Comparison Between the Different Definitions of the Essential Spectrum

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 operators 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 spectrum called the closed-range spectrum of an operator A, deﬁned by Goldberg in [45] by σec(A) = {λ ∈ C ; R(λI − A) is not closed}, 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 essential 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 applications. 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 spectrum 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 operators. 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 operators 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 subset 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 operators, semi-Fredholm operators, quasi-Fredholm operators and more recently B-Fredholm operators and generalized invertible operators (in particular, the Drazin and Koliha invertible 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. all bounded linear operators) from X into Y , for simplicity, we write C(X,X) (resp. L(X,X)) as C(X) (resp. L(X)). Let X∗ be the dual space of X. If A ∈ C(X,Y ), then

1Hermann Klaus Hugo Weyl, 9 November 1885 - 8 December 1955. German mathematician

1 A∗ ∈ C(Y ∗,X∗) denotes the adjoint operator of A. For A ∈ C(X,Y ), let R(A) and N(A) denote the range and kernel of A, respectively, and denote α(A) = dim N(A), β(A) = dim Y/R(A). If A ∈ C(X), the ascent a(A) of A is defined to be the smallest nonnegative integer k (if it exists) which satisfies that N(Ak) = N(Ak+1). If such k does not exist, then the ascent of A is defined as infinity. Similarly, the descent d(A) of A is defined as the smallest nonnegative integer k (if it exists) for which R(Ak) = R(Ak+1) holds. If such k does not exist, then d(A) is defined as infinity, too. If the ascent and the descent of A are finite, then they are equal. For A ∈ C(X), if R(A) is closed and α(A) < ∞, then A is said to be an upper semi-Fredholm operator, if β(A) < ∞, then A is said to be a lower semi-Fredholm operator. If A ∈ C(X) is either upper or lower semi-Fredholm operator, then A is said to be a semi-Fredholm operator. For semi-Fredholm operator A, its index ind (A) is defined as ind (A) = α(A) − β(A). Now, we introduce the following important operator classes: The sets of all invertible operators with bounded inverse, bounded below operators, surjective operators, left invertible operators, right invertible operators on X are defined, respectively, by G(X) := {A ∈ C(X): A is invertible and A−1 is bounded},

G+(X) := {A ∈ C(X): A is injective and R(A) is closed},

G−(X) := {A ∈ C(X): A is surjective},

Gl(X) := {A ∈ C(X): A is left invertible},

Gr(X) := {A ∈ C(X): A is right invertible}. The sets of all Fredholm operators, upper semi-Fredholm operators, lower semi-Fredholm operators, left semi-Fredholm operators, right semi-Fredholm operators on X are deﬁned, respectively, by Φ(X) := {A ∈ C(X): α(A) < ∞ and β(A) < ∞},

Φ+(X) := {A ∈ C(X): α(A) < ∞ and R(A) is closed},

Φ−(X) := {A ∈ C(X): β(A) < ∞},

Φl(X) := {A ∈ C(X): R(A) is a closed and complemented subspace of X and α(A) < ∞},

Φr(X) := {A ∈ C(X): N(A) is a closed and complemented subspace of X and β(A) < ∞}. The sets of all Weyl operators, upper semi-Weyl operators, lower semi-Weyl operators, left semi-Weyl operators, right semi-Weyl operators on X are deﬁned, respectively, by W(X) := {A ∈ Φ(X): ind(A) = 0},

W+(X) := {A ∈ Φ+(X): ind(A) ≤ 0},

W−(X) := {A ∈ Φ−(X): ind(A) ≥ 0},

Wlw(X) := {A ∈ Φl(X): ind(A) ≤ 0},

Wrw(X) := {A ∈ Φr(X): ind(A) ≥ 0}. The sets of all Browder operators, upper semi-Browder operators, lower semi-Browder operators, left semi-Browder operators, right semi-Browder operators on X are deﬁned, respectively, by B(X) := {A ∈ Φ(X): a(A) = d(A) < ∞},

B+(X) := {A ∈ Φ+(X): a(A) < ∞},

B−(X) := {A ∈ Φ−(X): d(A) < ∞},

Blb(X) := {A ∈ Φl(X): a(A) < ∞},

Brb(X) := {A ∈ Φr(X): d(A) < ∞}.

2 By the help of above set classes, for A ∈ C(X), we can deﬁne its corresponding spectra, respectively, as following:

the spectrum: σ(A) = {λ ∈ C : λI − A 6∈ G(X)}, the approximate point spectrum: σap(A) = {λ ∈ C : λI − A 6∈ G+(X)}, the defect spectrum: σsu(A) = {λ ∈ C : λI − A 6∈ G−(X)}, the left spectrum: σl(A) = {λ ∈ C : λI − A 6∈ Gl(X)}, the right spectrum: σri(A) = {λ ∈ C : λI − A 6∈ Gr(X)}, the Fredholm spectrum: σef (A) = {λ ∈ C : λI − A 6∈ Φ(X)}, the upper semi-Fredholm spectrum: σuf (A) = {λ ∈ C : λI − A 6∈ Φ+(X)}, the lower semi-Fredholm spectrum: σlf (A) = {λ ∈ C : λI − A 6∈ Φ−(X)}, the semi-Fredholm spectrum: σsf (A) = {λ ∈ C : λI − A 6∈ Φ+(X) ∪ Φ−(X)}, the left semi-Fredholm spectrum: σlef (A) = {λ ∈ C : λI − A 6∈ Φl(X)}, the right semi-Fredholm spectrum: σrf (A) = {λ ∈ C : λI − A 6∈ Φr(X)}, the Weyl spectrum: σew(A) = {λ ∈ C : λI − A 6∈ W(X)}, the upper semi-Weyl spectrum: σuw(A) = {λ ∈ C : λI − A 6∈ W+(X)}, the lower semi-Weyl spectrum: σlw(A) = {λ ∈ C : λI − A 6∈ W−(X)}, the left semi-Weyl spectrum: σlew(A) = {λ ∈ C : λI − A 6∈ Wl(X)}, the right semi-Weyl spectrum: σrw(A) = {λ ∈ C : λI − A 6∈ Wr(X)}, the Browder spectrum: σeb(A) = {λ ∈ C : λI − A 6∈ B(X)}, the upper semi-Browder spectrum: σub(A) = {λ ∈ C : λI − A 6∈ B+(X)}, the lower semi-Browder spectrum: σlb(A) = {λ ∈ C : λI − A 6∈ B−(X)}, the left semi-Browder spectrum: σleb(A) = {λ ∈ C : λI − A 6∈ Bl(X)}, the right semi-Browder spectrum: σrb(A) = {λ ∈ C : λI − A 6∈ Br(X)}, the compression spectrum: σco(A) := {λ ∈ C : R(λI − A) is not dense in X},

the third Kato spectrum: σK3 (A) = {λ ∈ C : λI − A 6∈ Φl(X) ∪ Φr(X)}. the Goldberg spectrum : σec(A) = {λ ∈ C ; R(λI − A) is not closed}.

It is well known that all of these spectra (except σec(A) ) are closed nonempty subsets of complex plane C (see [55, 62, 64, 69, 94, 95, 97, 101, 103, 108, 109, 118, 117, 122, 123]) and have the following relationships:

(1) σec(A) ⊆ σsf (A) ⊆ σuf (A) ⊆ σuw(A) ⊆ σub(A) ⊆ σeb(A), (2) σec(A) ⊆ σsf (A) ⊆ σlf (A) ⊆ σlw(A) ⊆ σlb(A) ⊆ σeb(A),

(3) σec(A) ⊆ σK3 (A) ⊆ σlef (A) ⊆ σlew(A) ⊆ σleb(A) ⊆ σeb(A),

(4) σec(A) ⊆ σK3 (A) ⊆ σrf (A) ⊆ σrw(A) ⊆ σrb(A) ⊆ σeb(A), (5) σec(A) ⊆ ∂(σeb(A)) ⊆ ∂(σew(A)) ⊆ ∂(σef (A)) ⊆ σsf (A) ⊆ σef (A) ⊆ σew(A) ⊆ σeb(A) ⊆ σ(A), (6) ∂(σ(A)) ⊆ σap(A) ∩ σsu(A) ⊆ σl(A) ∪ σri(A) ⊆ σ(A).

For a compact subset M of C, we use accM, ∂M, intM, M and isoM, respectively, to denote all the points of accumulation of M, the boundary of M, the interior of M, the closure of M and all isolated points of M. Recall that the discrete spectrum of self-adjoint operator A acting on Hilbert space consists of isolated eigenvalues of ﬁnite multiplicity, the remaining part of the spectrum is the essential spectrum and

σec(A) = σsf (A) = σef (A) = σew(A) = σeb(A).

3 If A is a compact operator in Banach space, then

σec(A) = σsf (A) = σef (A) = σew(A) = σeb(A) = {0}.

Let us mention the classical subdivision of the spectrum in the following three disjoint sets: The point spectrum: σp(A) := {λ ∈ C : λI − A is not injective}, The continuous spectrum: σc(A) := {λ ∈ C : λI−A is injective but R(A) is not closed }, The residual spectrum: σr(A) := {λ ∈ C : λI − A is injective but R(A) is not dense }.

If λ in the continuous spectrum σc(A) of A then R(λ − A) is not closed. Therefore λ ∈ σi(A), i ∈ Λ = {ec, lf, uf, ef, ew, uw, lw, eb, ub, lb}. Consequently we have \ σc(A) ⊂ σi(A). i∈Λ An operator A ∈ L(X) is said to be semi-regular if R(A) is closed and N(An) ⊆ R(A), for all n ≥ 0; A is said to be quasi-Fredholm if there exists d ∈ N such that 1. R(An) ∩ N(A) = R(Ad) ∩ N(A) for all n ≥ d.

