Stability of some essential spectra of closed linear relations on Banach spaces (original) (raw)

Some properties of theM − essential spectra of closed linear operator on a Banach space

2016

Abstract. In this paper, we study a detailed treatment of some subsets of M-essential spectra of closed linear operators subjected to additive perturbations not necessarily belonging to any ideal of the algebra of bounded linear operators and we investigate some properties of the M-essential spectra of 2 × 2 matrix operator acting on a Banach space. This study led us to generalize some well known results for essential spectra of closed linear operator.

Perturbation results in the Fredholm theory and m-essential spectra of some matrix operators

2021

In this paper, we will use some new properties of non-compactness measure, in order to establish a description of the M-essential spectrum for some matrix operators on Banach spaces. Note that in general L 0 is not closed or closable, even if its entries are closed. But the authors in [4], give some sufficient conditions under which L 0 is closable and describe its closure which we shall denote L. Remark that in the work [7], M. Faierman, R. Mennicken and M. Möller give a method for dealing with the spectral theory for pencils of the form L 0 − µM, where M is a bounded operator. To study the Wolf essential spectrum of the operator matrix L in Banach spaces, the authors in [4] (resp. in [12]) used the compactness condition for the operator (λ − A) −1 (resp. C(λ − A) −1 and ((λ − A) −1 B) *). Recently, in [1] the author describes the Fredholm essential spectra of L with the help of the measures of weak-noncompactness, where X is a Banach space which possess the Dunford-Pettis property. In this paper, we prove some localization results on the M-essential spectra of the matrix operator L via the concept of some quantities. The purpose of this work is to pursue the analysis started in [1, 4, 12]. Our paper is organized as follows : In Section 2, we recall some notations and definitions. In Section 3, we prove some results needed in the rest of the paper. In Section 4, we investigate the M-essential spectra of a general class of operators defined by a 2 × 2 block operator matrix by means of some quantities.

On the essential spectra of some matrix of linear relations

Mathematical Methods in the Applied Sciences, 2014

We introduce several essential spectra of a linear relation on a normed space. We investigate the closedness and the emptiness of such essential spectra. As an application we prove two results, the first of which characterizes the class of quotient indecomposable normed spaces in terms of F − and strictly cosingular linear relations, and the second gives conditions under which a linear relation on a complex quotient indecomposable normed space is a strictly cosingular perturbation of a multiple of the identity.

A New Stability of the S-Essential Spectrum of Multivalued Linear Operators

International Journal of Analysis and Applications, 2017

We unfold in this paper two main results. In the first, we give the necessary assumptions for three linear relations AAA, BBB and SSS such that sigmaeap,S(A+B)=sigmaeap,S(A)\sigma_{eap,S}(A+B)= \sigma _{eap,S}(A)sigmaeap,S(A+B)=sigmaeap,S(A) and sigmaedelta,S(A+B)=sigmaedelta,S(A)\sigma_{e\delta,S}(A+B)= \sigma_{e\delta,S}(A)sigmaedelta,S(A+B)=sigmaedelta,S(A) is true. In the second, considering the fact that the linear relations AAA, BBB and SSS are not precompact or relatively precompact, we can show that sigmaeap,S(A+B)=sigmaeap,S(A)\sigma_{eap,S}(A+B)= \sigma_{eap,S}(A)sigmaeap,S(A+B)=sigmaeap,S(A) is true.

Stability of essential B-spectra of unbounded linear operators and applications

Afrika Matematika, 2018

In this paper, we study the stability of some essential B-spectra of closed linear operators on a Banach space X , under polynomially finite rank operators and we give the characterization of some essential B-spectra of a 2 × 2 of unbounded matrix operator acting in the product of Banach spaces X × Y. Then, using the functional calculus, we prove that a spectral mapping type theorem holds for these essential B-spectra. As an application, we study the effect of the functional calculus on the class of meromorphic operators, and on the class of isoloid operators with sable sign index, satisfying generalized Weyl theorem.

On the Schechter Essential Spectrum on Banach Spaces and Application

2002

This paper is devoted to the investigation of the stability of the Schechter essential spectrum of closed densely defined linear operators A subjected to additive perturbations K such that (λ − A − K) −1 K or K(λ − A − K) −1 belonging to arbitrary subsets of L(X) (where X denotes a Banach spaces) contained in the set J (X). Our approach consists principally in considering the class of A-closable (not necessarily bounded) which contained in the set of Aresolvent Fredholm perturbations which zero index (see Definition 3.5). They are used to describe the Schechter essential spectrum of singular neutron transport equations in bounded geometries. Definition 1.1. An operator A ∈ L(X, Y) is said to be weakly compact if A(B) is relatively weakly compact in Y for every bounded subset B ⊂ X.