Axioms

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11 pages, 255 KiB

Open AccessArticle

Three Bounded Solutions for a Degenerate Nonlinear Dirichlet Problem

by Elisabetta Tornatore

Axioms 2025, 14(4), 289; https://doi.org/10.3390/axioms14040289 - 11 Apr 2025

Viewed by 138

The existence of at least three bounded weak solutions is established for a nonlinear elliptic equation with unbounded coefficients. The approach is based on variational methods and critical point theorems. Full article

(This article belongs to the Special Issue Recent Advances in Functional Analysis and Operator Theory)

18 pages, 273 KiB

Open AccessArticle

Strong k-Skew Commutativity Preserving Maps on Standard Operator Algebras

by Ting Zhang and Xiaofei Qi

Axioms 2025, 14(2), 93; https://doi.org/10.3390/axioms14020093 - 26 Jan 2025

Viewed by 426

Abstract

Let

A

be a self-adjoint standard operator algebra on a real or complex Hilbert space of dimension

\geq 2

, and let

k \in {1, 2, 3}

. The k-skew commutator for

A, B \in A

is [...] Read more.

Let

A

be a self-adjoint standard operator algebra on a real or complex Hilbert space of dimension

\geq 2

, and let

k \in {1, 2, 3}

. The k-skew commutator for

A, B \in A

is defined by

{}_{*}{[A, B]}_{1} = A B - B A^{*}

and

{}_{*}{[A, B]}_{k} = {}_{*}{[A, {}_{1}{[A, {}_{*}{[A, B]}_{k - 1}}]}_{1}

. Assume that

Φ : A \to A

is a map whose range contains all rank-one projections. In this paper, we prove that

Φ

is strong k-skew-commutativity preserving, that is,

{}_{*}{[Φ (A), Φ (B)]}_{k} = {}_{*}{[A, B]}_{k}

for all

A, B \in A

if and only if one of the following statements holds: (i)

Φ

is either the identity map or the negative identity map whenever

k \in {1, 3}

; (ii)

Φ

is the identity map whenever

k = 2

. Full article

(This article belongs to the Special Issue Recent Advances in Functional Analysis and Operator Theory)

19 pages, 294 KiB

Open AccessArticle

Quantum–Fractal–Fractional Operator in a Complex Domain

by Adel A. Attiya, Rabha W. Ibrahim, Ali H. Hakami, Nak Eun Cho and Mansour F. Yassen

Axioms 2025, 14(1), 57; https://doi.org/10.3390/axioms14010057 - 13 Jan 2025

Viewed by 651

Abstract

In this effort, we extend the fractal–fractional operators into the complex plane together with the quantum calculus derivative to obtain a quantum–fractal–fractional operators (QFFOs). Using this newly created operator, we create an entirely novel subclass of analytical functions in the unit disk. Motivated [...] Read more.

In this effort, we extend the fractal–fractional operators into the complex plane together with the quantum calculus derivative to obtain a quantum–fractal–fractional operators (QFFOs). Using this newly created operator, we create an entirely novel subclass of analytical functions in the unit disk. Motivated by the concept of differential subordination, we explore the most important geometric properties of this novel operator. This leads to a study on a set of differential inequalities in the open unit disk. We focus on the conditions to obtain a bounded turning function of QFFOs. Some examples are considered, involving special functions like Bessel and generalized hypergeometric functions. Full article

(This article belongs to the Special Issue Recent Advances in Functional Analysis and Operator Theory)

20 pages, 326 KiB

Open AccessArticle

Noncommutative Multi-Parameter Subsequential Wiener–Wintner-Type Ergodic Theorem

by Mu Sun and Yinmei Zhang

Axioms 2024, 13(9), 595; https://doi.org/10.3390/axioms13090595 - 31 Aug 2024

Viewed by 917

Abstract

This paper is devoted to the study of a multi-parameter subsequential version of the “Wiener–Wintner” ergodic theorem for the noncommutative Dunford–Schwartz system. We establish a structure to prove “Wiener–Wintner”-type convergence over a multi-parameter subsequence class

Δ

instead of the weight class case. In [...] Read more.

This paper is devoted to the study of a multi-parameter subsequential version of the “Wiener–Wintner” ergodic theorem for the noncommutative Dunford–Schwartz system. We establish a structure to prove “Wiener–Wintner”-type convergence over a multi-parameter subsequence class

Δ

instead of the weight class case. In our subsequence class, every term of

\underset{̲}{k} \in Δ

is one of the three kinds of nonzero density subsequences we consider. As key ingredients, we give the maximal ergodic inequalities of multi-parameter subsequential averages and obtain a noncommutative subsequential analogue of the Banach principle. Then, by combining the critical result of the uniform convergence for a dense subset of the noncommutative

L_{p} (M)

space and the noncommutative Orlicz space, we immediately obtain the main theorem. Full article

(This article belongs to the Special Issue Recent Advances in Functional Analysis and Operator Theory)

10 pages, 268 KiB

Open AccessArticle

Consistent Sampling Approximations in Abstract Hilbert Spaces

by Sinuk Kang, Kil Hyun Kwon and Dae Gwan Lee

Axioms 2024, 13(7), 489; https://doi.org/10.3390/axioms13070489 - 21 Jul 2024

Viewed by 741

Abstract

This paper considers generalized consistent sampling and reconstruction processes in an abstract separable Hilbert space. Using an operator-theoretical approach, quasi-consistent and consistent approximations with optimal properties, such as possessing the minimum norm or being closest to the original vector, are derived. The results [...] Read more.

