On some subsets of Schechter's essential spectrum of a matrix operator and application to transport operator

Ali, Naouel Ben; Jeribi, Aref; Moalla, Nedra

doi:10.1002/mana.200710081

Math. Nachr.

2010

DOI: 10.1002/mana.200710081

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On some subsets of Schechter's essential spectrum of a matrix operator and application to transport operator

Naouel Ben Ali

Aref Jeribi

Nedra Moalla

Abstract: This paper is devoted to the investigation of the essential approximate point spectrum and the essential defect spectrum of a 2 × 2 block operator matrix on a product of Banach spaces. The obtained results are applied to a two-group transport operators with general boundary conditions in the Banach space Lp([−a, a] × [−1, 1]) × Lp([−a, a] × [−1, 1]), a > 0, p ≥ 1.

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Cited by 20 publications

References 28 publications

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Introduction

Jeribi¹

2015

Spectral Theory and Applications of Linear Operators and Block Operator Matrices

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IntroductionThis book is devoted to some recent mathematical developments which cover several topics including Cauchy problem, Fredholm operators, spectral theory, and block operator matrices, both dealing with linear operators. Of course, these topics play a crucial role in many branches of mathematics and also in numerous applications as they are intimately related to the stability of the underlying physical systems.One of the objectives of this book is the study of the classical Riesz theory of polynomially compact operators, in order to establish the existence results of the second kind operator equations, hence allowing to describe the spectrum, multiplicities, and localization of the eigenvalues of polynomially compact operators. Fredholm theory and perturbation results are also widely investigated. The description of the large time behavior of solutions to an abstract Cauchy problem on Banach spaces without restriction on the initial data is studied. Further, the essential state of the art of research and essential pseudo-spectra of closed, densely defined, and linear operators subjected to additive perturbations is outlined. The spectral theory of block operator matrices is of major interest, since it describes coupled systems of partial differential equations of mixed order and type. For this reason, an important part of this book is devoted to develop essential spectra of 2 2 and 3 3 block operator matrices. Based on the spectral graph theory (which is an active research area), we are interested in the study of the adjacency matrix and the discrete Laplacian acting on forms. Most of the results of this book are motivated by physical transport problems for which we address our applications at the end of the book. Now, let us describe its contents.

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Introduction

Jeribi¹

2015

Spectral Theory and Applications of Linear Operators and Block Operator Matrices

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Essential Spectra of 2 × 2 Block Operator Matrices

Jeribi

2015

Spectral Theory and Applications of Linear Operators and Block Operator Matrices

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Let X and Y be Banach spaces. In the product space X Y , we consider an operator formally defined by a matrix and acts from Y into X (resp. from X into Y ). One of the problems in the study of such operators is that in general L 0 is not closed or even closable, even if its entries are closed. The aim of this chapter is to present some hypotheses which should allow the block operator matrix L 0 to be closable. For this purpose, it is interesting to present the suitable conditions for the entries A, B, C , and D. These conditions are mainly based on the FrobeniusSchur decomposition of L 0 . Then, our reasoning will be mainly based on the specific properties of the first entry A, hence allowing us to get a closable block matrix operator L 0 . It is interesting to notice that this reasoning based on A could also be applied for the other entries, namely B, C , or D. In fact our approach is based on three situations. In the first one, the resolvent of the operator A is a Fredholm perturbation. In the second situation, the operator A is closed, whereas this same operator is closable in the third situation. After that, we will give the essential spectra of the closure of the operator L 0 .

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Essential Spectra of 3 × 3 Block Operator Matrices

Jeribi

2015

Spectral Theory and Applications of Linear Operators and Block Operator Matrices

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On some subsets of Schechter's essential spectrum of a matrix operator and application to transport operator

Cited by 20 publications

References 28 publications

Introduction

Introduction

Essential Spectra of 2 × 2 Block Operator Matrices

Essential Spectra of 3 × 3 Block Operator Matrices

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