The Algerian Democratic and Popular Republic The Ministry of Higher Education and Scientific Research University of Oran Faculty of Sciences Department of Mathematics Thesis Submitted in partial fulfillment of the requirements for the degree of DOCTOR ES-SCIENCES–MATHEMATICS by Mohammed Benharrat Comparison between the different definitions of the essential spectrum and Applications. Thesis director Pr. Bekkai MESSIRDI Sustained on February 27th; 2013 before the exam committee: President: Mr. C. BOUZAR Prof. University of Oran. Supervisor: Mr. B. MESSIRDI Prof. University of Oran. Examiners: Mr. R. LABBAS Prof. University of Le Havre . Mr. B. BENDOUKHA Prof. University of Mostaganem. Mr. M. TERBECHE Prof. University of Oran. Mr. A. TALHAOUI MCA ENSET d’Oran. Invited: Mr. A. SENOUSSAOUI MCA University of Oran. Academic year 2012-2013 Comparison between the different definitions of the essential spectrum and Applications. Mohammed Benharrat Ecole Normale Supérieure de l’Enseignement Technologique d’Oran Département de Mathématiques et Informatique B.P. 1525 El M’Nouar. Oran Email: [email protected] Thesis submitted to The University of Oran for the degree of DOCTOR ES-SCIENCES–MATHEMATICS. Thesis director Pr. Bekkai MESSIRDI University of Oran, 2013 Acknowledgments I would like to thank first and foremost my supervisor, Professor Bekkai Messirdi, who has invested considerable time and energy into guiding me through my thesis, for his many suggestions and constant support during this research. Her scholarship and dedication has been an inspiration in my studies. I am very sensitive to the honor which makes me the Professor C. Bouzar by agreeing to chair this jury; and am him deeply grateful. I thank Professors B. Bendoukha, M. Terbeche, A. Talhaoui and A. Senoussaoui for their time and effort participating in may thesis committee. They are generously given their ex- pertise to improve my work. The presence of a specialist like Professor R. Labbas in the examen committee honors me and I would like to thank him. Of course, I am grateful to my family for their patience and love. Without them this work would never have come into existence. Abstract Our main objective in this thesis is to present the most various definitions of the essential spectrum founded in the mathematical literature, which beginning with the fundamental work of Weyl, is becoming more and more a special branch of spectral theory producing results and problems of its own. On other hand, we gives a remarkable various characteristic stability properties of the essential spectra under appropriate perturbations, as well as some equivalent descriptions of these spectra. Our contributions in this dissertation are: • We give a survey of results concerning various essential spectra of closed linear opera- tors in a Banach space and we give some relationships between this essential spectra and the SVEP theory. • We investigate a relationship between the Kato spectrum and another essential spec- trum called the closed-range spectrum of an operator A, defined by Goldberg in [45] by σec(A) = fλ 2 C ; R(λI − A) is not closedg, in the case of Banach spaces. This work is extended to the Banach spaces, the result was shown by J.P Labrousse [71] in the case of Hilbert spaces. • We show that the symmetric difference between the generalized Kato spectrum and the the closed-range spectrum is at most countable and we also give some relationships between the generalized Kato spectrum and the others essential by the use of the local spectral theory. • We present a survey of results of characterizations and perturbations for various essen- tial spectra and we consider their stability under some classes of perturbations. By the use of the Fredholm perturbations, we describe the various essential spectra of some transport operators. Key Words. Essential spectrum, Semi-Fredholm operators, Fredholm perturbations, Semi-regular operators, Quasi-Fredholm operators, Operators of Kato type, Generalized Kato spectrum, Closed-range spectrum, Local spectral theory, Transport operators. AMS Classification: 47A10, 47A53, 47A55, 47A60, 47B07, 47F05, 47G20. List of Publications • M. Benharrat, B. Messirdi. Essential spectrum: A Brief survey of concepts and appli- cations. Azerbaijan Journal of Mathematics. V. 2, No 1, 35-61 (2012). • M. Benharrat, B. Messirdi. On the generalized Kato spectrum. Serdica Math. J. 37 (2011), 283-294. • M. Benharrat, B. Messirdi. Relationship between the Kato essential spectrum and a variant of essential spectrum. To appear in General Mathematics. List of submitted Papers • M. Benharrat, B. Messirdi. Quasi-nilpotent perturbations of the generalized Kato spec- trum and Applications. (Submitted ). • M. Benharrat, B. Messirdi. B-Fredholm spectra of some transport operators. (Submit- ted ). List of Communications • M. Benharrat, B. Messirdi. Characterizations of Fredholm operators by quotient oper- ators. Congrés des Mathématiciens Algériens, CMA’2012, Annaba, 07-08 Mars 2012. Contents Introduction 1 1 Spectrum of an operator 9 1.1 Algebraic Properties . .9 1.1.1 Ascent and descent of an operator . .9 1.1.2 The nullity and the deficiency of an operator . 