A characterization of some subsets of S-essential spectra of a multivalued linear operator (original) (raw)

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.

On the essential spectra of some matrix of linear relations

Mathematical Methods in the Applied Sciences, 2014

In this paper, we investigate a detailed treatment of some subsets of essential spectra of a closed multivalued linear operator. On the following, we will establish some results on perturbation theory of 2 2 matrix of multivalued linear operators.

On the spectrum of linear operator pencils

Matematičnì studìï, 2019

We consider a linear operator pencil L(λ) = A − λB, λ ∈ C, where A and B are bounded operators on Hilbert space. The purpose of this paper is to study the conditions under which the spectrum of L(.) is the whole complex plane or empty. This leads to some criteria for the spectrum to be bounded.

A NOTE ON PRESERVATION OF SPECTRA FOR TWO GIVEN OPERATORS (Mb)

Mathematica Bohemica, 2020

We study the relationships between the spectra derived from Fredholm theory corresponding to two given bounded linear operators acting on the same space. The main goal of this paper is to obtain sufficient conditions for which the spectra derived from Fredholm theory and other parts of the spectra corresponding to two given operators are preserved. As an application of our results, we give conditions for which the above mentioned spectra corresponding to two multiplication operators acting on the space of functions of bounded ppp-variation in Wiener's sense coincide. Additional illustrative results are given too

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.

A note on preservation of spectra for two given operators

Mathematica Bohemica

We study the relationships between the spectra derived from Fredholm theory corresponding to two given bounded linear operators acting on the same space. The main goal of this paper is to obtain sufficient conditions for which the spectra derived from Fredholm theory and other parts of the spectra corresponding to two given operators are preserved. As an application of our results, we give conditions for which the above mentioned spectra corresponding to two multiplication operators acting on the space of functions of bounded p-variation in Wiener's sense coincide. Additional illustrative results are given too.

On the Spectra of Left-Definite Operators

Complex Analysis and Operator Theory, 2013

If A is a self-adjoint operator that is bounded below in a Hilbert space H; Littlejohn and Wellman showed that, for each r > 0; there exists a unique Hilbert space Hr and a unique self-adjoint operator Ar in Hr satisfying certain conditions dependent on H and A: The space Hr and the operator Ar are called, respectively, the r th left-de…nite space and r th left-de…nite operator associated with (H; A): In this paper, we show that the operators A; Ar; and As (r; s > 0) are isometrically isomorphically equivalent and that the spaces H; Hr; and Hs (r; s > 0) are isometrically isomorphic. These results are then used to reproduce the left-de…nite spaces and left-de…nite operators: Furthermore, we will see that our new results imply that the spectra of A and Ar are equal, giving us another proof of this phenomenon that was …rst established in . words and phrases. right-de…nite operator, left-de…nite operator, left-de…nite space, Hilbert scale, Sobolev space, self-adjoint operator, isometric isomorphism, similarity transformation, point spectrum, continuous spectrum. 1 2 LANCE L. LITTLEJOHN AND RICHARD WELLMAN