2. R(Ad) ∩ N(A) and R(Ad) + N(A) are closed in X. and A is admits a generalized Kato decomposition, GKD for short, if there exists a pair of A-invariant closed subspaces (M,N) such that X = M ⊕ N, where A|M is semi-regular and A|N is quasi-nilpotent. If we assume in the definition above that A|N is nilpotent, then there exists d ∈ N for which (A|N)d = 0. In this case A is said to be of Kato type of order d. An operator A is said to be essentially semi-regular if it admits a GKD(M,N) such that N is finite-dimensional and said is Saphar (resp.essentially Saphar) operator if A is semi-regular (resp. essentially semi regular) operator and has a generalized inverse. For every operator A ∈ L(X), let us define 2 the semi-regular spectrum : σse(A) := {λ ∈ C : λI − A is not semi-regular}, the essentially semi-regular spectrum: σes(A) := {λ ∈ C : λI−A is not essentially semi-regular}, the quasi-Fredholm spectrum: σqf (A) := {λ ∈ C : λI − A is not quasi-Fredholm}, the Kato type spectrum: σk(A) := {λ ∈ C : λI − A is not of Kato type}, the generalized Kato spectrum: σgk(A) := {λ ∈ C : λI − A does not admit a generalized Kato decomposition}, the Saphar spectrum: σsa(A) := {λ ∈ C : λI − A is not Saphar}, and the essentially Saphar spectrum: σesa(A) := {λ ∈ C : λI − A is not essentially Saphar}. Recall that all the seven sets defined above are always a compact subsets of the complex plane, (see [1, 8, 60, 70, 86, 89, 69, 108, 109, 89]) and ordered by :

σgk(A) ⊆ σk(A) ⊆ σes(A) ⊆ σse(A).

We also show in [1] and [60] that the sets σse(A) \ σgk(A), σse(A) \ σk(A), σes(A) \ σk(A), σes(A) \ σgk(A) and σk(A) \ σgk(A) are at most countable. Note that σgk(A) (resp. σk(A)) is not necessarily non-empty. For example, the quasi- nilpotent (resp. nilpotent) operator A has empty generalized Kato spectrum (resp. Kato spectrum).

2 In [98], the semi-regular spectrum σse(A) of A is also called as the regular spectrum and denoted by σg(A).

4 n n+1 0 n+1 n For each n ∈ N, we set cn(A) = dim R(A )/R(A ) and cn(A) = dim N(A )/N(A ). the essential descent and the essential ascent of A ∈ L(X) are

0 de(T ) = inf{n ∈ N : cn(T ) < ∞} and ae(T ) = inf{n ∈ N : cn(T ) < ∞}. (the infimum of an empty set is defined to be ∞). Now, we continue to introduce the following spectra which were discussed in [11, 12, 49, 51, 62, 69, 78]: The descent spectrum: σd(A) = {λ ∈ C : d(λI − A) = ∞}, e The essential descent spectrum: σd(A) = {λ ∈ C : de(λI − A) = ∞}, a(A)+1 The ascent spectrum : σa(A) = {λ ∈ C : a(λI − A) = ∞ or R(A ) is not closed }, e ae(A)+1 The essential ascent spectrum: σa(A) = {λ ∈ C : ae(λI−A) = ∞ or R(A ) is not closed}. For A ∈ L(X), if a(A) < ∞ and R(Aa(A)+1) is closed, then A is said to be left Drazin invertible. If d(A) < ∞ and R(Ad(A)) is closed, then A is said to be right Drazin invertible. If a(A) = d(A) < ∞, then A is said to be Drazin invertible. Clearly, A ∈ L(X) is both left and right Drazin invertible if and only if A is Drazin invertible. If ae(A) < ∞ and ae(A)+1 R(A ) is closed, then A is said to be left essentially Drazin invertible. If de(A) < ∞ and R(Ade(A)) is closed, then A is said to be right essentially Drazin invertible. A is said to be essentially Drazin invertible (resp. semi-essentially Drazin invertible) if A is left essentially Drazin invertible and (resp. or) right essentially Drazin invertible. Now, we can define the left Drazin spectrum, the right Drazin spectrum, the Drazin spectrum, the left essentially Drazin spectrum, right essentially Drazin spectrum, essentially Drazin spectrum and semi-essentially Drazin spectrum the of A respectively, as following:

σLD(A) = {λ ∈ C : λI − A is not a left Drazin invertible operator}, σRD(A) = {λ ∈ C : λI − A is not a right Drazin invertible operator}, σD(A) = {λ ∈ C : λI − A is not a Drazin invertible operator}, e σLD(A) = {λ ∈ C : λI − A is not a left essentially Drazin invertible operator}, e σRD(A) = {λ ∈ C : λI − A is not a right essentially Drazin invertible operator}, e σD(A) = {λ ∈ C : λI − A is not a essentially Drazin invertible operator}, e σSD(A) = {λ ∈ C : λI − A is not a semi-essentially Drazin invertible operator}. These spectra have been extensively studied by several authors, see e.g [6, 18, 37]. n Given n ∈ N, we denote by An the restriction of A ∈ L(X) on the subspace R(A ). According Berkani [18], A is said to be semi B-Fredholm (resp. B-Fredholm, upper semi B-Fredholm, lower semi B-Fredholm), if for some integer n ≥ 0 the range R(An) is closed n and An, viewed as a operator from the space R(A ) in to itself, is a semi-Fredholm operator (resp. Fredholm, upper semi-Fredholm, lower semi-Fredholm). Analogously, A ∈ L(X) is said to be B-Browder (resp. upper semi B-Browder, lower semi B-Browder, B-Weyl, upper semi B-Weyl, lower semi B-Weyl ), if for some integer n ≥ 0 the range R(An) is closed and An is a Browder operator (resp. upper semi-Browder, lower semi-Browder, Weyl, upper semi-weyl, lower semi-Weyl). If A ∈ L(X) is upper or lower semi-B-Weyl (resp. upper or lower semi-B-Browder), then A is called semi-B-Weyl (resp. semi-B-Browder). For A ∈ L(X), let us deﬁne the upper semi-B-Fredholm spectrum, the lower semi-B- Fredholm spectrum, the semi-B-Fredholm spectrum, the B-Fredholm spectrum, the upper semi-B-Weyl spectrum, the lower semi-B-Weyl spectrum, the semi-B-Weyl spectrum, the B- Weyl spectrum, the upper semi-B-Browder spectrum, the lower semi-B-Browder spectrum, the semi-B-Browder spectrum, the B-Browder spectrum, and the quasi-Fredholm spectrum of A as follows respectively: σubf (A) := {λ ∈ C : λI − A is not upper semi B-Fredholm},

5 σlbf (A) := {λ ∈ C : λI − A is not lower semi B-Fredholm}, σsbf (A) := {λ ∈ C : λI − A is not semi B-Fredholm}, σbf (A) := {λ ∈ C : λI − A is not B-Ferdholm}, σubw(A) := {λ ∈ C : λI − A is not upper semi B-Weyl}, σlbw(A) := {λ ∈ C : λI − A is not lower semi B-Weyl}, σsbw(A) = {λ ∈ C : λI − A is not a semi-B-Weyl operator}, σbw(A) := {λ ∈ C : λI − A is not B-Weyl}, σubb(A) := {λ ∈ C : λI − A is not upper semi B-Browder}, σlbb(A) := {λ ∈ C : λI − A is not lower semi B-Browder}, σsbb(A) = {λ ∈ C : λI − A is not a semi-B-Browder operator}, σbb(A) := {λ ∈ C : λI − A is not B-Browder}.

All this spectra are closed and may be empty (see [6, 18, 19, 20, 21, 22, 23, 25, 26, 28, 29, 30]).

For any A ∈ L(X), Berkani have found in [18, Theorem 3.6] , the following elegant equalities: σLD(A) = σubb(A), σRD(A) = σlbb(A); e e σLD(A) = σubf (A), σRD(A) = σlbf (A);

σD(A) = σbb(A). Now, we continue to introduce the following essential spectrum which recently discussed in [18, 61] and originated from the work of Grabiner [50]:

σud(A) = {A ∈ L(X): λI − A does not have eventual topological uniform descent}, where A ∈ L(X) is said to have a topological uniform descent if there exists d ∈ N such that R(A) + N(Ad) is closed and R(A) + N(An) = R(A) + N(Ad) for all n ≥ d. The aim of this thesis is to present a survey of results concerning various types of essential spectra which exists in the form of research papers scattered throughout the literature. To achieve this goal addressed the following tasks:

• Provides definitions of semi-regular and essentially semi-regular operators are described by their properties and relationships to other known classes of operators. Provides definitions of essential spectra: Goldberg, quasi-Fredholm, Kato, B-Fredholm, Fredholm, Weyl and Browder, as well as introduce a definition of the generalized Kato spectrum and significant relationships are established between this essential spectra.

• Identifies conditions and additional constraints under wich the spectrum splits into Fredholm spectra and eigenvalues or to divide into Weyl spectrum and eigenvalues which are both topologically isolated in the spectrum, and geometrically of finite multiplicity, with finite dimensional eigenspaces.

• Proves that both the symmetric diﬀerence, between the closed range spectrum and the Kato spectrum and between the closed range spectrum and the generalize Kato spectrum are at most countable. We study the relationships between the generalized Kato spectrum and some other spectra originated from Fredholm theory and B-Fredholm theory. This study is enriching by the use of the localized single valued property. In

6 pariticular, we show that many spectra coincide in two case. The ﬁrst is the bounded operator A, or its adjoint A∗, or both, admits the SVEP 3. The second is that the approximate spectrum, or the surjective spectrum, coincide with the connected boundary spectrum.

