Aref Jeribi

University of Sfax | US · Department of Mathematics

NEW RESULTS OF SPECTRA AND PSEUDOSPECTRA OF MULTIVALUED LINEAR OPERATORS

In this note, we introduce a notion of the J -kernel of a bounded linear operator on a Krein space and study the J -Fredholm theory for Krein space operators. Using J -Fredholm theory, we discuss and study the J -essential pseudospectra of bounded linear operators on Krein spaces, and some of their properties are studied. Furthermore, we discuss th...

This paper is devoted to the study of the S-eigenvalue of finite type of a bounded right quaternionic linear operator acting in a right quaternionic Hilbert space. The study is based on the different properties of the Riesz projection associated with the connected part of the S-spectrum. Furthermore, we introduce the left and right Browder S-resolv...

In this paper we extend the results obtained by X. Dong and D. Wu in [1] to 3×3 Lipschitz continuous nonlinear operator matrices. In this work, the Kachurovskij spectrum of 3 × 3 Lipschitz continuous nonlinear operator matrices are studied. Firstly, some connections between the Kachurovskij spectrum of certain 3 × 3 Lipschitz continuous nonlinear o...

In this paper, we present some results concerning the weakly quasi-compact and lower characteristic operators. An application to Markov chains, is given.

This work establishes a connection between the class of generalized lower characteristic operators and [ ⋅ ] a {\left[\cdot ]}_{a} acting on a Banach space involving measures of non-strict singularity. This study presents findings on the Jeribi essential spectra of the sum of two bounded linear operators and the Jeribi essential spectra of each ope...

Aqzzouz and Elbour proved that an operator $T$ on a Banach lattice $E$ is $b$-weakly compact if and only if $\|Tx_{n}\|\rightarrow 0$ as $n\rightarrow \infty$ for each $b$-order bounded weakly null sequence $\{x_{n}\}$ in $E_{+}$. In this present paper, we introduce and study new concept of operators that we call $b$-weakly demicompact, use it to g...

Let $(S(t))_{t\geq0}$ and $(T(t))_{t\geq0}$ denote the strongly continuous semigroups of operators in a Banach space $X$. In this paper, we give a sufficient condition guaranteeing that $(S(t))_{t\geq0}$ can be embedded in a $C_{0}$-group on $X$. Moreover, we characterize the demicompactness of $I-(S(t)-T(t))$ for $t>0$. Our theoretical results wil...

UDC 517.9 We extend the concept of generalized weakly demicompact and relatively weakly demicompact operators on linear relations and present some outstanding results. Moreover, we address the theory of Fredholm and upper semi-Fredholm relations attempting to establish a connection with them.

The purpose of this paper is to introduce and study some basic proprieties of the pseudospectra of linear operator pencils (or S-pseudospectra of linear operators) defined by non-strict inequality in a Hilbert space. Inspired by A. Böttcher’s result [3], we show that the S-resolvent of a bounded operator acting in Hilbert space cannot have constant...

This paper is devoted to the study of the S-eigenvalue of finite type of a bounded right quaternionic linear operator acting in a right quaternionic Hilbert space. The study is based on the different properties of the Riesz projection associated with the connected part of the S-spectrum. Furthermore, we introduce the left and right Browder S-resolv...

In this paper, we establish some new variants of fixed point theorems for a large class of countably nonexpansive multi-valued mappings. Some fixed point theorems for the sum and the product of three multi-valued mappings defined on nonempty, closed convex set of Banach algebras are also presented. These results improve and complement a number of e...

In this paper, we give new results on demicompact linear relations, we study some properties and some results on Fredholm and upper semi-Fredholm relations involving demicompact relations. Our results are used to establish a ne description of the essential spectrum and essential pseudospectrum of a linear relation. Subject Classications (2000) : 47...

In this research paper, we introduce the concept of pseudo ellipsoid spectrum in a right quaternionic Hilbert space and display some properties about this notion. Furthermore, we give a characterization for the Weyl pseudo ellipsoid spectrum in Hilbert space.

This paper deals with the existence of integrable solutions for an initial value problem involving Riemann-Liouville-type fractional derivatives. To this end, we transform the posed problem to a sum of two integral operators, then we apply a variant of Krasnoselskii’s fixed point theorem under weak topology to conclude the existence of integrable s...

With the identification of new mutations in the coronavirus with greater transmissibility and pathogenicity, the number of infected people with COVID-19 worldwide has increased as from 22 June 2021, and a new wave has been created. Since the spread of the coronavirus, many studies have been conducted on different groups. The current research was ad...

