Spectra of some block operator matrices and application to transport operators

Abstract

In this paper a new concept for a 3 × 3 block operator matrix is studied on a Banach space.It is shown that, under certain conditions, it defines a closable operator and its essential spectra are determined. Application to transport operators in L 1 -space is given.

On Relative Essential Spectra of Block Operator Matrices and Application

“…This together with Lemma 4.2-(ii), Proposition 2 in [9] and Eq. (9), make us conclude that (11) G(λ)(S λ − λM 4 ) −1 := G(λ)(T…”

“…In this direction some issues may be found in the literature, we can quote for example [1,2,10,16,24]. Recently, an account research and a wide panorama of methods to investigate the spectral theory of block operator matrices is given in [3,4,5,11,14,25]. More precisely, the description of various essential spectra of a block operator matrix L appears in [3,5,11,14] to improve and generalize some results given by [1,2,24] for block operator matrices in Banach spaces under some compactness assumptions.…”

“…Recently, an account research and a wide panorama of methods to investigate the spectral theory of block operator matrices is given in [3,4,5,11,14,25]. More precisely, the description of various essential spectra of a block operator matrix L appears in [3,5,11,14] to improve and generalize some results given by [1,2,24] for block operator matrices in Banach spaces under some compactness assumptions. However, it should be noted that several results for the authors cited in the papers of [1,2,4,14,24] are aimed at providing methods for dealing with spectral theory for operator in the form L 0 − λM where M = I.…”

Essential Spectra of a 3 × 3 Operator Matrix and an Application to Three-Group Transport Equations

“…In [20], Moalla, Damak and Jeribi extended these results to a large class of operators, described their essential spectra, and applied these results to describe the essential spectra of two-group transport operators with general boundary conditions in L p -spaces. In [16], Jeribi, Moalla and Walha treated a 3 × 3 block operator matrix (1.1) on a Banach space. It was shown that under certain conditions, this block operator matrix defines a closable operator and its essential spectra are determined.…”

Variation of the Spectrum of Operators in Infinite Dimensional Spaces

“…Essential spectra of some matrix operators on Banach spaces (see [1]) and spectra of some block operator matrices (see [2]) were investigated, with applications to differential and transport operators. This paper investigates the variations of the unit spectrum as T varies over the space of all bounded operator on the Banach space…”