• Give a special attention for the properties of stability of the essential spectra of closed linear operators under (additive) perturbations , such as operators of finite rank, compact operators, small in norm and quasinilpotent operators, because they have different useful applications and which can be used to obtain information on the location in the complex plane of the essential spectrum for large classes of linear operators arising in applications. For example, differential, integral, integro-differential operators, difference, and pseudo-differential operators, in particular operators of all these types on unbounded domains.

The thesis is organized in five chapters. The first chapter discusses the elements of spectral theory of operators. The topics discussed include spectral decomposition theorems, Riesz projections, functional calculus, the singularities of the resolvent and eigenvalues of finite type. The beginning of this chapter contains an important proprieties of some classical algebraic quantities associated with an operator, such as the ascent, the descent, the nullity and the deficiency of an operator. This quantities are the basic bricks in the construction of the most important classes of linear operators. In the second chapter we gives a survey of results concerning various types of essential spectra, Fredholm and Browder operators etc. A section of this chapter is also devoted to study of some perturbation ideals which occur in Fredholm theory. In particular we study the ideal of inessential operators, the ideal of strictly singular operators and the ideal of strictly cosingular operators, which are a generalization of the class of the compact perturbations. The third chapter provides definitions of semi-regular and essentially semi-regular operators are described by their properties and relationships to other known classes of operators, as well as introduce a various spectra which involves the concept of semiregularity: the semi- regular spectrum and its essential version, the quasi-Fredholm spectrum, the Kato spectrum, the generalized Kato spectrum, Saphar spectrum and essentially Saphar spectrum. In particular we study the generalized spectrum which is characterized by the generalized Kato decomposition. This decomposition first appeared in the classic work of Kato [63] perturbation of linear operators, and its has greatly benefited from the work of many authors in the last years, in particular from the work of Mbekhta [81, 83, 85], Aiena [1] and Q. Jiang-H. Zhong [60]. The operators which satisfy this property form a class which includes the class of quasi-Fredholm, semi-regular, Kato type, semi-Fredholm and B-Fredholm operators. This concept leads in a natural way to the generalized Kato spectrum σgk(A), an important subset of the ordinary spectrum which is defined as the set of all λ ∈ C for which λI − A does not admit a generalized Kato decomposition. It is shown in [71], in the Hilbert space case, that the symmetric difference between the essential Goldberg spectrum defined in [45] and the essential quasi-Fredholm spectrum is at most countable, which is of course, in this case, a quasi-Fredholm operators equivalent to A is of Kato type, but in the case of Banach spaces the Kato type operator is also quasi-Fredholm, the inverse is not true. We generalize this results for the Kato spectrum and the generalized Kato spectrum.

The next chapter deals with spectral theory, we focus on the study of several spectra that originating from B-Fredholm theory. We shall also introduce some special classes of

3Abbreviation of the Single-Valued Extension Property

7 operators having nice spectral properties. These operators include those for which Brow- der’s theorem and Weyl’s theorem hold. We also consider some variations of both theorems and the corresponding perturbation theory. In this chapter we also introduce the elegant interaction between the localized SVEP and Fredholm theory. This interaction is studied in the more general context of operators admits a generalized Kato decomposition. We give a summary of the Weyl-Browder type theorems and properties, in their classical and more recently in their generalized form.

The last chapter is devoted to the investigation of some essential spectra of the one- dimensional transport operator with abstract boundary conditions on Lp-spaces, with p ∈ [1, ∞). More precisely, we consider the unbounded operator

Z 1 ∂ψ 0 0 0 AH ψ(x, ξ) = −ξ (x, ξ) − σ(ξ)ψ(x, ξ) + κ(x, ξ, ξ )ψ(x, ξ ) dξ ∂x −1

= TH ψ(x, ξ) + Kψ(x, ξ) with the boundary conditions ψi = H(ψo) where H is bounded linear operator deﬁned on suitable boundary spaces and σ(.) ∈ L∞(−1, 1). Here x ∈ (−a, a) and ξ ∈ (−1, 1) and ψ(x, ξ) represents the angular density of particles (for instance gas molecules, photons, or neutrons) in a homogeneous slab of thickness 2a. The functions σ(.) and κ(., ., .) are called, respectively, the collision frequency and the scattering kernel. Our analysis is based essentially on the perturbations results and the knowledge of the essential spectra of T0 where T0 (i.e., H = 0) denotes the streaming operator with vacuum boundary conditions. We prove that, if the classes of boundary and collision operators operators is in appropriate class of Fredholm perturbation, then the Fredholm spectra of the operators T0 and AH coincide. Also, we apply the same technic to study the essential spectra of the following singular neutron transport operator

Z 0 0 0 Aψ(x, ξ) = −ξ · ∇xψ(x, ξ) − σ(ξ)ψ(x, ξ) + κ(x, ξ )ψ(x, ξ ) dξ (x, ξ) ∈ Ω × V, Rn with vacuum boundary conditions, i.e., φ|Γ− (x, ξ) = 0

Γ− = {(x, ξ) ∈ ∂Ω × V ; ξ · n(x) < 0} where n(x) stands for the outward normal unit at x ∈ ∂Ω. Here Ω is an open bounded subset of Rn and dµ(.) is a positive Radon measure on Rn. All chapters (except the first and the last chapter ) are concluded by a table where we give further information and discuss some of the more recent developments in the theory previously developed. In general, all the results established in these final sections are presented without proofs. However, we always give appropriate references to the original sources, where the reader can find the relative details.

8 Chapter 1

Spectrum of an operator

In this chapter we recall the basic properties of the spectrum of a bounded and unbounded linear operator in a Banach spaces. Let us start by setting the stage, introducing the basic notions necessary to study linear operators. Through this monograph, an operator means a linear transformation defined on Banach space. Although many of the results in these monograph are valid for real Banach spaces, we always assume that all Banach spaces are complex infinite-dimensional Banach spaces. First we present some classical quantities associated with an operator. These quantities, such as the ascent, and the descent of an operator are defined in the first section and are the basic bricks in the construction of one of the most important branches of spectral theory, the theory of Fredholm operators.

1.1 Algebraic Properties

Let X and Y be two vector spaces over the real or complex numbers (the scalars) and L(X,Y ) the set of all linear operators from X into Y , if X = Y we put L(X) = L(X,X) For A ∈ L(X,Y ) we denote by D(A) ⊆ X its domain, N(A) = {x ∈ D(A), Ax = 0} its kernel and R(A) = {Ax, x ∈ D(A)} its range. For all n ∈ N domain, kernel and the range of power operator An are deﬁned by : If n ≥ 1: D(An) := {x ∈ D(A): Akx ∈ D(A), k = 1, . . . , n − 1},

N(An) := {x ∈ D(An): Anx = 0} and R(An) := {Anx : x ∈ D(An)} If n = 0: A0 = I,D(A0) = X,N(A0) = {0}. In general, we have D(An) ⊆ D(An−1), the inclusion my be proper. Following deﬁnitions and well know results are relevant to our context, see [1, 49, 50, 51, 62, 63, 78, 112] .

1.1.1 Ascent and descent of an operator The kernels and the ranges of the iterates of a linear operator A, deﬁned on a vector space X, form the following two increasing and decreasing chains (sequences of subspaces), respectively: N(A0) = {0} ⊆ N(A) ⊆ N(A2) ⊆ ...

9 and R(A0) = Y ⊇ R(A) ⊇ R(A2) ⊇ ... Generally all these inclusions are strict. In this subsection we shall consider operators for which one, or both, of these chains becomes constant at some n ∈ N. Deﬁnition 1.1 Let A ∈ L(X) . The ascent of A, is the smallest positive integer p = a(A) such that

N(Ap) = N(Ap+1).

If there is no such integer we set a(A) = ∞. The descent of A is the smallest positive integer q = d(A) such that

R(Aq) = R(Aq+1).

If such an integer does not exist, we put d(A) = ∞.

Clearly, a(A) = 0 if and only if A is injective and d(A) = 0 if and only if A is surjective.

Lemma 1.1 Let A ∈ L(X). For every m ∈ N we have 1. a(A) ≤ m < ∞ if and only if R(Am) ∩ N(An) = {0} for some ( equivalently, for all ) n ≥ 1.

2. d(A) ≤ m < ∞ if and only if R(An) + N(Am) = X for some ( equivalently, for all ) n ≥ 1.

3. If both a(A) and d(A) are ﬁnite then a(A) = d(A).

Proof. Recall that, Whenever A and B are linear operators on a vector space, we have

A(N(BA)) = R(A) ∩ N(B), (1.1)

A−1(R(AB)) = R(B) + N(A), (1.2) By (1.1), we get

m n+m m n A (N(A )) = R(A ) ∩ N(A ), for all n, m ∈ N, (1.3) and since also A−m(R(Am) ∩ N(An)) = N(Am+n), we see that N(Am) = N(An+m) if and only if R(Am)∩N(An) = {0}. This proves (1). Similarly, A−m(R(Am+n)) = R(An)+N(Am) and Am(R(An) + N(Am)) = R(Am+n) imply (2).

For (3), set p = a(A) et q = d(A). Assume ﬁrst that p ≤ q with q > 0, so that R(Aq) ⊆ R(Ap). By (2) we have X = N(Aq) + R(Aq), so every element y = Aqx ∈ R(Aq) admits the decomposition y = z + Aqw with z ∈ N(Aq). From z = y − Aqw ∈ R(Aq), we then obtain that z ∈ N(Aq) ∩ R(Aq), and hence the last intersection is {0}. Therefore z = 0 and y = Aqw ∈ R(Aq)and this shows R(Aq) = R(Ap), from whence we obtain p ≥ q, so that p = q. Assume now that q ≤ p and p > 0, so N(Aq) ⊆ N(Ap). Again from (2) we have X = N(Aq) + R(Ap), then if x ∈ N(Ap), there exists u ∈ N(Aq) such that x = u + Apv, since Apx = Apu = 0, then A2pv = 0, hence v ∈ N(A2p) = N(Ap) and Apv = 0, consequently x = u ∈ N(Aq). This shows that N(Aq) = N(Ap) and q ≥ p, we conclude p = q.

