M. ACHOUR Dahmane

Prof

Directory of teachers

Department

Mathematics Department

Research Interests

Functional Analysis

Contact Info

University of M'Sila, Algeria

Email : dahmane.achour@univ-msila.dz

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Recent Publications

2025-02-10

Building Ideals of Two-Lipschitz Operators Between Metric and Banach Spaces

In this paper, we present and characterize the injective hull of a two-Lipschitz operator ideal and the definition of two-Lipschitz dual operator ideal. Also we introduce two methods for creating ideals of two-Lipschitz operators from a pair of Lipschitz operator ideals. Namely, Lipschitzization and factorization method. We show the closedness, the injectivity and the symmetry of these two-Lipschitz ideals according to the closedness, injectivity and symmetry of the corresponding Lipschitz operator ideals. Some illustrative examples are given.
Citation

M. ACHOUR Dahmane, (2025-02-10), "Building Ideals of Two-Lipschitz Operators Between Metric and Banach Spaces", [national] Mediterranean Journal of Mathematics , Springer Nature Link

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2024-12-12

Compactness properties of Lipschitz operators

The theory of operator ideals has long been recognized as a powerful framework for studying and
classifying linear operators between Banach spaces. It now serves as a foundational approach for
addressing new challenges involving non-linear operators. This linear theory has been extended to
Lipschitz operators, introducing the concept of Lipschitz operator ideals. Farmer and Johnson (2009)
first presented an initial framework for this Lipschitz theory. Later, in 2016, Achour et al. developed
an axiomatic theory of Lipschitz operator ideals for Lipschitz mappings taking values in Banach
spaces. These new Lipschitz operator ideals provide a basis for exploring specific properties of non-
linear operators, marking a potential new direction in non-linear functional analysis.
The significance of compact operators in functional analysis is well-known. In this work, we
demonstrate the feasibility of defining these characteristics for Lipschitz operators. We extend certain
characterizations from the linear case to the Lipschitz case. Furthermore, we present a Lipschitz
version of the well-known characterization theorem by Terzioglu for compact maps.
Citation

M. ACHOUR Dahmane, (2024-12-12), "Compactness properties of Lipschitz operators", [international] 5th International Conference on Mathematical, Engineering, and Management Sciences ( 5th ICMEMS-2024) , India

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2024-10-23

The closed injective hull of a two Lipschitz operatorsideals

In this talk we present two methods for establishing a linear op-
erators ideals: The Closed Injective hull and the interpolative method, related

to an operator ideal, in order to get an new classes of linear operators ideals,
some examples are given, modeled the class of (p, σ)-absolutely continuous
operators.
Citation

M. ACHOUR Dahmane, (2024-10-23), "The closed injective hull of a two Lipschitz operatorsideals", [national] 8 th M’Sila conference on Mathematical Analysis , University of M'sila

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Characterization of multilinear compact operators

In this paper, we focus on the compactness of multilinear operators, which is an
extension of the well-known concept of compact linear operators, Understanding the compact-
ness of such operators is crucial, as it generalizes many important properties observed in the
linear to nonlinear caces. We begin by examining the relationship between compact multilin-
ear operators and compact linear operators. For linear operators, compactness implies that
they map bounded sets to relatively compact sets, meaning their image has compact closure.
In the multilinear context, this behavior becomes more intricate, but similar The paper also
explores the composition theorem, which provides conditions under which the composition of
multilinear operators preserves compactness. This theorem is a natural extension of its linear
analog. we aim to shed light on the broader structure of multilinear operators and how they
interact in different functional spaces. Throughout the work, we will provide examples and
proofs that illustrate the key ideas, demonstrating how the theory of multilinear operators
fits within the broader framework of functional analysis.
Citation

M. ACHOUR Dahmane, (2024-10-23), "Characterization of multilinear compact operators", [national] 8th National M'sila conference on mathematical analysis , University of M'sila

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2024-07-06

Ideal of multilinear -factorable operators and applications

In the present paper we introduce a method for generating ideals of linear and multilinear operators from what we call generalized left and right operator ideals, that we discuss with p-th power factorable, p-convex and q-concave operators. Then we combine this method with the Factorization Ideal method, that construct multilinear operators, in order to introduce the ideal of multilinear -factorable operators as an example of an ideal generated by means of our method. Finally, we investigate its relation with multilinear summing operators.
Citation

M. ACHOUR Dahmane, D., Galdames-Bravo, , (2024-07-06), "Ideal of multilinear -factorable operators and applications", [national] Advances in Operator Theory , Springer International Publishing