In this paper, we introduce and study the new concept of demi KB-operators. Let E be a Banach lattice. An operator T : E −→ E is said to be a demi KB-operator if, for every positive increasing sequence {xn} in the closed unit ball BE of E such that {xn − T xn} is norm convergent to x ∈ E , there is a norm convergent subsequence of {xn}. If the latt...

Let C be an invertible bounded linear operator in Banach space X. In this paper, we use the concept of relative demicompactness in order to study some properties of an exponentially bounded C-semigroup (T(t))t ≥ 0. More precisely, we prove that the relative demicompactness of T(t) at some positive values of t is equivalent to relative demicompactne...

In this paper, we extend some aspects of the essential spectra theory of linear operators acting in non-Archimedean (or p-adic) Banach spaces. In particular, we establish sufficient conditions for the relations between the essential spectra of the sum of two bounded linear operators and the union of their essential spectra. Moreover, we give essent...

In this paper, we use the concept of weak demicompactness in order to give some properties for the uniformly continuous cosine families. Our theoretical results will be illustrated by investigating the spectral inclusion for a uniformly continuous cosine family for an upper semi-Fredholm spectrum.

In this paper we are concerned with existence results for a coupled system of quadratic functional differential equations. This system is reduced to a fixed point problem for a block operator matrix with nonlinear inputs. To prove the existence we are established some fixed point theorem of Dhage's type for the block matrix operator acting in parti...

Making use of the Boyd-Wong fixed point theorem, we establish a new existence and uniqueness result and an approximation process of the fixed point for the product of two nonlinear operators in Banach algebras. This provides an adequate tool for deriving the existence and uniqueness of solutions of two interesting type of nonlinear functional equat...

In this work, we establish a sequential characterization of the notion of relatively weak compactness of Banach algebras introduced recently by J. Banaś and L. Olszowy. Moreover, we show that this structure is one of the most important properties which could be lifted from a Banach algebra X to C(K,X) and L1(μ,X). In addition, fixed point theorems...

In this work, we investigate the S-pseudospectra of closed linear operators defined by non-strict inequality in Banach space. We begin the analysis by studying some of this basic properties. After that, we characterize the S-pseudospectra of closed linear operator by means the S-spectra of all perturbed operators with perturbations that have norms...

Using Krasnoselskii type fixed point theorem under the weak topology, we establish some sufficient conditions to ensure the existence of the weak solutions for kinds of initial value problems of fractional differential equations, involving Riemann-Liouville fractional derivative.

This paper is devoted to the investigation of the Weyl and the essential S-spectra of a bounded right quaternionic linear operator in a right quaternionic Hilbert space. Using the quaternionic Riesz projection, the S-eigenvalue of finite type is both introduced and studied. In particular, we have shown that the Weyl and the essential S-spectra do n...

In this paper, we begin with the definition of the S-resolvent set of a linear relation. Throughout this paper, X will denote a normed linear space over the complex field C. Operator S plays the role of a transition multivalued linear operator from X. It is the main goal of the present note to study the basic spectral properties of T linked to the...

In this research paper, we develop some aspects of the theory of Fredholm of linear operators acting in p-adic (or non-Archimedean) Banach spaces. In this regard, we establish sufficient conditions for p-adic Fredholmeness of the algebraic sum of unbounded linear operators. Next, we study the perturbation of p-adic upper semi-Fredholm operators und...

In the present paper, we establish rst the relation between the perturbation of upper Fredholm and strictly singular, and then the relation between lower semi-Fredholm and strictly cosingular linear relations. Most importantly in Theorem 2.1, we show that P(F−(X, Y )) coincides with SC(X, Y ). We bring to light, the relationship between the essenti...

The paper is devoted to some new sufficient conditions to ensure the upper-Fredholmness and Fredholmness of an unbounded densely defined linear operator T acting on a Banach space. Some characterizations of the relative essential spectra of T are also given. These characterizations are developed by introducing new classes of perturbations containin...

In this work, the notion of extended eigenvalues of a 2 × 2 lower triangular operator matrix has been researched. More precisely, the relations between the extended spectrum of a 2 × 2 lower triangular operator matrix with the spectrum, the point spectrum, and the extended spectrum of its diagonal entries have been investigated. The obtained result...