10 Lemma 1.2 For a linear operator A on a vector space X the following statements are equivalent:

1. N(A) ⊆ R(Am), for all m ∈ N.

2. N(An) ⊆ R(A), for all n ∈ N.

3. N(An) ⊆ R(Am), for all n, m ∈ N.

4. N(An) = Am(N(An+m)), for all n, m ∈ N. Proof. 1. ⇒ 2. If we apply the inclusion in 1. to the operator An, we obtain N(An) ⊆ R(An+m) ⊆ R(A). 2. ⇒ 3. We apply the inclusion in 2. to the operator Am and the fact that N(An) ⊆ N(An+m), we obtain 3. 3. ⇒ 4. Follows from (1.3). 4. ⇒ 1. For n = 1 we have N(A) = Am(N(A1+m)) ⊆ R(Am).

Given n ∈ N, we denote by An = A|R(An) the restriction of A ∈ L(X) on the subspace R(An). Then

n+1 n N(An+1) = N(A) ∩ R(A ) ⊆ N(A) ∩ R(A ) = N(An) for all n ∈ N, (1.4) and m m+n n R(An ) = R(A ) = R(Am) for all m, n ∈ N, (1.5)

Lemma 1.3 Let A be a linear operator on a vector space X. Then the following statements are equivalent: (i) a(A) < ∞;

(ii) there exists k ∈ N such that Ak is injective;

(iii) there exists k ∈ N such that a(Ak) < ∞. p Proof. (i) ⇔ (ii). If p = a(A) < ∞, by Lemma 1.1, then N(Ap) = R(A ) ∩ N(A) = {0}. k+1 k Conversely , suppose that N(Ak) = {0}, for some k ∈ N. If x ∈ N(A ) then A(A x) = 0, so k k A x ∈ N(A) ∩ R(A ) = N(Ak) = {0}. Hence x ∈ N(Ak). This shows that N(Ak+1) ⊆ N(Ak), thus N(Ak+1) = N(Ak) and consequently a(A) ≤ k. (ii) ⇔ (iii). The implication (ii) ⇒ (iii) is obvious. To show the opposite implication, suppose m = a(Ak) < ∞. By Lemma 1.1, and equality 1.5 we have

m k m m {0} = N(Ak) ∩ R(Ak ) = (N(A) ∩ R(A )) ∩ R(Ak ) = N(A) ∩ R(Ak ) m+k = N(A) ∩ R(A ) = N(Am+k)

so that the equivalence is proved.

A similar result holds for the descent:

11 Lemma 1.4 Let A be a linear operator on a vector space X. Then the following statements are equivalent:

(i) d(A) < ∞;

(ii) there exists k ∈ N such that Ak is onto;

(iii) there exists k ∈ N such that d(Ak) < ∞. Proof. (i) ⇔ (ii). Suppose that q = d(A) < ∞, then

q q+1 q R(A ) = R(A ) = A(R(A )) = R(Aq).

Hence Aq is onto. Conversely , if Ak is onto for some k ∈ N, then

k+1 k k R(A ) = A(R(A )) = R(Ak) = R(A ),

thus d(A) ≤ k. The implication (ii) ⇒ (iii) is obvious. we show the opposite implication, let m = m m+1 m+k m+k+1 d(Ak) < ∞ for some k ∈ N. Then R(Ak ) = R(Ak ), so R(A ) = R(A ), hence d(A) ≤ k + m.

As observed in the proof of Lemma 1.3 if p = a(A) < ∞ then N(Ap) = {0} and from inclusion (1.4) it is obvious that N(Aj) = {0} for all j ≥ p. Conversely, if N(Ak) = {0} for some k ∈ N then a(A) < ∞ and a(A) ≤ k. Hence, if a(A) < ∞ then

a(A) = inf{k ∈ N : Ak is injective }.

Analogously, if q = d(A) < ∞ then Aj is onto for all j ≥ q. Conversely, if Ak is onto for some k ∈ N then d(A) ≤ k, so that

d(A) = inf{k ∈ N : Ak is onto }. The ﬁniteness of the ascent and the descent of a linear operator A is related to a certain decomposition of X. This follows by combining Lemma 1.1, Lemma 1.3 and Lemma 1.4,

Theorem 1.1 Suppose that A ∈ L(X). If both a(A) and d(A) are ﬁnite then a(A) = d(A) = m, and we have the decomposition X = R(Am) ⊕ N(Am). Conversely, if for a natural number m we have the decomposition X = R(Am)⊕N(Am) then a(A) = d(A) ≤ m. In this case Am is bijective. Now we give examples of descent and ascent of operators deﬁned on `p ( 1 ≤ p ≤ ∞), the Banach space of of all p-summable sequences (bounded sequences for p = ∞) of complex numbers under the the stander p-norm on it.

Example 1.1 let A be deﬁned by A(x) = y, where x = (xn)n and y = (yn)n are related by xn+1 if n is even yn = xn if n is odd

2 p Then N(A) = N(A ) = {(xn)n ∈ ` : x2n = 0 for each n ∈ N}. Hence a(A) = 1. Also 2 p R(A) = R(A ) = {(yn)n ∈ ` : y2n+1 = y2n for each n ∈ N}. Threfore d(A) = 1.

12 Example 1.2 let B be deﬁned by

Bx = (x1, x2, x3,... )

n As B is surjective, so d(B) = 0. Further we note that for each n ≥ 1, en ∈ N(B ) but n−1 j en ∈/ N(B ). Note that the sequence en = δn. Hence a(B) = ∞.

Example 1.3 let C be deﬁned by C(x) = y, where x = (xn)n and y = (yn)n are related by x0 if n = 0, 1 yn = xn if n ≥ 1

n p C is injective . Hence a(C) = 0. Further for each n ≥ 1, R(C ) = {(yn)n ∈ ` : y0 = y1 = n n+1 ... = yn}. Thus R(C ) 6= R(C ). Hence d(C) = ∞.

Example 1.4 Consider the operator D be deﬁned by D(x) = y, where x = (xn)n and y = (yn)n are related by   xn+2 if n is odd yn = x0 if n = 0, 2  xn−2 if n is even and n ≥ 4

n+1 n+2 Then for each n ∈ N, e2n+1 ∈ N(D ) but e2n+1 ∈/ N(D ). Hence a(D) = ∞. Further p 2 p we note that R(D) = {(yn)n ∈ ` : y0 = y2}, R(D ) = {(yn)n ∈ ` : y0 = y2 = y4} and so on. Thus R(Dn) 6= R(Dn+1) for each n ≥ 1. Hence d(D) = ∞.

1.1.2 The nullity and the deﬁciency of an operator Let A an operator on a vector space X. The nullity of A is the positive integer

α(A) = dim N(A).

The deﬁciency of A is the positive integer

β(A) = codimR(A).

Let ∆(X) denote the set of all linear operators on vector space X for which α(A) and β(A) are both ﬁnite. The index of A ∈ ∆(X) is the integer

ind(A) = α(A) − β(A)

By the index theorem we have

ind(AB) = ind(A) + ind(B), ∀A, B ∈ ∆(X)

In the next theorem we establish the basic relationships between the quantities α(A), β(A), a(A) and d(A).

Theorem 1.2 If A is a linear operator on a vector space X then the following properties hold:

1. If a(A) < ∞ then α(A) ≤ β(A).

13 2. If d(A) < ∞ then β(A) ≤ α(A).

3. If a(A) = d(A) < ∞ then α(A) = β(A).

4. If α(A) = β(A) < ∞ and if either a(A) < ∞ or d(A) < ∞ then a(A) = d(A).

Proof. 1. Let p = a(A) < ∞, if β(A) = ∞ the inequality is obvious. suppose that β(A) < ∞, so β(An) < ∞, from Lemma 1.1 we have N(A) ∩ R(Ap) = {0} and this implies that α(A) < ∞, and for all n ≥ p we have

nind(A) = ind(An) = α(Ap) − β(An)

Now assume that q = d(A) < ∞. For all integers n ≥ max{p, q} the quantity nind(A) = α(Ap)−β(Aq) is then constant, so that ind(A) = 0 and α(A) = β(A). Consider the other case q = ∞, then β(An) → ∞ as n → ∞, so nind(A) eventually becomes negative, α(A) < β(A).

2. Set q = d(A) < ∞. Also here we can assume that α(A) < ∞, so β(An) < ∞ and by Lemme 1.1 we have X = Y ⊕ R(A) with Y ⊆ N(Aq). From this it follows that β(A) = dim Y ≤ α(Aq) < ∞ and β(A) < ∞. For all n ≥ q we have

nind(A) = ind(An) = α(Ap) − β(Aq)

If p = a(A) < ∞, then for all n ≥ max{p, q} the quantity nind(A) = α(Ap) − β(Aq) is constant, hence ind(A) = 0 and α(A) = β(A) = 0. Now if p = ∞, then α(An) → ∞ as n → ∞, and nind(A) > 0, so β(A) < α(A).

3. Consequence of (1) and (2).

4. This is an immediate consequence of the equality α(An) − β(An) = ind(An) = nind(A) = 0, for all n ∈ N.

The hyper-kernel of A is the subspace [ N ∞(A) = N(An). n∈N The hyper-range of A is the subspace \ R∞(A) = R(An). n∈N Both of N ∞(A) and R∞(A) are A-invariant subspaces, but Generally are not closed.