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2024-04-02

Strongly Lipschitz (l p, l q)-factorable mappings

In this paper we study the space of strongly Lipschitz (ip, iq)-factorable operators between metric spaces and a Banach spaces. In particular, a factorization of this class through ip and iq spaces is given. We show that this type of operators fits in the theory of composition α-Banach Lipschitz operator ideal. As a special case, we get a Lipschitz version of weakly p-nuclear operators.
Citation

M. ACHOUR Dahmane, (2024-04-02), "Strongly Lipschitz (l p, l q)-factorable mappings", [national] Applied General Topology , Universitat Politècnica de València

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2022

Lipschitz Operator Ideals

Farmer and Johnson introduce the Lipschitz p-summing operator ideals between metric spaces. Their work motivated many authors to study different classes of Lipschitz mappings that extend, in some sense, linear operators ideals, leading to the recent notion of Banach Lipschitz operator ideals. In this work we will give some basics of the theory of Lipschitz and two Lipschitz operator ideals, also we introduce the concept of p-nuclear two Lipschitz operator (1≤ p≤∞) with respect to a Banach ideal of two Lipschitz functions..
Citation

M. ACHOUR Dahmane, (2022), "Lipschitz Operator Ideals", [international] INTERNATIONAL E-CONFERENCE ON MATHEMATICAL AND STATISTICAL SCIENCES: A SELCUK MEETING , Turkey

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Banach space of strongly (p,q,σ) -summable sequences and applications

We introduce the Banach space of strongly (p,q,σ)-summable sequences with values in a Banach space obtaining in this way some characterizations of the two classes of already known operators: the strongly (p,σ)-continuous operators and the class called (p,σ,q,ν)-nuclear operators, which is a particular case of the (p,σ,q,ν)-dominated operators. As an application, we show that (p,σ,q,ν)-nuclear linear operators are compact under some requirements and we give a Dvoretzky–Rogers and Schauder type theorems for this class of operators.
Citation

M. ACHOUR Dahmane, Elhadj Dahia, Rachid Soualmia, , (2022), "Banach space of strongly (p,q,σ) -summable sequences and applications", [national] Rendiconti del Circolo Matematico di Palermo Series 2 , Springer Milan

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The ideal of Lipschitz classical-compact operators and its injective hull

We introduce and investigate the injective hull of the strongly Lipschitz classical p-compact operator ideal defined between a pointed metric space and a Banach space. As an application we extend some characterizations of the injective hull of the strongly Lipschitz classical p-compact from the linear case to the Lipschitz case. Also, we introduce the ideal of Lipschitz unconditionally quasi p-nuclear operators between pointed metric spaces and show that it coincides with the Lipschitz injective hull of the ideal of Lipschitz classical p-compact operators.
Citation

M. ACHOUR Dahmane, Toufik Tiaib, , (2022), "The ideal of Lipschitz classical-compact operators and its injective hull", [national] Moroccan Journal of Pure and Applied Analysis , Sidi Mohamed Ben Abdallah University.

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2021

Operator ideals generated by strongly Lorentz sequence spaces

We introduce and study the new ideal of strongly Lorentz summing operators between Banach spaces generated by strongly Lorentz sequence spaces to study the adjoints of the Lorentz summing operators. We also prove the related dual result: an operator is Lorentz summing if and only if its adjoint is strongly Lorentz summing. Some examples, counterexamples and connections with the theory of absolutely summing operators are given.
Citation

M. ACHOUR Dahmane, Aldjia Attallah, , (2021), "Operator ideals generated by strongly Lorentz sequence spaces", [national] Advances in Operator Theory , Springer International Publishing

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Vector-valued Lorentz sequence spaces and their dual spaces

We introduce and investigate the Banach space of vector-valued strongly Lorentz sequences. We prove that this space is a topological dual of a class of weakly Lorentz sequences; moreover, we prove that also the class of weakly Lorentz sequences is a topological dual of a class of strongly Lorentz sequences. We apply these results to obtain a characterization of the bidual of Lorentz summing linear operators.
Citation

M. ACHOUR Dahmane, Aldjia Attallah, , (2021), "Vector-valued Lorentz sequence spaces and their dual spaces", [national] Colloquium Mathematicum , Instytut Matematyczny Polskiej Akademii Nauk

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2020

Strongly (p, q)-summable sequences

In this paper we provide a detailed study of the Banach space of strongly (p, q)-summable sequences. We prove that this space is a topological dual of a class of mixed (s, p)-summable sequences,
showing in this way new properties of this space. We apply these results to obtain the characterization of the adjoints of (r, p, q)-summing operators.
Citation

