scholarly journals Essential pseudospectra involving demicompact and pseudodemicompact operators and some perturbation results

Filomat ◽  
2019 ◽  
Vol 33 (8) ◽  
pp. 2519-2528 ◽  
Author(s):  
Fatma Brahim ◽  
Aref Jeribi ◽  
Bilel Krichen
Keyword(s):  
Banach Space ◽  
Linear Operators ◽  
Fredholm Operators ◽  
The Stability ◽  
Stability Results ◽  
Densely Defined

In this paper, we study the essential and the structured essential pseudospectra of closed densely defined linear operators acting on a Banach space X. We start by giving a refinement and investigating the stability of these essential pseudospectra by means of the class of demicompact linear operators. Moreover, we introduce the notion of pseudo demicompactness and we study its relationship with pseudo upper semi-Fredholm operators. Some stability results for the Gustafson essential pseudospectrum involving pseudo demicompact operators is given.

A characterization of the essential approximation pseudospectrum on a Banach space

Filomat ◽  
2017 ◽  
Vol 31 (11) ◽  
pp. 3599-3610 ◽  
Author(s):  
Aymen Ammar ◽  
Aref Jeribi ◽  
Kamel Mahfoudhi
Keyword(s):  
Banach Space ◽  
Approximation Theory ◽  
Linear Operators ◽  
The Stability ◽  
Densely Defined

One impetus for writing this paper is the issue of approximation pseudospectrum introduced by M. P. H.Wolff in the journal of approximation theory (2001). The latter study motivates us to investigate the essential approximation pseudospectrum of closed, densely defined linear operators on a Banach space. We begin by defining it and then we focus on the characterization, the stability and some properties of these pseudospectra.

scholarly journals The essential approximate pseudospectrum and related results

Filomat ◽  
2018 ◽  
Vol 32 (6) ◽  
pp. 2139-2151 ◽  
Author(s):  
Aymen Ammar ◽  
Aref Jeribi ◽  
Kamel Mahfoudhi
Keyword(s):  
Banach Space ◽  
Compact Operators ◽  
Linear Operators ◽  
The Stability ◽  
Densely Defined

In this paper, we introduce and study the essential approximate pseudospectrum of closed, densely defined linear operators in the Banach space. We begin by the definition and we investigate the characterization, the stability by means of quasi-compact operators and some properties of these pseudospectrum.

scholarly journals Closed linear operators with domain containing their range

1984 ◽  
Vol 27 (2) ◽  
pp. 229-233 ◽  
Author(s):  
Schôichi Ôta
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Linear Operator ◽  
Linear Operators ◽  
Closed Linear Operator ◽  
Unbounded Operators ◽  
Densely Defined ◽  
Closed Linear Operators

In connection with algebras of unbounded operators, Lassner showed in [4] that, if T is a densely defined, closed linear operator in a Hilbert space such that its domain is contained in the domain of its adjoint T* and is globally invariant under T and T*,then T is bounded. In the case of a Banach space (in particular, a C*-algebra) weshowed in [6] that a densely defined closed derivation in a C*-algebra with domaincontaining its range is automatically bounded (see the references in [6] and [7] for thetheory of derivations in C*-algebras).

scholarly journals Relatively compact-like perturbations, essential spectra and application

2004 ◽  
Vol 77 (1) ◽  
pp. 73-90 ◽  
Author(s):  
Khalid Latrach ◽  
J. Martin Paoli
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Neutron Transport ◽  
Linear Operators ◽  
Essential Spectra ◽  
Graph Norm ◽  
Bounded Linear ◽  
Neutron Transport Equations ◽  
Densely Defined ◽  
Fredholm Perturbations

AbstractThe purpose of this paper is to provide a detailed treatment of the behaviour of essential spectra of closed densely defined linear operators subjected to additive perturbations not necessarily belonging to any ideal of the algebra of bounded linear operators. IfAdenotes a closed densely defined linear operator on a Banach spaceX, our approach consists principally in considering the class ofA-closable operators which, regarded as operators in ℒ(XA,X) (whereXAdenotes the domain ofAequipped with the graph norm), are contained in the set ofA-Fredholm perturbations (see Definition 1.2). Our results are used to describe the essential spectra of singular neutron transport equations in bounded geometries.

scholarly journals Generic stability of the spectra for bounded linear operators on Banach spaces

1999 ◽  
Vol 12 (1) ◽  
pp. 31-33
Author(s):  
Luo Qun
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Residual Subset ◽  
Bounded Linear ◽  
Generic Stability ◽  
The Stability

In this paper, we study the stability of the spectra of bounded linear operators B(X) in a Banach space X, and obtain that their spectra are stable on a dense residual subset of B(X).

scholarly journals UNBOUNDED B-FREDHOLM OPERATORS ON HILBERT SPACES

2008 ◽  
Vol 51 (2) ◽  
pp. 285-296 ◽  
Author(s):  
M. Berkani ◽  
N. Castro-González
Keyword(s):  
Fredholm Operator ◽  
Hilbert Spaces ◽  
Closed Operator ◽  
Linear Operators ◽  
Fredholm Operators ◽  
Spectral Mapping ◽  
Nilpotent Operator ◽  
Densely Defined ◽  
Closed Linear Operators

AbstractThis paper is concerned with the study of a class of closed linear operators densely defined on a Hilbert space $H$ and called B-Fredholm operators. We characterize a B-Fredholm operator as the direct sum of a Fredholm closed operator and a bounded nilpotent operator. The notion of an index of a B-Fredholm operator is introduced and a characterization of B-Fredholm operators with index $0$ is given in terms of the sum of a Drazin closed operator and a finite-rank operator. We analyse the properties of the powers $T^m$ of a closed B-Fredholm operator and we establish a spectral mapping theorem.