Corollary 1.1 The statements of lemma 1.2 equivalent to each of the following inclusions:

1. N(A) ⊆ R∞(A).

2. N ∞(A) ⊆ R(A).

3. N ∞(A) ⊆ R∞(A).

Theorem 1.3 Let A ∈ L(X). If one of the following conditions holds:

1. α(A) < ∞

14 2. β(A) < ∞

3. N(A) ⊆ R(An), for all n ∈ N. Then there exists m ∈ N such that

m m+k N(A) ∩ R(A ) = N(A) ∩ R(A ), for all k ∈ N. (1.6)

Proof. 1. If α(A) < ∞ or N(A) ⊆ R(An), for all n ∈ N, then the relation (1.6) is obvious. Now, suppose that X = F ⊕ R(A) with dim F < ∞. Let Dn = N(A) ∩ R(A), we have Dn+1 ⊆ Dn, foe every n ∈ N. Suppose that there exist k ∈ N such that Di 6= Di+1, i i = 1 . . . k, Then for every one of these i , we can ﬁnd an element wi such that A wi ∈ Di i and A wi ∈/ Di+1. By means of the decomposition of X, we also ﬁnd ui ∈ F and vi ∈ R(A) such that wi = ui +vi. We claim that the vectors u1, . . . , uk are linearly independent. Indeed, Pk if i=1 λiui = 0, then k k X X λiwi = λivi = 0 i=1 i=1 k k and therefore from the equalities A w1 = ... = A wk−1 = 0, we deduce that

k k k X k k X k+1 A ( λiwi) = λkA wk = A ( λivi) ∈ R(A ) i=1 i=1

k k k From A wk ∈ N(A), we obtain λkA wk ∈ Dk+1, but λkA ∈/ Dk+1, this is possible only if λk = 0. Analogously we have λk−1 = ... = λ1 = 0, so the vectors u1, . . . , uk are linearly independent and we deduce that k ≤ dimF , But then for a suﬃciently large m we obtain the equality (1.6).

Lemma 1.5 Let A ∈ L(X). If there exists m ∈ N such that (1.6) holds, then N(An) ∩ R(Am) ⊆ R∞(A), for all n ≥ 1, (1.7) and A(R∞(A)) = R∞(A). Proof. To prove (1.7), we proceed by induction, the hypotheses of lemma implies that

N(An) ∩ R(Am) ⊆ R(Am+k), for all k.

n m m i n m ∞ i On other hand N(A )∩R(A ) ⊆ ∩i=0R(A ), hence N(A )∩R(A ) ⊆ ∩i=0R(A ), this proved the case n = 1. Now assume that the equality 1.7 is vitriﬁed for n. Let x ∈ N(An+1)∩R(Am) et k ≥ m, then

x ∈ N(An+1) ∩ R(Am) ⇒ x ∈ N(An+1) and x ∈ R(Am) ⇒ Ax ∈ N(An) and Amy = x, y ∈ X ⇒ Ax ∈ N(An) and Am+1y = Ax, y ∈ X ⇒ Ax ∈ N(An) ∩ R(Am)

and by the hypotheses of induction we have N(An) ∩ R(Am) ⊆ R(Ak+1), hence Ax = Ak+1y, y ∈ X and x − Aky ∈ N(A), so x = Aky + u, u ∈ N(A), since k ≥ m then u ∈ R(Am), so x ∈ R(Ak) + (R(Am) ∩ N(A)) ⊂ R(Ak). Hence N(An+1) ∩ R(Am) ⊂ R(Ak), for all k ≥ m. This proves (1.7) The fact that R∞(A) is invariant by A, then the proof is done if we show that R∞(A)) ⊆

15 ∞ ∞ n A(R (A)). Let y ∈ R (A) , then y ∈ R(A ), for every n ∈ N, so there exists xk ∈ X such m+k that y = A xk, for every k ∈ N. If we set

m m+k−1 zk = A x1 − A x, k ∈ N.

m m+1 m+k Then zk ∈ R(A ) and since Azk = A x1 − A xk = y − y = 0, we also have zk ∈ N(A), m m+k m+k−1 thus zk ∈ N(A) ∩ R(A ) and since N(A) ∩ R(A ) ⊆ N(A) ∩ R(A ) we deduce that m+k−1 zkR(A ). This implies that

m m+k−1 m+k−1 y = A x1 = zk + A xk ∈ R(A ), for all k ∈ N, and therefore y ∈ R∞(A), we may conclude that R∞(A) ⊆ A(R∞(A)).

Lemma 1.6 Let A ∈ L(X) and λ, µ ∈ C. We have

i. R((λI − A)n) + R(Am) = X, for all n, m ∈ N and λ 6= 0.

ii. (λI − A)(N(An)) = N(An), for all n ∈ N and λ 6= 0.

iii. N((λI − A)n) ⊆ R((µI − A)n), for all n ∈ N and λ 6= µ.

Proof. i. Consider also the polynomials p(z) = (λ − z)n and q(z) = zm. Since p and q have no common divisors then there exist two polynomials u and v such that 1 = p(z)u(z) + q(z)v(z) for all z ∈ C. Hence I = (λI−A)nu(A)+Amv(A) and so X = R((λI−A)n)+R(Am). ii. By the same argument in i. with n = 1 and n = m, we obtain I = (λI −A)u(A)+Anv(A). If x ∈ N(An) and since p(A)x ∈ N(An) this implies N(An) ⊆ (λI−A)(N(An)). The converse inclusion is obvious. iii. By assumption µ − λ 6= 0, so by part ii. we obtain that

(µI − A)(N((λI − A)n)) = ((µ − λ)I + λI − A)(N((λI − A)n)) = N((λI − A)n).

From this it follows that

n n (µI − A)(N((λI − A) )) = N((λI − A) ), for all n ∈ N, and consequently N((λI − A)n) ⊆ R((µI − A)n).

Corollary 1.2 Let A ∈ L(X) and Am = A|R(Am). For all λ ∈ C and λ 6= 0, we have

n n 1. β((λI − A) ) = β((λI − Am) ), for all n ∈ N.

n n 2. α((λI − Am) ) ≤ α((λI − A) ), for all n ∈ N. Proof. 1. by part i. of Lemma 1.6, we have

β((λI − A)n) = dim(X/R(λI − A)n) = dim(R((λI − A)n) + R(Am)/R(λI − A)n) = dim(R(Am)/R(Am) ∩ R((λI − A)n)) n = β((λI − Am) ).

2. Follows from N((λI − Am)) ⊆ N((λI − A)).

16 1.2 Generalities about Closed operators

If X,Y are Banach spaces, we says that an operator A from X into Y is bounded (or continuous) if there is a constant c ≥ 0 such that

kAxk ≤ c kxk for all x ∈ X

We denote the Banach space of all bounded linear operators from X into Y by L(X,Y ), L(X,X) is also denoted L(X). Recall that if A ∈ L(X,Y ), the norm of A is deﬁned by

||Ax|| ||A|| := sup . x6=0 ||x||

(unless further speciﬁcation is necessary, k.k will always denote the norm in an appropriate space). For linear operators the concepts of continuity at a point, uniform continuity and boundedness coincide. But when one deals with diﬀerential operators, one discovers the need to consider also unbounded linear operators. For example,

2 Example 1.5 The differential operator of first order A = i∂x on L ([0, 1]). It readily seen to 2 inx be unbounded since one can find a sequence of functions ϕn ∈ L ([0, 1]), given by ϕn = e for n ∈ N, satisfying kϕnkL2([0,1]) = 1 and kAϕnkL2([0,1]) = n −→ ∞ as n −→ ∞. We shall adopt the following definition of (possibly unbounded) operators.

Deﬁnition 1.2 An unbounded linear operator from X into Y is a linear map A : D(A) ⊂ X −→ Y, deﬁned on a linear subspace D(A). The set D(A) is called the domain of A.

Of special interest are the operators with dense domain in X (i.e. D(A) = X), where M denote, in the sequel, the closure of subset M of X . When A is bounded and densely deﬁned, it extends by continuity to an operator in L(X,Y ), but when it is not bounded, there is no such extension. For such operators, another property of interest is the property of being closed:

Deﬁnition 1.3 The operator A with the domain D(A) is called a closed operator if and only if for any sequence (xn)n ⊂ D(A) such that xn −→ x ∈ X and Axn −→ y ∈ Y it follows that x ∈ D(A) and Ax = y.

Deﬁnition 1.4 Let A : X −→ Y be a linear operator with the domain D(A). The graph of A is the linear subspace of X × Y deﬁned by

G(A) = {(x, Ax); x ∈ D(A)} .

The graph norm of A is deﬁned by

kxkA = kxk + kAxk .

We write XA if D(A) is equipped with the graph norm. Clearly XA is a normed vector space and A ∈ L(XA,Y ). The graph norm on D(A) is clearly stronger than the X-norm on D(A); the norms are equivalent if and only if A is a bounded operator. An equivalent deﬁnition of Deﬁnition 1.3 is given by

Proposition 1.1 ([64]) Let A : X −→ Y be a linear operator with the domain D(A). Then the following assertions are equivalent

17 1. A is a closed operator;

2. The graph G(A) of A is closed in X × Y .

3. XA is a Banach space.

Remark 1.1 1. If D(A) = X, then A is closed if and only if A is bounded, by the closed Graph Theorem.

2. The inverse of a closed injective operator is closed.

3. The continuity of A does not necessarily imply that A is closed. Conversely, A closed does not necessarily imply that A is continuous.

4. If A is closed, then N(A) is closed; however, R(A) need not be closed.

The natural operations sum, product and limits are well deﬁned on L(X,Y ). This is thanks to the domain of the bounded operators which is always taken to be the whole Banach space X. However, one has to be careful with those manipulations when dealing with unbounded operators, this is essentially due to the domains. If A : D(A) ⊂ X → Y and B : D(B) ⊂ X → Y , their sum A + B is deﬁned by

(A + B)x = Ax + Bx for all x ∈ D(A + B) = D(A) ∩ D(B),.

and when C is an operator from Y to Z with domain D(C), the product (or composition) CA is deﬁned by

(CA)x = C(Ax) for all x ∈ D(CA) = {x ∈ D(A): Ax ∈ D(C)}.