In this paper, by establishing a new characterization of the notion of upper semi-continuity of multi-valued mappings in generalized Banach spaces, we prove some Perov type fixed point theorems for multi-valued mappings with closed graphs. Moreover, we derive some Krasnoselskii's fixe point results for multi-valued mappings in generalized Banach sp...

In this paper, we establish some properties for a uniformly continuous cosine family (C(t))t∈R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(C(t))_{t\in \mathbb {R}}$...

n this work, we introduce and study the S-pseudospectra of linear operators defined by non-strict inequality in a Hilbert space. Inspired by A. Böttcher’s result [3], we prove that the S-resolvent norm of bounded linear operators is not constant in any open set of the S-resolvent set. Beside, we find a characterization of the S-pseudospectrum of bo...

In this research paper, we provide firstly the necessary and sufficient conditions for the algebraic sum to become a closed and closable linear relation. Secondly, we investigate the stability of the essential approximate point spectrum σeap(.) as well as the essential defect spectrum σeδ(T), in terms of linear relations on Banach spaces, which wer...

On the Left (Right) Condition Pseudospectrum of Linear Operators

In this paper, we show that an unbounded weakly S0-demicompact linear operator T, introduced in [16], acting on a Banach space, can be character- ized by some measures of weak noncompactness. Moreover, some other quantities related to these measures provide su�cient conditions to the operator T to be S0- demicompact. Our results are illustrated to...

We first attempt to determine conditions on a linear relation T such that µT becomes demicompact linear relation for each µ ∈ [0, 1) (see Theorems 2.1 and 2.2). Second, we display some results on Fredholm and upper semi-Fredholm linear relations involving a demicompact one (see Theorems 3.1 and 3.2). Finally, we provide some results in which a bloc...

In this paper, we introduce and investigate a new concept that we call demicompact elements in Banach algebras as a generalization of demicompact linear operators acting on Banach spaces. Our concept is used to construct a new class of Fredholm perturbations with respect to a given Banach subalgebra B , that contains an inessential ideal k B {k_{B}...

One of the fundamental ideas investigated in A. Ammar, A. Jeribi and K. Mahfoudhi in [?] is that of providing conditions under which the essential approximate pseudospectrum of closed, densely defined linear operators have a relationship with Fredholm theory and perturbation theory. In this paper the approximate pseudospectrum and the essential appr...

In this paper, we establish some new variants of fixed point theorems for a large class of countably nonexpansive multi-valued mappings. Some fixed point theorems for the sum and the product of three multi-valued mappings defined on nonempty, closed convex set of Banach algebras are also presented. These results improve and complement a number of e...

In the present paper, we study some properties of the generalized Drazin-meromorphic pseudospectrum for a bounded linear operator on a Banach space. We also make several observations about the level set of the generalized Drazin-meromorphic pseudospectrum. Further, it has been shown that pseudospectrum has no isolated points, has a finite number of...

In this paper, we are concerned with a 3 × 3 block operator matrices acting in a Banach or Hilbert space X 1 × X 2 × X 3 given by A 1 B 1 C 1 A 2 B 2 C 2 A 3 B 3 C 3 , where the linear entries are assumed to be unbounded. We study the closure as well as the self-adjointness in the case where the linear operators A 2 and A 3 are A 1 -bounded, B 1 an...

In this paper, a general hybrid fixed point theorem for the contractive mappings in generalized Banach spaces is proved via measure of weak non-compactness and it is further applied to fractional integral equations for proving the existence results for the solutions under mixed Lipschitz and weakly sequentially continuous conditions. Finally, an ex...

In this paper, we introduce and study new concepts of L-weakly and M-weakly demicompact operators. Let E be a Banach lattice. An operator T : E −→ E is called L-weakly demicompact if for every norm bounded sequence (x n) in B E such that {x n − T x n , n ∈ N} is L-weakly compact subset of E, we have {x n , n ∈ N} is L-weakly compact subset of E, an...

In this paper, our central focus is upon introducing the class of demi Dunford-Pettis operators. The paper rests essentially on two parts. In the first part we study the connection of this new class of operators with classical notions of operators, such as Dunfort-Pettis operators , strictly singular operators and demicompact operators. In the seco...

In this paper, the existence and uniqueness of the fixed point for the product of two nonlinear operator in Banach algebra is discussed. In addition, an approximation method of the fixed point of hybrid nonlinear equations in Banach algebras is established. This method is applied to two interesting different types of functional equations. In additi...

