[ι_{1}, ι_{2}] \subset R

. If

ζ \in C ([ι_{1}, ι_{2}])

,

ζ \geq 0

and

ζ

is symmetric with respect [...] Read more.

We let

υ

be a convex function on an interval

[ι_{1}, ι_{2}] \subset R

. If

ζ \in C ([ι_{1}, ι_{2}])

,

ζ \geq 0

and

ζ

is symmetric with respect to

\frac{ι_{1} + ι_{2}}{2}

, then

υ (\frac{1}{2} \sum_{j = 1}^{2} ι_{j}) \int_{ι_{1}}^{ι_{2}} ζ (s) d s \leq \int_{ι_{1}}^{ι_{2}} υ (s) ζ (s) d s \leq \frac{1}{2} \sum_{j = 1}^{2} υ (ι_{j}) \int_{ι_{1}}^{ι_{2}} ζ (s) d s .

The above estimates were obtained by Fejér in 1906 as a generalization of the Hermite–Hadamard inequality (the above inequality with

ζ \equiv 1

). This work is focused on the study of right-side Fejér-type inequalities in one- and two-dimensional cases for new classes of differentiable functions

υ

. In the one-dimensional case, the obtained results hold without any symmetry condition imposed on the weight function

ζ

. In the two-dimensional case, the right side of Fejer’s inequality is extended to the class of subharmonic functions

υ

on a disk. Full article

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

16 pages, 345 KiB

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

Viewed by 765

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)

12 pages, 283 KiB

Open AccessArticle

On the Solution of Generalized Banach Space Valued Equations

by Ramandeep Behl and Ioannis K. Argyros

Mathematics 2022, 10(1), 132; https://doi.org/10.3390/math10010132 - 02 Jan 2022

Viewed by 1048

Abstract

We develop a class of Steffensen-like schemes for approximating solution of Banach space valued equations. The sequences generated by these schemes are, converging to the solution under certain hypotheses that are weaker than in earlier studies. Hence, extending the region of applicability of [...] Read more.

We develop a class of Steffensen-like schemes for approximating solution of Banach space valued equations. The sequences generated by these schemes are, converging to the solution under certain hypotheses that are weaker than in earlier studies. Hence, extending the region of applicability of these schemes without additional hypotheses. Benefits include: more choices for initial points; the computation of fewer iterates to reach a certain accuracy in the error distances, and a more precise knowledge of the solution. Technique is applicable on other schemes our due to its generality. Full article

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

28 pages, 555 KiB

Open AccessArticle

Geometric Invariants of Surjective Isometries between Unit Spheres

by Almudena Campos-Jiménez and Francisco Javier García-Pacheco

Mathematics 2021, 9(18), 2346; https://doi.org/10.3390/math9182346 - 21 Sep 2021

Cited by 5 | Viewed by 1731

Abstract

In this paper we provide new geometric invariants of surjective isometries between unit spheres of Banach spaces. Let

X, Y

be Banach spaces and let

T : S_{X} \to S_{Y}

be a surjective isometry. The most relevant geometric invariants under [...] Read more.

In this paper we provide new geometric invariants of surjective isometries between unit spheres of Banach spaces. Let

X, Y

be Banach spaces and let

T : S_{X} \to S_{Y}

be a surjective isometry. The most relevant geometric invariants under surjective isometries such as T are known to be the starlike sets, the maximal faces of the unit ball, and the antipodal points (in the finite-dimensional case). Here, new geometric invariants are found, such as almost flat sets, flat sets, starlike compatible sets, and starlike generated sets. Also, in this work, it is proved that if F is a maximal face of the unit ball containing inner points, then

T (- F) = - T (F)

. We also show that if

[x, y]

is a non-trivial segment contained in the unit sphere such that

T ([x, y])

is convex, then T is affine on

[x, y]

. As a consequence, T is affine on every segment that is a maximal face. On the other hand, we introduce a new geometric property called property P, which states that every face of the unit ball is the intersection of all maximal faces containing it. This property has turned out to be, in a implicit way, a very useful tool to show that many Banach spaces enjoy the Mazur-Ulam property. Following this line, in this manuscript it is proved that every reflexive or separable Banach space with dimension greater than or equal to 2 can be equivalently renormed to fail property P. Full article

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

9 pages, 314 KiB

Open AccessArticle

Composition Vector Spaces as a New Type of Tri-Operational Algebras

by Omid Reza Dehghan, Morteza Norouzi and Irina Cristea

Mathematics 2021, 9(18), 2344; https://doi.org/10.3390/math9182344 - 21 Sep 2021

Viewed by 1481

Abstract

The aim of this paper is to define and study the composition vector spaces as a type of tri-operational algebras. In this regard, by presenting nontrivial examples, it is emphasized that they are a proper generalization of vector spaces and their structure can [...] Read more.