Note that the operators A + B and CB can just not make any sense because D(A + B) or D(CA) may become trivial, i.e it reduces to zero even if strong conditions are imposed on A, B and C. In practice, most unbounded operators are closed and are densely deﬁned. In the sequel we denote by C(X,Y ) the set of all closed, densely deﬁned linear operators from X into Y . If X = Y we write C(X) = C(X,X).

1.2.1 Closable operators When A and B are operators from X to Y and D(B) ⊂ D(A) with Bx = Ax for x ∈ D(B), we say that A is an extension of B and B is a restriction of A, and we write B ⊂ A. Equivalently, B ⊂ A if and only if G(A) ⊂ G(B). One often wants to know whether a given operator A has a closed extension. If A is bounded, this always holds, since we can simply take the operator A with graph G(A); here G(A) is a graph since xn −→ 0 ∈ X implies Axn −→ 0 ∈ Y . But when A is unbounded, one cannot be certain that it has a closed extension. But if A has a closed extension A1, then G(A1) is a closed subspace of X × Y containing G(A), hence also containing G(A). In that case G(A) is a graph. It is in fact the graph of the smallest closed extension of A, we call it the closure of A and denote it A. (Observe that when A is unbounded, then D(A) is a proper subset of D(A).)

Deﬁnition 1.5 An operator A is called closable if it has a closed extension. The smallest closed extension of A whose graph equals G(A) is denoted by A and called the closure of A. Every closable operator has a closure.

18 Proposition 1.2 ([64]) Let A : X −→ Y be an operator. The following conditions are equivalent: 1. A is closable.

2. G(A) is a graph of an operator.

3. If (0, y) ∈ G(A) then y = 0.

4. If for any sequence (xn)n ⊂ D(A) such that xn −→ 0 ∈ X and Axn −→ y ∈ Y implies y = 0.

Concerning closures of products of operators, we have Theorem 1.4 Let A be closed (resp. closable) operator from X to Y and B ∈ L(Z,X). Then the operator AB is closed (resp. closable) with D(AB) = {x ∈ D(B): Ax ∈ D(A)}.

Proof. Let zn ∈ D(AB), n ∈ N, and z ∈ Z, y ∈ Y such that zn → z in Z and ABzn → y in Y as n → ∞. Since B is bounded, then xn = Bzn converges to Bz and so Axn → y as n → ∞. Since A is closed, we deduce that Bz ∈ D(A) and ABz = y. Now if A is closable then AB has a closed extension AB.

Note that the product of two closed operator need not be closed operator. Example 1.6 Let X = C([0, 1]), A = f 0 with D(A) = C1([0, 1]) and ϕ ∈ C([0, 1]) such that ϕ = 0 on [0,1/2]. Deﬁne B ∈ L(X) by Bf = ϕf for all f ∈ X. Then the operator BA with D(BA) = D(A) is not closed. To see this, take functions fn ∈ D(A) such that fn = 1 on 1 0 [1/2; 1] and fn → f in X with f∈ / C ([0, 1]). Then, BAfn = ϕfn = 0 converges to 0, but f∈ / D(A).

1.2.2 Adjoint operator.

Recall that when X is a Banach space, the dual space X∗ := L(X, C), consists of the bounded linear functionals x∗ on X; it is a Banach space with the norm

∗ kxkX∗ = inf{|x (x)| : x ∈ X, kxk = 1}

When A : X −→ Y is densely deﬁned, we can deﬁne the adjoint operator A∗ : Y ∗ −→ X∗ as follows: The domain D(A∗) consists of the y∗ ∈ Y ∗. for which the linear functional

x 7−→ y∗(Ax), x ∈ D(A) (1.8) is continuous (from X to C). This means that there is a constant c (depending on y∗ ) such that ∗ |y (Ax)| ≤ ckxkX , for all x ∈ D(A) Since D(A) is dense in X, the mapping extends by continuity to X, so there is a uniquely determined x∗ ∈ X∗ so that

y∗(Ax) = x∗(x), for x ∈ D(A) (1.9)

Since x∗ is determined from y∗, we can deﬁne the operator A∗ from Y ∗ to X∗ by:

A∗y∗ = x∗, for y∗ ∈ D(A∗) (1.10)

19 Theorem 1.5 ([64]) Let A ∈ C(X,Y ). Then there is an adjoint operator A∗ : Y ∗ −→ X∗, uniquely deﬁned by (1.8)-(1.10). Moreover, A∗ is closed.

If A is a bounded operator then A∗ is also a bounded operator from Y ∗ into X∗ and, moreover, kA∗k = kAk.

For nonempty sets M ⊆ X and N ⊆ X∗ we deﬁne the annihilators

M ⊥ = {f ∈ X∗ : f(x) = 0 for all x ∈ M}

N ⊥ = {x ∈ X : f(x) = 0 for all x ∈ N}. Even if M and N are not subspaces, and M ⊥ and M ⊥ are closed subspaces of X∗ and X respectively. We have M ⊥ = X∗ (resp. N ⊥ = X) if and only if M = {0} (resp. N = {0}). Proposition 1.3 ([45]) Let A ∈ C(X,Y ). Then we have

N(A) = R(A∗)⊥,N(A∗) = R(A)⊥, R(A) = N(A∗)⊥ and R(A∗) ⊂ N(A)⊥.

Recall that if X and Y are Hilbert spaces with scalar product h., .i. We can deﬁne the adjoint of A ∈ C(X,Y ) as follows

D(A∗) = {y ∈ Y : ∃z ∈ Y ∀x ∈ D(A): hAx, yi = hx, zi},A∗y = z.

Moreover, if A is a bounded, we have

for all x ∈ X, y ∈ Y : hAx, yi = hx, A∗yi .

Theorem 1.6 Suppose that H is a Hilbert space and A is a densely deﬁned operator from H to itself. Then A is closable if and only if A∗ is densely deﬁned. In this case, A = A∗∗.

Deﬁnition 1.6 Let H is a Hilbert space and A is a densely deﬁned operator from H to itself. • If A∗ is an extension of A, that is, A ⊂ A∗, then A is called a symmetric operator.

• If A = A∗, then A is called a self-adjoint operator.

• If A = A∗, then A is called essentially self-adjoint.

Remark 1.2 • If A is a symmetric operator, then for any x, y ∈ D(A)

hAx, yi = hx, Ayi .

If A is a symmetric bounded operator, then A is self-adjoint.

• If A is a symmetric operator, then D(A) ⊂ D(A∗). So A∗ is densely deﬁned. By Theorem 1.6, A is closable and A = A∗∗ ⊂ A∗. Thus if A is essentially self-adjoint, then A = A∗∗ = A∗.

• Let A be a self-adjoint operator and B be a symmetric operator such that A ⊂ B, then A = B . This is because A ⊂ B ⊂ B∗ ⊂ A∗ = A. We see that a symmetric operator can have diﬀerent self-adjoint extension.

20 1.3 Operators with closed range

The main result concerning operators with closed range is the following. Theorem 1.7 ([45]) Let A ∈ C(X,Y ). The following properties are equivalent: 1. R(A) is closed,

2. R(A∗) is closed,

3. R(A) = N(A∗)⊥,

4. R(A∗) = N(A)⊥ The property of R(A) being closed may be characterized by means of a suitable number associated with A. Deﬁnition 1.7 the reduced minimal modulus of A ∈ C(X) is deﬁned to be kAxk γ(A) = inf x/∈N(A) dist(x, N(A))

where dist(x, N(A)) = infy∈N(A) kx − yk. If A = 0 then we take γ(A) = ∞. Note that ( see [64]): γ(A) > 0 ⇔ R(A) is closed

Proposition 1.4 ([64]) Let A ∈ C(X) with closed range and Y a subspace of X (not necessarily closed). If Y + N(A) is closed then A(Y ) is closed.

Proof. Let us denote by xb the equivalence class x + N(A) in the quotient space X/N(A) and by Ab : X/N(A) → X the canonical injection deﬁned by Ab(xb) = Ax, where x ∈ xb. Since −1 A(X) is closed Ab has a bounded inverse Ab : R(A) → X/N(A). Let Yb = yb : y ∈ Y . Clearly A(Y ) = Ab(Yb) is the inverse image of Yb under the continuous map Ab−1, so A(Y ) is closed if Yb is closed . It remains to show that Yb is closed if Y + N(A) is closed. Suppose that the sequence (xcn) of Yb converges to xb ∈ X/N(A) . This implies that there exists a sequence (xn) with xn ∈ xcn such that dist(xn − x, N(A)) converges to zero, and so there exists a sequence (zn) ⊂ N(A) such that xn − x − zn → 0. Then the sequence (xn − zn) ⊂ Y + N(A) converges to x and since by assumption Y + N(A) is closed, we have x ∈ Y + N(A). This implies xb ∈ Yb; thus Yb is closed .

Theorem 1.8 ([45]) Let A ∈ C(X,Y ). If there is a closed subspace Y0 of Y for which R(A) ⊕ Y0 is closed, then A has closed range. In particular, if R(A) is complemented in Y or β(A) < ∞ then R(A) is closed. To see the importance of Theorem 1.8 note that for a subspace M of a Banach space Y ,

Y = M ⊕ Y0 does not imply that M is closed.

Take a non-continuous linear functional f on Y and put M = N(f). Then there exists a one-dimensional subspace Y0 such that Y = M ⊕Y0 (recall that Y/N(f) is one-dimensional). But M = N(f) cannot be closed because f is continuous if and only if f −1(0) is closed.

