Aref Jeribi

A characterization of the Schechter essential spectrum on Banach spaces and applications

Abstract In a recent article by the author (C. R. Acad. Sci. Paris Ser. I 331 (2000) 525–530; Boll. Un. Mat. Ital. (2002), to appear) the Schechter spectrum of closed, densely defined linear operators has been characterized on spaces, which possess the Dunford–Pettis property or which are isomorphic to one of the spaces L p (Ω) , p>1. The purpose of the present work is to extend this analysis to the case of Banach spaces. Further we apply the obtained results to investigate the Schechter essential spectrum of one-dimensional transport equations with different boundary conditions.

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

Quelques remarques sur le spectre de Weyl et applications

Resume Dans cette Note, nous donnons une caracterisation du spectre de Weyl au moyen des operateurs reguliers. Ensuite, nous l’appliquons a l’equation de transport.

Une nouvelle caractérisation du spectre essentiel et application

In this Note we give a characterization of the essential spectrum on a spaces which possess the Dunford–Pettis property or which isomorphic to one of the spaces Lp(Ω) , p>1 , and we apply thus result in transport equation.

On the essential spectrum of transport operators on L1‐spaces

In a recent article by the first author [J. Math. Phys. 35, 6199–6212 (1994)] the essential spectrum of transport operator was analyzed in Lp‐spaces for p∈(1,+∞). The purpose of the present work is to extend this analysis to the case of L1‐spaces. After establishing preliminary results we define the notion of the weak spectrum which we characterize by means of Fredholm operators. We show, in particular, that in L1‐spaces the weak spectrum is nothing else but the essential spectrum. Using the same techniques as in the above‐mentioned work, we prove the stability of the essential spectrum of a one‐dimensional transport operator with general boundary conditions where an abstract boundary operator relates the incoming and the outgoing fluxes. Sufficient conditions are given in terms of boundary and collision operators, assuring the stability of the essential spectrum. We show also that our results remain valid for neutron transport operators in arbitrary dimension.

Some Fixed Point Theorems and Application to Biological Model

In this paper, we establish some results regarding the existence of solution on L 1 spaces to a nonlinear boundary value problem originally proposed by Rotenberg (J. Theo. Biol. 1983; 103:181–199) to model the growth of cell population. Our strategy consists in establishing new variants of fixed point theorems in general Banach spaces. These topological results can be used to resolve some open problems posed by Latrach and Jeribi (Nonlinear Anal. T.M.A. 1999; 36:843–862).

Some new properties in Fredholm theory, Schechter essential spectrum, and application to transport theory.

The theory of measures of noncompactness has many applications on topology, functional analysis, and operator theory. In this paper, we consider one axiomatic approach to this notion which includes the most important classical definitions. We give some results concerning a certain class of semi-Fredholm and Fredholm operators via the concept of measures of noncompactness. Moreover, we establish a fine description of the Schechter essential spectrum of a closed densely defined operators. These results are exploited to investigate the Schechter essential spectrum of a multidimensional neutron transport operator.

Time asymptotic behaviour for unbounded linear operator arising in growing cell populations

Abstract This paper is concerned with the spectral analysis of a class of unbounded, linear operators, originally proposed by M. Rotenberg (J. Theor. Biol. 96 (1982) 495; J. Theor. Biol. 103 (1983) 181). After a detailed spectral analysis it is shown that the associated Cauchy problem is governed by a C 0 -semigroup. Next, we discuss the irreducibility of the transport semigroup. In particular, we show that the transport semigroup is irreducible if the boundary operator is strictly positive. Finally, a spectral decomposition of the solutions into an asymptotic term and a transient one which will be estimated for smooth initial data is given.

Stability of the S -essential spectra on a Banach space

In this paper, we give the characterization of S-essential spectra, we define the S-Riesz projection and we investigate the S-Browder resolvent. Finally, we study the S-essential spectra of sum of two bounded linear operators acting on a Banach space.

Riesz Basis Property of Families of Nonharmonic Exponentials and Application to a Problem of a Radiation of a Vibrating Structure in a Light Fluid

This article deals with the Riesz basis property of families of nonharmonic exponentials. The exponents coincide with the eigenvalues of a specific perturbation of a closed linear operator. The key idea of this work is based on the estimate given by Nagy [6] using the spectral analysis method. Furthermore, it is also shown that the system of a sequence of exponentials families of the operator forms a Riesz basis in L 2(0, T), T > 0, where K is the integral operator with kernel the Hankel function of the first kind and order 0.