M. ACHOUR Dahmane, Rachid Soualmia, Elhadj Dahia, , (2020), "Strongly (p, q)-summable sequences", [national] Filomat , Published by Faculty of Sciences and Mathematics, University of Niˇs, Serbia

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Two-Lipschitz operator ideals

We introduce and investigate the concept of two-Lipschitz operator ideal between pointed metric spaces and Banach spaces. We show the basics of this new theory and we give a procedure to create a two-Lipschitz operator ideal from a linear operator ideal. We apply our result to the ideals of strongly p-summing and compact linear operator to obtain their corresponding two-Lipschitz operator ideal. Also, we establish a natural relation between two-Lipschitz and bilinear maps and show that the two-Lipschitz factorable p-dominated operators are those which are associated to the well-known p-semi-integral bilinear operators.
Citation

M. ACHOUR Dahmane, K Hamidi, E Dahia, A Tallab, , (2020), "Two-Lipschitz operator ideals", [international] Journal of Mathematical Analysis and Applications , Academic Press

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The Lipschitz injective hull of Lipschitz operator ideals and applications

We introduce and study the Lipschitz injective hull of Lipschitz operator ideals defined between metric spaces. We show some properties and apply the results to the ideal of Lipschitz p-nuclear operators, obtaining the ideal of Lipschitz quasi p-nuclear operators. Also, we introduce in a natural way the ideal of Lipschitz Pietsch p-integral operators and show that its Lipschitz injective hull coincide with the ideal of Lipschitz p-summing operators defined by Farmer and Johnson. Finally, we consider both ideals as Lipschitz operator ideals between a metric space and a Banach space, showing that these ideals are not of composition type. Their maximal hull and minimal kernel are also studied.
Citation

M. ACHOUR Dahmane, Elhadj Dahia, Pablo Turco, , (2020), "The Lipschitz injective hull of Lipschitz operator ideals and applications", [national] Banach Journal of Mathematical Analysis , Springer International Publishing

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2019

Factorable strongly p-nuclear m-homogeneous polynomials

We characterize in terms of summabiility those homogeneous polynomials whose linearization is p-nuclear. This characterization provides a strong link between the theory of p-nuclear linear operators and the (non linear) homogeneous p-nuclear polynomials that significantly improves former approaches. The deep connection with Grothendieck-integral polynomials is also analyzed.
Citation

M. ACHOUR Dahmane, P. Rueda, , (2019), "Factorable strongly p-nuclear m-homogeneous polynomials", [national] Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas , Springer International Publishing

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Lipschitz p-compact mappings

We introduce the notion of Lipschitz p-compact operators. We show that they can be seen as a natural extension of the linear p-compact operators of Sinha and Karn and we transfer some properties of the linear case into the Lipschitz setting. Also, we introduce the notions of Lipschitz-free p-compact operators and Lipschitz locally p-compact operators. We compare all these three notions and show different properties. Finally, we exhibit examples to show that these three notions are different.
Citation

M. ACHOUR Dahmane, Elhadj Dahia, Pablo Turco, , (2019), "Lipschitz p-compact mappings", [national] Monatshefte für Mathematik , Springer Milan

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2018

Factorization of Lipschitz operators on Banach function spaces

Let (X,d) be a pointed metric space. Let T : X ¿ Y1(µ) and S : X ¿ Y2(µ) be two Lipschitz operators into two Banach function spaces Y1 and Y2 over the same finite measure µ. We show which are the vector norm inequalities that characterize those T and S for which T = Mg ¿ S, for some multiplication operator Mg : Y2 ¿ Y1. Our ideas give rise to Maurey-Rosenthal type factorization results for Lipschitz operators. We provide some applications on the Lipschitz structure of metric subsets of Banach function spaces.
Citation

M. ACHOUR Dahmane, E Dahia, P Rueda, Enrique Alfonso Sánchez Pérez, , (2018), "Factorization of Lipschitz operators on Banach function spaces", [national] Mathematical Inequalities & Applications , Element doo

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Multilinear mixing operators and Lipschitz mixing operator ideals

In [15], EA Sánchez Pérez introduced the class of (s; q, θ)-mixing operators, as a generalization of (s; q)-mixing operators. We investigate analogous concepts here for the case of multilinear operators between Banach spaces and Lipschitz mappings between metric spaces, introducing the class of (s, q; p1,..., pm; θ)-mixing multilinear operators and the Lipschitz Banach ideal of (s, q, θ)-mixing mappings show that our approach provides a multilinear and Lipschitz extension of quotient theorem like the linear case. Several characterizations of these mappings are presented, especially, every Lipschitz (s, q)-mixing map is Lipschitz (s, q, θ)-mixing map and a result relies on the duality theory for (q, θ)-absolutely Lipschitz operators are given.
Citation