21 Consequently, we don’t guarantee that

dim(Y/M) < ∞ ⇒ M is closed. (1.11)

However Theorem 1.8 asserts that if M is a range of a closed linear operator then (1.11) is true. Of course, it is true that

M is closed; dim(Y/M) < ∞ ⇒ M is complemented.

A very important class of operators is the class of injective operators having closed range.

Deﬁnition 1.8 An operator A ∈ C(X,Y ) is said to be bounded below if A is injective and has closed range.

Theorem 1.9 ([64]) A ∈ C(X,Y ) is bounded below if and only if there exists c > 0 such that kAxk ≥ ckxk for all x ∈ D(A) (1.12)

The next result shows that the properties to be bounded below or to be surjective are dual each other.

Theorem 1.10 Let A ∈ C(X,Y ), then

1. A is bounded below ( respectively, surjective) if and only if T ∗ is surjective (respectively, bounded below).

2. If A is bounded below ( respectively, surjective) then λI − A is surjective (respectively, bounded below) for all |λ| < γ(A).

Proof. 1. Suppose that A is bounded below, then A is injective and from the equality R(A∗) = N(A)⊥ = X, we conclude that A∗ is surjective. Conversely, suppose that A∗ is surjective. Then A∗ has closed range and therfore also A has closed range. By the equality R(A∗)⊥ = N(A) = {0}, we conclude that A∗ is injective and hence bounded below. The proof that A being surjective if and only if A∗ is bounded below is analogous. 2. Suppose that A is injective with closed range. Then γ(A) > 0 and from the deﬁnition of γ(A) we obtain

γ(A)dist(x, N(A)) = γ(A)kxk ≤ kAxk for all x ∈ D(A)

From we obtain kAxk ≥ kAxk − |λ| kxk ≥ (γ(A) − |λ|)kxk thus for all |λ| < γ(A), the operator λI − A is bounded below. The case that A is surjective follows now easily by considering the adjoint A∗.

1.4 Compact operators

Deﬁnition 1.9 An operator A : X → Y is said to be compact if A(B) is relatively compact in Y for every bounded subset B ⊂ X. Equivalently, for every bounded sequence (xn)n in X there exists a convergent subsequence of (Axn)n in Y .

22 The set of all compact operators from X into Y is denoted by K(X,Y ). If X = Y , we write K(X) := K(X,X). If A is compact, then A(B(0, 1)) is bounded and thus A is bounded; i.e. K(X,Y ) ⊂ L(X,Y ). A special class of compact operators is the space of operators of ﬁnite-rank deﬁned by

F(X,Y ) = {A ∈ L(X,Y ) : dim R(A) < ∞}.

If X = Y , we write F(X) := K(X,X).

Proposition 1.5 ([64]) K(X,Y ) is a closed linear subspace of L(X,Y ). Let A ∈ L(X,Y ) and B ∈ L(Y,Z). If one of the operator A or B is compact, then BA is compact.

Strong limits of compact operators need not be compact. Consider the following operators on X = `2 deﬁned by

Anx = (x1, . . . , xn, 0, 0,... ) for all n ∈ N.

Then An ∈ F(X) but Anx → Ix for every x ∈ X and I is not compact operator. Theorem 1.11 (Schauder, [64]) . An operator A ∈ L(X,Y ) is compact if and only if A∗ ∈ L(Y ∗,X∗) is compact.

The following classical theorem extends fundamental results on matrices known from Linear Algebra.

Theorem 1.12 ([64]) [Riesz 1918, Schauder 1930]. Let K ∈ K(X). Then

1. R(I − K) is closed.

2. a(I − K) = d(I − K) < ∞.

3. α(I − K) = β(I − K) < ∞.

The easiest non-trivial example of a compact operator would have to be an integral operator:

Example 1.7 (Hilbert-Schmidt Operators) Let (X, µ) and (Y, ν) be measure spaces and let k(x, y) be a measurable function on X × Y with Z k(x, y)dµ(x)dν(y) < ∞ X×Y

Then Z (Kf)(x) = k(x, y)f(y)dν(y) Y deﬁnes a compact operator from L2(Y ; dν(y)) to L2(X; dµ(x)). Such an operator is called Hilbert-Schmidt.

Example 1.8 (Nuclear Operators) Let X and Y be Banach spaces and denote by X∗. 0 ∗ If (xn) is a bounded sequence in X ,(yn) is a bounded sequence in Y and (cn) is a set of P∞ complex numbers obeying n=0 |cn| < ∞, then

∞ X 0 Kx = cnxn(x)yn k=0

23 is called a nuclear operator from X to Y . Since

∞ ∞ X 0 0 X |cn| |xn(x)| |yn| ≤ (sup kxnkX∗ sup kynkY |cn|) kxkX n n k=0 n=0 the series deﬁning Kx converges strongly and K is a bounded operator of norm at most

∞ 0 X sup kxnkX∗ sup kynkY |cn| . n n n=0 1.5 Perturbations of closed operators

The following theorem shows the stability of closedness under a bounded perturbation.

Theorem 1.13 Let A be closed operator from X to Y and B ∈ L(X,Y ). Then the operator A + B is closed with D(A + B) = D(A).

Proof. Let xn ∈ D(A + B), n ∈ N, and x ∈ X, y ∈ Y such that xn → x in X and Axn + Bxn → y in Y as n → ∞. Since B is bounded, then limn→∞ Bxn = Bx and so Axn → y − Bx as n → ∞. Since A is closed, we deduce that x ∈ D(A) = D(A + B) and (A + B)x = y. The following example show that closedness can be lost when taking sums of closed operators

2 2 Example 1.9 Let X = Cb(R ) = {f : R → R bounded }. and Ak = ∂k with

D(Ak) = {f ∈ X the partial derivative ∂kf exists and belongs to X}, for k = 1, 2. Set B = ∂1 + ∂2 on

1 2 1 2 D(B) = D(A1) ∩ D(A2) = Cb (R ) = {f ∈ C (R ): f, ∂1f, ∂2f ∈ X}.

1 We have A1 and A2 are closed, But B is not closed. Take ϕn ∈ Cb (R) converging uniformly 1 2 to some Cb(R) \ C (R). Set fn(x, y) = ϕn(x − y) and f(x, y) = ϕ(x − y) for (x, y) ∈ R and n ∈ N. We then obtain f ∈ X, fn ∈ D(B), kfn − fk∞ = kϕn − ϕk∞ → 0 and 0 0 Bfn = ϕn − ϕn = 0, but f∈ / D(B). The extension of the stability Theorem to a not necessarily bounded perturbation is based on the notion of a relatively bounded perturbation.

Deﬁnition 1.10 Let A, B : X −→ Y . We say that B is bounded relatively to A or simply A-bounded if D(A) ⊂ D(B) and there exist nonnegative constants a, b such that

kBxk ≤ akAxk + bkxk (1.13)

In particular, if B is bounded, then is bounded relatively to any operator A with D(A) ⊂ D(B).If B is an operator from X to Y with D(A) ⊂ D(B), the restriction of of B to D(A) can be regarded as an operator B0 on XA to Y . It is easily seen that B is A-bonded if and only if B0 is bounded. Theorem 1.14 Let A be closed and let B be bounded relatively to A with a < 1 . Then A + B with the domain D(A) is closed.

24 Proof. We know that (1.13) hold for some a < 1 and b. Hence k(A + B)xk + kxk ≤ (1 + a)kAxk + (1 + b)kxk and (1 − a)kAxk + kxk ≤ kAxk − kBxk + (1 + b)kxk ≤ k(A + B)xk + (1 + b)kxk Hence the norms kAxk + kxk and k(A + B)xk + kxk are equivalent on D(A). Theorem 1.15 Let A be injective and D(A) ⊂ D(B). Then B is A-bounded with a = kBA−1k in (1.13). If, moreover, kBA−1k < 1, then A + B with domain D(A) is closed, invertible and

∞ X (A + B)−1 = (−1)nA−1(BA−1)n n=0 Proof. By the estimate kBxk ≤ kBA−1kkAxk, x ∈ D(A) we have a = kBA−1k. Assume now kBA−1k < 1. Let n X i −1 −1 i Cn = (−1) A (BA ) i=0

Then limn→∞ Cn = C exists. Let y ∈ Y . Clearly; limn→∞ Cny = Cy, and

n −1 −1 n+1 (A + B)Cny = y + (−1) A (BA ) y → y. But A + B is closed, hence Cy ∈ D(A + B) and (A + B)Cy = y. If x ∈ D(A + B), then

n −1 −1 n Cn(A + B)x = x + (−1) A (BA ) Ax → x. Hence C(A + B)x = x. Proposition 1.6 Let A and B be invertible and D(A) ⊂ D(B). Then B−1 − A−1 = B−1(A − B)A−1. Now we give the important Kato theorem’s on the stability of the propriety a linear operator having closed range. Theorem 1.16 ([63]) Let A, B : X −→ Y and Let A be a closed operator with closed range ( so that γ(A) > 0 ) and with least one of α(A) and β(A) ﬁnite. If B is bounded operator such that D(A) ⊂ D(B) and kBk < γ(A) then A+B is closed and has closed range (γ(A + B) > 0)and α(A + B) ≤ α(A), β(A + B) ≤ β(A), ind(A + B) = ind(A). This theorem can be extended to unbounded operator B in the following Theorem 1.17 ([63]) Theorem 1.16 is true if B is A-bonded with a < (1−b)γ(A) in (1.13). A notion analogous to relative boundedness is that of relative compactness, Deﬁnition 1.11 An operator B : X −→ Y is called compact relative to A or simply A- compact if D(A) ⊂ D(B) and from any sequence (xn) ⊂ D(A) such that

kAxnk + kxnk ≤ c. (1.14)

the sequence (Bxn) contains a convergent subsequence.