Using the technique of measures of weak non-compactness we obtain the existence of fixed points of a 2 × 2 block operator matrix involving nonlinear maps in non-separable Banach space. These theorems are created in terms of weak sequential continuity and the theory of countably condensing maps. Our results generalize, improve and complement a numbe...

In this paper, we devote our research to the essential spectra of linear relations defined on a Banach space. We extend the main results of paper [1] to linear relations.

In this paper, a general hybrid fixed point theorem for the contractive mappings in generalized Banach spaces is proved via measure of weak non-compactness and it is further applied to fractional integral equations for proving the existence results for the solutions under mixed Lipschitz and weakly sequentially continuous conditions. Finally, an ex...

In this paper, we investigate the behaviour of the index of upper and lower semi Fredholm multivalued linear operators in Banach spaces under strictly singular perturbation.

The aim of this concept is to study the existence of eigenvalues far from the essential S-spectra on quaternionic Hilbert spaces and also to search the instability of the essential S-spectra under every perturbation. Besides, this paper is devoted to the investigation of the stability of the Weyl essential S-spectrum of the linear operator $A$ subj...

This paper presents new fixed point theorems for 2 × 2 block operator matrix with countably condensing or countably D-set-contraction multi-valued inputs. Our theory will then be used to establish some new existence theorems for coupled system of functional differential inclusions in general Banach spaces under weak topology. Our results generalize...

In this manuscript, by removing the domain convexity hypothesis, the existence of fixed set results for the sum and the product of (p + 1)-multi-valued operators acting on Banach algebras under some suitable conditions on the operators A, B1, . . . , Bp. Applications to self-similarity theory are also given.

The main goal of this paper is to elaborate some results on the spectral properties of 2 × 2 block matrix linear relations with unbounded entries and with a domain consisting of vector which satisfy certain relations between their components. We present some conditions to prove some Frobenius-Schur decompositions for linear relations and characteri...

In this paper, we determine some properties of extended eigenvalues for operators pair. Furthermore, the relationship between this kind of operators pair and the operators pencils in Hilbert space is established.

In this paper, a new concept for a 3 * 3 block relation matrix is studied in a Banach space. It is shown that, under certain condition, we can investigate the Frobenius-Schur decomposition of relation matrices. Furthermore, we present some conditions which should allow the multivalued 3 * 3 matrices linear operator to be closable.

The main purpose of this article is to give a relationship between Fredholm multivalued linear operators and the demicompact linear relation; we provide some sufficient conditions on the inputs of a closable block multivalued linear operator matrix to ensure the generalized demicompactness of its closure. Our results generalize many known ones in t...

This paper is devoted to the investigation of the Weyl and the essential $S-$spectrum of a bounded right quaternionic linear operator in a right quaternionic Hilbert space. Using the quaternionic Riesz projection, the $S-$eigenvalue of finite type is introduced and studied. In particular, it is shown that the Weyl and the essential $S-$spectra does...

This chapter deals with Schauder and Riesz bases of eigenvectors of a family of non-normal operators. Indeed, we generalize some results due to Nagy in [29] and we extend the main result in [7] to Schauder basis in a separable Banach space. Second, we investigate under sufficient conditions assuring the existence of a Riesz basis in a Hilbert space...

In this chapter, we focus on the study of the asymptotic behavior of the eigenvalues of an analytic operator in the sense of Kato. More precisely, we investigate the behavior of the spectrum of the perturbed operator \(T(\varepsilon )\) under a finite rank perturbation and we develop perturbation theory for these new type conditions.

We begin this chapter by presenting the background material that is needed in our work. Most of this background is drawn from functional analysis and operator theory by giving some definitions, notations, and basic information on functional analysis that underlies most of the concepts presented in this book. The aim of this chapter is to introduce...

The purpose of this chapter is to formulate some new supplements to perturbation theory of linear operators [21] by considering a non-analytic perturbation “analytic operators in Feki-Jeribi-Sfaxi’s sense” involving more than one perturbation parameter. On the one hand, under a relative boundedness condition, we show the invariance of the closure a...

In this chapter, we recall some facts related to the evolutionary problem such as Hille-Yosida theorem, differentiability of the semigroup. We also give some properties of fractional operators and expansion of solution on generalized eigenvectors of operators in Hilbert space. In this chapter some elementary properties of semigroups are given.

This chapter is concerned with operators with Carleman-class and the spectral theory of compact operators. The focus of this part aims at defining the singular values of a compact operator and at introducing the Carleman-class of operators \({\mathscr {C}}_p\).