M. ACHOUR Dahmane, Elhadj Dahia, MAS Saleh, , (2018), "Multilinear mixing operators and Lipschitz mixing operator ideals", [national] OPERATORS AND MATRICES , ELEMENT

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2017

(p,σ)-Absolutely Lipschitz operators

Due to recent advances in the theory of ideals of Lipschitz mappings, we introduce (p,σ)-absolutely Lipschitz mappings as an interpolating class between Lipschitz mappings and Lipschitz absolutely p-summing mappings. Among other results, we prove a factorization theorem that provides a reformulation to the one given by Farmer and Johnson for Lipschitz absolutely p-summing mappings.
Citation

M. ACHOUR Dahmane, P. Rueda, , (2017), "(p,σ)-Absolutely Lipschitz operators", [national] Annals of Functional Analysis , Tusi Mathematical Research Group

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2016

Domination spaces and factorization of linear and multilinear summing operators

It is well known that not every summability property for multilinear operators leads to a factorization theorem. In this paper we undertake a detailed study of factorization schemes for summing linear and nonlinear operators. Our aim is to integrate under the same theory a wide family of classes of mappings for which a Pietsch type factorization theorem holds. Our construction includes the cases of absolutely p-summing linear operators, (p, σ)-absolutely continuous linear operators, factorable strongly p-summing multilinear operators, (p1, . . . , pn)-dominated multilinear operators and dominated (p1, . . . , pn; σ)-continuous multilinear operators.
Citation

M. ACHOUR Dahmane, Elhadj Dahia, Pilar Rueda, Enrique A Sánchez-Pérez, , (2016), "Domination spaces and factorization of linear and multilinear summing operators", [national] Quaestiones Mathematicae , Taylor & Francis

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Absolutely summing Lipschitz conjugates

The aim of this paper is to contribute to the study of summability of Lipschitz mappings by characterizing those Lipschitz mappings whose Lipschitz conjugates are absolutely p-summing, namely the classes of Lipschitz strongly p-summing mappings (1 < p ≤ ∞).
Citation

M. ACHOUR Dahmane, P Rueda, , (2016), "Absolutely summing Lipschitz conjugates", [national] Mediterranean Journal of Mathematics , Springer International Publishing

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Lipschitz operator ideals and the approximation property

We establish the basics of the theory of Lipschitz operator ideals with the aim of recovering several classes of Lipschitz maps related to absolute summability that have been introduced in the literature in the last years. As an application we extend the notion and main results on the approximation property for Banach spaces to the case of metric spaces.
Citation

M. ACHOUR Dahmane, P. Rueda, EA Sánchez-Pérez, , (2016), "Lipschitz operator ideals and the approximation property", [national] Journal of Mathematical Analysis and Applications , Academic Press

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2015

Virtually (r; r₁,…, rn; s)-nuclear multilinear operators

In this paper, the space of virtually (r; r₁,…, rn; s)-nuclear multilinear operators between Banach spaces is introduced, some of its properties are described and its topological dual is characterized as a Banach space of multiple absolutely (r′ ; r₁′ ,..., rn' ; s′ )-summing multilinear operators.
Citation

M. ACHOUR Dahmane, A. Belacel, , (2015), "Virtually (r; r₁,…, rn; s)-nuclear multilinear operators", [national] Extracta Mathematica , Universidad de Extremadura

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2011

Multilinear extensions of absolutely (p; q; r)-summing operators

In this paper we introduce and study a new class containing the class of absolutely summing multilinear mappings, which we call absolutely (p;q 1,…,q m ;r)-summing multilinear mappings. We investigate some interesting properties concerning the absolutely (p;q 1,…,q m ;r)-summing m-linear mappings defined on Banach spaces. In particular, we prove a kind of Pietsch’s Domination Theorem and a multilinear version of the Factorization Theorem.
Citation

M. ACHOUR Dahmane, (2011), "Multilinear extensions of absolutely (p; q; r)-summing operators", [national] Rendiconti del Circolo Matematico di Palermo , Springer Milan

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2010

A polynomial characterization of Hilbert spaces

In this paper, we obtain a new characterization of Hilbert spaces by means of polynomial mappings, extending the linear result of Kwapień.
Citation

M. ACHOUR Dahmane, (2010), "A polynomial characterization of Hilbert spaces", [national] Collectanea mathematica , Springer Link

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