This paper considers generalized consistent sampling and reconstruction processes in an abstract separable Hilbert space. Using an operator-theoretical approach, quasi-consistent and consistent approximations with optimal properties, such as possessing the minimum norm or being closest to the original vector, are derived. The results are illustrated with several examples. Full article

(This article belongs to the Special Issue Recent Advances in Functional Analysis and Operator Theory)

21 pages, 316 KiB

Open AccessArticle

Multiplication Operators on Weighted Zygmund Spaces of the First Cartan Domain

by Zhi-Jie Jiang

Axioms 2023, 12(12), 1131; https://doi.org/10.3390/axioms12121131 - 17 Dec 2023

Cited by 1 | Viewed by 1354

Abstract

Inspired by some recent studies of the multiplication operators on holomorphic function spaces of the classical domains such as the open unit disk, the unit ball and the unit polydisk, the purpose of the present paper is to study just the operators that [...] Read more.

Inspired by some recent studies of the multiplication operators on holomorphic function spaces of the classical domains such as the open unit disk, the unit ball and the unit polydisk, the purpose of the present paper is to study just the operators that are defined on weighted Zygmund spaces of the first Cartan domain. We obtain some necessary conditions and sufficient conditions for the operators to be bounded and compact. Full article

(This article belongs to the Special Issue Recent Advances in Functional Analysis and Operator Theory)

15 pages, 333 KiB

Open AccessArticle

Metrical Boundedness and Compactness of a New Operator between Some Spaces of Analytic Functions

by Stevo Stević

Axioms 2023, 12(9), 851; https://doi.org/10.3390/axioms12090851 - 31 Aug 2023

Viewed by 1343

Abstract

The metrical boundedness and metrical compactness of a new operator from the weighted Bergman-Orlicz spaces to the weighted-type spaces and little weighted-type spaces of analytic functions are characterized. Full article

(This article belongs to the Special Issue Recent Advances in Functional Analysis and Operator Theory)

11 pages, 307 KiB

Open AccessArticle

L^p-Mapping Properties of a Class of Spherical Integral Operators

by Laith Hawawsheh, Ahmad Qazza, Rania Saadeh, Amjed Zraiqat and Iqbal M. Batiha

Axioms 2023, 12(9), 802; https://doi.org/10.3390/axioms12090802 - 22 Aug 2023

Cited by 3 | Viewed by 1111

Abstract

In this paper, we study a class of spherical integral operators

I_{Ω} f

. We prove an inequality that relates this class of operators with some well-known Marcinkiewicz integral operators by using the classical Hardy inequality. We also attain the boundedness of [...] Read more.

In this paper, we study a class of spherical integral operators

I_{Ω} f

. We prove an inequality that relates this class of operators with some well-known Marcinkiewicz integral operators by using the classical Hardy inequality. We also attain the boundedness of the operator

I_{Ω} f

for some

1 < p < 2

whenever

Ω

belongs to a certain class of Lebesgue spaces. In addition, we introduce a new proof of the optimality condition on

Ω

in order to obtain the

L^{2}

-boundedness of

I_{Ω}

. Generally, the purpose of this work is to set up new proofs and extend several known results connected with a class of spherical integral operators. Full article

(This article belongs to the Special Issue Recent Advances in Functional Analysis and Operator Theory)

22 pages, 338 KiB

Open AccessArticle

Bombieri-Type Inequalities and Their Applications in Semi-Hilbert Spaces

by Najla Altwaijry, Silvestru Sever Dragomir and Kais Feki

Axioms 2023, 12(6), 522; https://doi.org/10.3390/axioms12060522 - 26 May 2023

Cited by 1 | Viewed by 1315

Abstract

This paper presents new results related to Bombieri’s generalization of Bessel’s inequality in a semi-inner product space induced by a positive semidefinite operator A. Specifically, we establish new inequalities that generalize the classical Bessel inequality and extend previous results in this area. [...] Read more.

This paper presents new results related to Bombieri’s generalization of Bessel’s inequality in a semi-inner product space induced by a positive semidefinite operator A. Specifically, we establish new inequalities that generalize the classical Bessel inequality and extend previous results in this area. Furthermore, our findings have applications to the study of operators on positive semidefinite inner product spaces, also known as semi-Hilbert spaces, and contribute to a deeper understanding of their properties and applications. Our work has implications for various fields, including functional analysis and operator theory. Full article

(This article belongs to the Special Issue Recent Advances in Functional Analysis and Operator Theory)

12 pages, 287 KiB

Open AccessArticle

Norms of a Product of Integral and Composition Operators between Some Bloch-Type Spaces

by Stevo Stević

Axioms 2023, 12(5), 491; https://doi.org/10.3390/axioms12050491 - 18 May 2023

Viewed by 1190

Abstract

We present some formulas for the norm, as well as the essential norm, of a product of composition and an integral operator between some Bloch-type spaces of analytic functions on the unit ball, in terms of given symbols and weights. Full article

(This article belongs to the Special Issue Recent Advances in Functional Analysis and Operator Theory)

9 pages, 247 KiB

Open AccessArticle

Norm of Hilbert Operator’s Commutants

by Hadi Roopaei

Axioms 2023, 12(5), 422; https://doi.org/10.3390/axioms12050422 - 26 Apr 2023

Viewed by 1399

Abstract

In this study, we obtain the

ℓ_{p}

-norms of six classes of operators that commute with the infinite Hilbert operators. Full article

(This article belongs to the Special Issue Recent Advances in Functional Analysis and Operator Theory)

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