This chapter concens a perturbation method for the Gribov operator in Bargmann space. We treat the Gribov operator in Bargmann space in the cases of finite and infinite sum on null transverse dimension and we confirm the existence of Riesz basis of subspaces, Schauder basis, and Basis with parentheses. It is worth mentioning that each section has i...

The basic idea of this chapter is to derive a precise description, on a separable Hilbert space X, to the behavior of the spectrum of a self-adjoint operator \(T_0\) after a perturbation by an infinite sum of operators where \(\varepsilon \in \mathbb {C}\) and \(T_1\), \(T_2\), \(T_3\ldots \) are linear operators on the space X having the same doma...

In this chapter, we apply the results of Chaps. 6–12 to two examples: to the problem of radiation of a vibrating structure in a light fluid and perturbation method for sound radiation by a vibrating plate in a light fluid. This chapter contains two sections. However, it is worth mentioning that each section has its own equations, notations, and sym...

In this chapter, we recall some Keldysh results that are not only specific for Hilbert spaces, but they have been formulated for operators in Banach spaces. We also discuss some theorems on denseness of the generalized eigenvectors of a compact operator or an operator with compact resolvent and different conditions assuring the completeness of the...

This chapter concentrate on a selection of applications in mathematical physics and mechanics to which the results of the preceding chapters are applied. This chapter contains some applications in mathematical physics and mechanics to investigate the expansion of solution in terms of generalized eigenvectors for a rectilinear transport equation and...

This chapter is devoted to study the Riesz basis of finite-dimensional invariant subspaces for a class of unbounded perturbations of unbounded normal operators, we study the change of the spectrum and we establish the existence of a Riesz basis of finite-dimensional invariant subspaces under an additional a priori assumption on the spectrum of the...

This chapter is devoted to study the convergence of series of complex terms.

In this chapter, we give some fundamentals and elementary results about projections, particular functions such as the function of finite-order and the sine-type function, which are used in the sequel. We also recall Phragmén-Lindelöf theorems, spectral properties of holomorphic operator functions. After that, we introduce some definitions about the...

In this chapter, considerable attention has been devoted to bases on Hilbert and Banach spaces. We derive their basic properties, one of the most important of which is the fact that the coefficient functionals associated with a basis are automatically continuous.

The main goal of this paper is to introduce the condition pseudospectrum of multivalued linear operators and prove several relations to the usual spectrum. We start by giving the definition then we focus on the characterization, the stability and some of their properties.

A complex number ? is an extended eigenvalue of an operator A if there is a nonzero operator B such that = ?BA. In this case, B is said to be an eigenoperator. This research paper is devoted to the investigation of some results of extended eigenvalues for a closed linear operator on a complex Banach space. The obtained results are explored in terms...

In this paper, we show that an unbounded weakly $S_{0}$-demicompact linear operator $T$, introduced in \cite{Bilel-O'Regan}, acting on a Banach space, can be characterized by some measures of weak noncompactness. Moreover, some other quantities related to these measures provide sufficient conditions to the operator $T$ to be $S_{0}$-demicompact. Ou...

We work with the notion of trace pseudospectra for an element in the matrix algebra.~Many new interesting properties of the trace pseudospectrum have been discovered.~In addition, we show an analogue of the spectral mapping theorem for trace pseudospectrum in the matrix algebra. Among other things, we illustrate the applicability of this concepts b...

Motivated by Aiena, Trapani and Triolo’s work (Filomat 28(2), 263–273, 2014), we introduced some concepts of the local spectral theory and the single-valued extension property abbreviated SVEP of the closed linear relations on a Banach space. After that, we analyzed basic proprieties of these notions and established a relationship between the analy...

We develop a general framework for perturbation analysis of matrix. More specifically, the C-determinant pseudospectrum \(\mathrm{Det}_{\varepsilon}^{C}(T)\) for an element in the matrix algebra \(\mathcal{M}_n(\mathbb{C})\) is studied. We also make several observations on the C-determinant pseudospectrum.

In this paper, we introduce and study the structured essential approximate and defect pseudospectrum of closed, densely defined linear operators in a Banach space. Beside that, we discuss some results of stability and some properties of these essential pseudospectra. Finally, we will apply the results described above to investigate the essential ap...

This book discusses the important aspects of spectral theory, in particular, the completeness of generalised eigenvectors, Riesz bases, semigroup theory, families of analytic operators, and Gribov operator acting in the Bargmann space. Recent mathematical developments of perturbed non-self-adjoint operators are discussed with the completeness of th...