The aim of this paper is to define and study the composition vector spaces as a type of tri-operational algebras. In this regard, by presenting nontrivial examples, it is emphasized that they are a proper generalization of vector spaces and their structure can be characterized by using linear operators. Additionally, some related properties about foundations, composition subspaces and residual elements are investigated. Moreover, it is shown how to endow a vector space with a composition structure by using bijective linear operators. Finally, more properties of the composition vector spaces are presented in connection with linear transformations. Full article

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

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7 pages, 270 KiB

Open AccessArticle

Second-Order PDE Constrained Controlled Optimization Problems with Application in Mechanics

by Savin Treanţă

Mathematics 2021, 9(13), 1472; https://doi.org/10.3390/math9131472 - 23 Jun 2021

Cited by 6 | Viewed by 1474

Abstract

The present paper deals with a class of second-order PDE constrained controlled optimization problems with application in Lagrange–Hamilton dynamics. Concretely, we formulate and prove necessary conditions of optimality for the considered class of control problems driven by multiple integral cost functionals involving second-order [...] Read more.

The present paper deals with a class of second-order PDE constrained controlled optimization problems with application in Lagrange–Hamilton dynamics. Concretely, we formulate and prove necessary conditions of optimality for the considered class of control problems driven by multiple integral cost functionals involving second-order partial derivatives. Moreover, an illustrative example is provided to highlight the effectiveness of the results derived in the paper. In the final part of the paper, we present an algorithm to summarize the steps for solving a control problem such as the one investigated here. Full article

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

Review

Jump to: Research

17 pages, 342 KiB

Open AccessReview

Convexity, Markov Operators, Approximation, and Related Optimization

by Octav Olteanu

Mathematics 2022, 10(15), 2775; https://doi.org/10.3390/math10152775 - 04 Aug 2022

Cited by 3 | Viewed by 1037

Abstract

The present review paper provides recent results on convexity and its applications to the constrained extension of linear operators, motivated by the existence of subgradients of continuous convex operators, the Markov moment problem and related Markov operators, approximation using the Krein–Milman theorem, related [...] Read more.

The present review paper provides recent results on convexity and its applications to the constrained extension of linear operators, motivated by the existence of subgradients of continuous convex operators, the Markov moment problem and related Markov operators, approximation using the Krein–Milman theorem, related optimization, and polynomial approximation on unbounded subsets. In many cases, the Mazur–Orlicz theorem also leads to Markov operators as solutions. The common point of all these results is the Hahn–Banach theorem and its consequences, supplied by specific results in polynomial approximation. All these theorems or their proofs essentially involve the notion of convexity. Full article

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

26 pages, 434 KiB

Open AccessReview

On the Moment Problem and Related Problems

by Octav Olteanu

Mathematics 2021, 9(18), 2289; https://doi.org/10.3390/math9182289 - 17 Sep 2021

Cited by 1 | Viewed by 1475

Abstract

Firstly, we recall the classical moment problem and some basic results related to it. By its formulation, this is an inverse problem: being given a sequence

{(y_{j})}_{j \in ℕ^{n}}

of real numbers and a closed subset [...] Read more.

Firstly, we recall the classical moment problem and some basic results related to it. By its formulation, this is an inverse problem: being given a sequence

{(y_{j})}_{j \in ℕ^{n}}

of real numbers and a closed subset

F \subseteq ℝ^{n}

,

n \in {1, 2, \dots},

find a positive regular Borel measure

μ

on

F

such that

\int_{F}^{} t^{j} d μ = y_{j}, j \in ℕ^{n} .

This is the full moment problem. The existence, uniqueness, and construction of the unknown solution

μ

are the focus of attention. The numbers

y_{j}, j \in ℕ^{n}

are called the moments of the measure

μ .

When a sandwich condition on the solution is required, we have a Markov moment problem. Secondly, we study the existence and uniqueness of the solutions to some full Markov moment problems. If the moments

y_{j}

are self-adjoint operators, we have an operator-valued moment problem. Related results are the subject of attention. The truncated moment problem is also discussed, constituting the third aim of this work. Full article

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

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