The adjacency matrix and the discrete Laplacian acting on forms

We study the relationship between the adjacency matrix and the discrete Laplacian acting on 1-forms. We also prove that if the adjacency matrix is bounded from below it is not necessarily essentially self-adjoint. We discuss the question of essential self-adjointness and the notion of completeness

A characterization of the essential pseudospectra and application to a transport equation

In this paper, we introduce and study the essential pseudospectra of closed, densely defined linear operators in the Banach space. We start by giving the definition and we investigate the characterization, the stability and some properties of this essential pseudospectra. The obtained results are used to describe the essential pseudospectra of transport operators.peerReviewe

Fredholm theory for demicompact linear relations

[EN] We first attempt to determine conditions on a linear relation T such that µT becomes a demicompact linear relation for each µ ∈ [0, 1) (see Theorems 2.4 and 2.5). 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 block matrix of linear relations becomes a demicompact block matrix of linear relations (see Theorems 4.2 and 4.3).Ammar, A.; Fakhfakh, S.; Jeribi, A. (2022). Fredholm theory for demicompact linear relations. Applied General Topology. 23(2):425-436. https://doi.org/10.4995/agt.2022.1694042543623

Spectral properties for polynomial and matrix operators involving demicompactness classes

The first aim of this paper is to show that a polynomially demicompact operator satisfying certain conditions is demicompact. Furthermore, we give a refinement of the Schmoëger and the Rakocević essential spectra of a closed linear operator involving the class of demicompact ones. The second aim of this work is devoted to provide some sufficient conditions on the inputs of a closable block operator matrix to ensure the demicompactness of its closure. An example involving the Caputo derivative of fractional of order α is provided. Moreover, a study of the essential spectra and an investigation of some perturbation results.peerReviewe

Essential spectra of some matrix operators and application to two-group transport operators with general boundary conditions

AbstractIn this article we investigate the essential spectra of a 2×2 block operator matrix on a Banach space. Furthermore, we apply the obtained results to determine the essential spectra of two-group transport operators with general boundary conditions in the Banach space Lp([−a,a]×[−1,1])×Lp([−a,a]×[−1,1]), a>0

ESSENTIALLY SEMI-REGULAR LINEAR RELATIONS

In this paper, we study the essentially semi-regular linear relation operators everywhere defined in Hilbert space. We establish a Kato-type decomposition of essentially semi-regular relations in Hilbert spaces. The result is then applied to study and give some properties of the Samuel-multiplicity

The Essential Spectrum of a Sequence of Linear Operators in Banach Spaces

In this work we introduce some essential spectra $(\sigma_{ei}, i=1,...,5)$ of a sequence of closed linear operators $(T_{n})_{n\in\mathbb{N}}$ on Banach space, we prove that if $(T_{n})_{n\in\mathbb{N}}$ converges in the generalized sense to a closed linear operator $T$, then there exists $n_{0}\in \mathbb{N}$ such that, for every $n\geq n_{0}$, we have $\sigma_{ei}(\lambda _{0}-(T_{n}+B))\subseteq \sigma _{ei}(\lambda_{0}-(T+B)), i=1,...,5$, where $B$ is a bounded linear operator, and $\lambda _{0}\in \mathbb{C}$. The same treatment is made when $(T_{n}-T)$ converges to zero compactly.</p

Spectral theory and applications of linear operators and block operator matrices

Examining recent mathematical developments in the study of Fredholm operators, spectral theory and block operator matrices, with a rigorous treatment of classical Riesz theory of polynomially compact operators, this volume covers both abstract and applied developments in the study of spectral theory. These topics are intimately related to the stability of underlying physical systems and play a crucial role in many branches of mathematics as well as numerous interdisciplinary applications. By studying classical Riesz theory of polynomially compact operators in order to establish the existence results of the second kind operator equations, this volume will assist the reader working to describe the spectrum, multiplicities and localization of the eigenvalues of polynomially compact operators