13 1.2 Generalities about Closed operators . 17 1.2.1 Closable operators . 18 1.2.2 Adjoint operator. 19 1.3 Operators with closed range . 21 1.4 Compact operators . 22 1.5 Perturbations of closed operators . 24 1.6 The spectrum of closed operators . 26 1.7 Approximate point spectrum . 33 1.8 The Riesz projection and the singularities of the resolvent . 35 2 Essential Fredholm, Weyl and Browder spectra 42 2.1 Essential Fredholm spectra . 42 2.2 Fredholm perturbations . 45 2.3 Browder and Weyl spectra . 54 2.3.1 The Browder resolvent . 57 2.3.2 The essential spectral radius . 58 2.4 Characterizations of the essential spectra . 59 2.5 Left-right Fredholm and Left-right Browder spectra . 63 2.6 Invariance of the essential spectra . 67 3 Generalized Kato spectrum 70 3.1 The semi-regular spectrum and its essential version . 70 3.2 Closed-range spectrum . 75 3.3 Quasi-Fredholm spectrum . 76 3.4 Generalized Kato spectrum . 78 3.5 Saphar operators, essentially Saphar operators and corresponding spectra . 83 4 Essential spectra defined by means of restrictions 86 4.1 Descent spectrum and essential descent spectrum . 86 4.2 Ascent spectrum and essential ascent spectrum . 89 4.3 Essential spectrum and Drazin invertible operators . 91 4.4 B-Fredholm, B-Browder and B-Weyl spectra . 94 4.5 Essential spectra and The SVEP theory . 99 4.6 Weyl’s theorem and Browder’s theorem . 103 iv 5 Applications 105 5.1 One-dimensional transport equation . 105 5.1.1 Application of the Fredholm perturbations to transport equations . 108 5.1.2 Application of the quasi-nilpotent perturbations to transport equations 110 5.2 Singular transport operators . 111 Conclusion and perspectives 114 Bibliography 115 v Introduction The theory of the essential spectra of linear operators in Banach space is a modern section of the spectral analysis widely used in the mathematical and physical sense when resolving a number of applications that can be formulated in terms of linear operators. Within the spectral theory lie a vast number of essential spectra defined for an individual operator, that have been introduced and investigated extensively. The original definition of the essential spectrum goes back to H. Weyl1 [119] around 1909, when he defined the essential spectrum of a self-adjoint operator A on a Hilbert space as the set of all points of the spectrum of A except isolated eigenvalues of finite multiplicity and he proved that the addition of a compact operator to A does not affect the essential spectrum, today this classical result is known as Weyl’s theorem, this theorem is very important in the description of the essential spectrum of the Schrodinger operators for a large class of two-body potentials. When A is not self-adjoint bounded operator (or just assumed to be closed and densely defined in an arbitrary Banach space), in analogy with Weyl’s theorem, one would like that the essential spectrum to be invariant under arbitrary compact perturbations. The definition given above is not suitable in this direction and the situation is considerably more compli- cated, because it is possible for the unperturbed operator to have only a discrete spectrum while the point spectrum of the perturbed operator is the whole complex plane, and some op- erators have point eigenvalues which are not isolated and are carried into the resolvent under a compact perturbation. However, there are applications in which one would like to know that certain types of singularities are not introduced under compact perturbations, even though such singularities lie outside the essential spectrum. This motivates another several possible definitions of the essential spectrum (for an arbitrary operator) as the largest sub- set of the spectrum remaining invariant under arbitrary compact perturbations. From this moment, essential spectrum and their properties of stability under (additive) perturbations in appropriate class of operators, have been a research interest of many authors. The theory has been examined in connection with various classes of linear operators defined by means of kernels and ranges, the most important of this classes Fredholm op- erators, semi-Fredholm operators, quasi-Fredholm operators and more recently B-Fredholm operators and generalized invertible operators (in particular, the Drazin and Koliha invert- ible operators). With these different classes of operators associated essential spectra, which are qualitatively different subsets of the spectrum of an operator. However, the search for different subsets of the spectrum, satisfying certain properties, is so far, which confirms the relevance of research presented in this thesis. In this thesis, we give out at least 55 kind spectra and at least 46 kind essential spectra of an operator. Throughout this monograph, let X and Y be complex infinite dimensional Banach spaces and C(X; Y ) (resp. L(X; Y )) be the set of all closed, densely linear operators (resp.