Continuity of the generalized kernel and range of semi-Fredholm operators

1989 ◽  
Vol 105 (3) ◽  
pp. 513-522 ◽  
Author(s):  
M. Ó Searcóid ◽  
T. T. West
Keyword(s):  
Banach Space ◽  
Banach Algebra ◽  
Linear Operators ◽  
Complex Field ◽  
Fredholm Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Generalized Kernel

Let X denote a Banach space over the complex field ℂ and let B(X) be the Banach algebra of all bounded linear operators on X. If T ε B(X), we write n(T) = dim ker (T) and d(T) = codim T(X). Suppose that Y is a subspace invariant under T; then TY will denote the restriction of T to Y and Y the operator on X/Y defined byY: x/Y →(Tx)/Y

scholarly journals Extended Weyl-Type Theorems for Direct Sums

2014 ◽  
Vol 47 (2) ◽  
Author(s):  
M. Berkani ◽  
M. Kachad ◽  
H. Zariouh
Keyword(s):  
Banach Space ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Direct Sums ◽  
Bounded Linear ◽  
Direct Sum ◽  
Weyl Type Theorems ◽  
The Stability

AbstractIn this paper, we study the stability of extended Weyl and Browdertype theorems for orthogonal direct sum S⊕T, where S and T are bounded linear operators acting on Banach space. Two counterexamples shows that property (ab), in general, is not preserved under direct sum. Nonetheless, and under the assumptions that Π

Stability and Regularization of Difference Schemes in Banach and Hilbert Spaces

2013 ◽  
Vol 13 (2) ◽  
pp. 139-160
Author(s):  
Ivan P. Gavrilyuk ◽  
Volodymyr L. Makarov
Keyword(s):  
Banach Space ◽  
Adjoint Operator ◽  
Sufficient Conditions ◽  
Iteration Scheme ◽  
Difference Schemes ◽  
Stability Conditions ◽  
Initial Operator ◽  
The Stability ◽  
Necessary And Sufficient ◽  
Stability Results

Abstract. The necessary and sufficient conditions for stability of abstract difference schemes in Hilbert and Banach spaces are formulated. Contrary to known stability results we give stability conditions for schemes with non-self-adjoint operator coefficients in a Hilbert space and with strongly positive operator coefficients in a Banach space. It is shown that the parameters of the sectorial spectral domain play the crucial role. As an application we consider the Richardson iteration scheme for an operator equation in a Banach space, in particulary the Richardson iteration with precondition for a finite element scheme for a non-selfadjoint operator. The theoretical results are also the basis when using the regularization principle to construct stable difference schemes. For this aim we start from some simple scheme (even unstable) and derive stable schemes by perturbing the initial operator coefficients and by taking into account the stability conditions. Our approach is also valid for schemes with unbounded operator coefficients.

Epitaxial Growth in Dislocation-Free Strained Alloy Films

2002 ◽  
Vol 715 ◽  
Author(s):  
Zhi-Feng Huang ◽  
Rashmi C. Desai
Keyword(s):  
Deposition Rate ◽  
Elastic Moduli ◽  
Composition Dependence ◽  
Critical Thickness ◽  
Lattice Misfit ◽  
Misfit Strain ◽  
Joint Stability ◽  
Alloy Films ◽  
The Stability ◽  
Stability Results

AbstractThe morphological and compositional instabilities in the heteroepitaxial strained alloy films have attracted intense interest from both experimentalists and theorists. To understand the mechanisms and properties for the generation of instabilities, we have developed a nonequilibrium, continuum model for the dislocation-free and coherent film systems. The early evolution processes of surface pro.les for both growing and postdeposition (non-growing) thin alloy films are studied through a linear stability analysis. We consider the coupling between top surface of the film and the underlying bulk, as well as the combination and interplay of different elastic effects. These e.ects are caused by filmsubstrate lattice misfit, composition dependence of film lattice constant (compositional stress), and composition dependence of both Young's and shear elastic moduli. The interplay of these factors as well as the growth temperature and deposition rate leads to rich and complicated stability results. For both the growing.lm and non-growing alloy free surface, we determine the stability conditions and diagrams for the system. These show the joint stability or instability for film morphology and compositional pro.les, as well as the asymmetry between tensile and compressive layers. The kinetic critical thickness for the onset of instability during.lm growth is also calculated, and its scaling behavior with respect to misfit strain and deposition rate determined. Our results have implications for real alloy growth systems such as SiGe and InGaAs, which agree with qualitative trends seen in recent experimental observations.

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