25 If B is A-compact, B is A-bounded. In fact, if B is not A-bounded, there is a sequence (xn) such that kAxnk + kxnk = 1 but kBxnk ≥ n for all n ∈ N. It follows that (Bxn) hes no convergent subsequence.

Remark 1.3 Let A be closed operator and D(A) ⊂ D(B). The restriction B0 of B to D(A) as an operator on XA to Y is A-compact if and only if B0 is compact. More generally the relative to strictly singular operators.

Deﬁnition 1.12 Let X and Y be two Banach spaces. An operator A ∈ L(X,Y ) is called strictly singular if, for every inﬁnite-dimensional subspace M, the restriction of A to M is not a homeomorphism.

Let S(X,Y ) denote the set of strictly singular operators from X into Y , if X = Y , S(X) := S(X,X) is a closed two-sided ideal of L(X) containing K(X). If X is a Hilbert space then K(X) = S(X).

Deﬁnition 1.13 An operator A ∈ L(X,Y ) is said to be weakly compact if A(B) is relatively weakly compact in Y for every bounded subset B ⊂ X.

The family of weakly compact operators from X into Y is denoted by Θ(X,Y ). If X = Y , the family of weakly compact operators on X, Θ(X) := Θ(X,X), is a closed two-sided ideal of L(X) containing K(X) (cf. [43, 45]).

The class of weakly compact operators on L1-spaces (resp. C(K)-spaces with K a compact Haussdorﬀ space) is nothing else but the family of strictly singular operators on L1- spaces (resp. C(K)-spaces) (see [93, Theorem 1]). The concept of strictly singular operators was introduced in the pioneering paper by Kato [63] as a generalization of the notion of compact operators. For a detailed study of the properties of strictly singular operators we refer to [45, 63]. For our own use, let us recall the following facts.

Theorem 1.18 ([63]) Let A, B : X −→ Y and Let A be a closed operator with closed range ( so that γ(A) > 0 ) and with α(A) < ∞. If B is bounded strictly singular operator then A + B is closed and has closed range with α(A + B) < ∞.

Theorem 1.18 can be extended to unbounded case.

Deﬁnition 1.14 Let A, B : X −→ Y . We say that B is strictly singular relative to A if D(A) ⊂ D(B) and there is no inﬁnite-dimensional subspace M such that

kBxk ≥ γ > 0 for all x ∈ M. (1.15) kAxk + kxk

Theorem 1.19 ([63]) Theorem 1.18 is true if B is is strictly singular relative to A.

1.6 The spectrum of closed operators

Let X be a complex Banach space and A : D(A) −→ X a closed operator with domain D(A) ⊂ X. The identity operator on X will be denoted by IX , or simply I if no confusion can arise. Frequently, even if A is unbounded, it might have a bounded inverse. In that case, we may use properties and theorems on bounded operators to study A. For λ ∈ C, if the

26 operator λI − A has an inverse, which is linear, we denote it R(λ, A) the inverse operator, that is R(λ, A) = (λI − A)−1 (1.16) and call it the resolvent operator of A at λ. The name resolvent is appropriate, since R(λ, A) a helps to solve the equation (λI −A)x = y. Thus, x = R(λ, A)y provided R(λ, A) exists. More important, the investigation of properties of R(λ, A) will be basic for an understanding of the operator A itself. Naturally, many properties of (λI − A) (or simply (λ − A)) and R(λ, A) depend on λ, and spectral theory is concerned with those properties. For instance, we shall be interested in the set of all λ in the complex plane such that R(λ, A) exists and bounded. For our investigation of R(λ, A) , we shall need some basic concepts in the spectral theory which are given as follows : The resolvent set of A is,

n −1 o ρ(A) := λ ∈ C : R(λ − A) = X and (λ − A) : R(λ − A) −→ D(A) exists and bounded (1.17) its complement σ(A) = C \ ρ(A) (1.18) is called the spectrum of A. Finally, the number r(A) := sup {|λ| ; λ ∈ σ(A)} (1.19) is called the spectral radius of A. For further reference [43, 45, 64, 105], we collect some important facts about the spectrum, resolvent operator, and spectral radius in the following theorem. Theorem 1.20 Let A be a closed operator on X. The operator (1.16), the sets (1.17) and (1.18) have the following properties: 1. The resolvent identities:

∀λ, µ ∈ ρ(A) R(λ, A) − R(µ, A) = (µ − λ)R(λ, A)R(µ, A)

moreover R(λ, A)and R(µ, A) commute for λ, µ ∈ ρ(A).

−1 2. If λ ∈ ρ(A) and |λ − µ| ≤ kR(λ, A)k then µ ∈ ρ(A). Thus ρ(A) is open in C.

3. The resolvent is an analytic map from ρ(A) to L(X,XA). Moreover

∞ X n n n+1 R(λ, A) = (−1) (λ − λ0) (R(λ0,A)) , n=0

−1 for all λ0 ∈ ρ(A) such that |λ − λ0| < kR(λ0,A)k .

4. σ(A) is closed in C. Theorem 1.21 Let A ∈ L(X). Then

1. σ(A) is nonempty compact set in C. 2. The spectral radius is given by the Gel’fand formula

1 1 r(A) = lim kAnk n = inf kAnk n n→+∞ n∈N

27 3. We have r(A) ≤ kAk , equality holds, for example, if X is a Hilbert space and A is normal, i.e. commutes with its adjoint.

4. The Neumann series

∞ 1 1 X 1 R(λ, A) = (I − A)−1 = An λ λ λn+1 n=0

converges in L(X) for each λ ∈ C with |λ| > r(A). 5. For every λ with |λ| > kAk we have λ ∈ ρ(A) and 1 kR(λ, A)k ≤ (1.20) |λ| − kAk

Note that the spectrum of bounded operator is never empty nor equal to C and in the unbounded case we have also, if σ(A) 6= C then A is closed, whereas the following example shows that there exist closed unbounded operators with spectrum may be empty or it may be unbounded.

d 2 Example 1.10 Let A = i dx on L [0, 1] and

2 2 D0 = {f : f ∈ AC [0, 1] and f(0) = 0},D1 = {f/f ∈ AC [0, 1]}

where AC2[0, 1] denotes the set of absolutely continuous functions on [0, 1] whose derivatives are in L2[0, 1].

The operators A0 = A/D0 and A1 = A/D1 are closed and σ(A0) = ∅ and σ(A1) = C.

Example 1.11 Let X = `1(Z) be the space of all summable complex sequences

x = (xn)n = (. . . , x−2, x−1, x0, x1, x2,...),

indexed by the integers, with the usual norm. For any ∈ R, let A ∈ L(X) be deﬁned by A(x) = y, where x = (xn)n and y = (yn)n are related by

x if k 6= 0 y = k−1 x−1 if k = 0 Then we have σ(A0) = D where D denotes the open complex unit disc. On the other hand,

σ(A) = S = ∂D = {λ ∈ C : |λ| = 1} ( 6= 0). So, here the spectrum collapses when changes from zero to a nonzero value.

The spectrum σ(A) is partitioned into three disjoint sets as follows:

• The point spectrum σp(A) of A, is the set of λ ∈ C such that λ − A is not injective. λ ∈ σp(A) is called an eigenvalue of A and for this λ there exists a non zero vector x such that Ax = λx called an eigenvector corresponding to λ.

28 • The continuous spectrum σc(A) of A, is the set of λ ∈ C such that λ − A is injective but its range is not closed.

• The residual spectrum σr(A) of A, is the set of λ ∈ C such that λ − A is injective but its range is not dense in X.

The following table reﬁnes this subdivision. The residual spectrum is split into two disjoint parts, σr(A) = σr1(A) ∪ σr2(A), and the point spectrum is split into four disjoint parts, 4 σp(A) = ∪i=1σpi (A).

N(λ − A) = {0} N(λ − A) 6= {0} R(λ, A) exists R(λ, A) exists R(λ, A) does not and is bounded and is unbounded exists

R(λ − A) R(λ − A) is dense in X λ ∈ ρ(A) ∅ λ ∈ σp1(A) is closed R(λ − A) is not dense in X λ ∈ σr1(A) ∅ λ ∈ σp2(A) R(λ − A) R(λ − A) is not dense in X ∅ λ ∈ σr2(A) λ ∈ σp3(A) is not closed R(λ − A) is dense in X ∅ λ ∈ σc(A) λ ∈ σp4(A) Table 1.1:

Remark 1.4 1. If X is ﬁnite dimension, then σc(A) = σr(A) = ∅.

2. If X is a Hilbert space and A is a self-adjoint operator then σr(A) = ∅.

∗ 3. λ ∈ σr(A) means that λ is an eigenvalue of A , but not of A, i.e λ − A is injective but λ − A∗ is not: there exists then x∗ ∈ X∗ such that (λ − A∗)x∗ = 0 hence x∗(λ − A) = 0 which implies that R(λ − A) ∈ N(x∗). Then R(λ − A) is not dense in X.

∗ 4. If λ ∈ σc(A), then λ is not eigenvalue of A or of A .

29 If we define β(A) as the codimension of the closure of R(A), then the following table gives a useful characterizations of the different parts of the spectrum defined in Table 1.1 in terms of the quantities defined in Section 1.1 and β(A).

α(λI − A) a(λI − A) β(λI − A) β(λI − A) d(λI − A)

ρ(A) 0 0 0 0 0

σp1(A) 6= 0 ∞ 0 0 0

σp2(A) 6= 0 6= 0 6= 0 6= 0 6= 0 (β = β)

σp3(A) 6= 0 6= 0 ∞ 6= 0 ∞

σp4(A) 6= 0 6= 0 ∞ 0 ∞

σc(A) 0 0 ∞ 6= 0 ∞

σr1(A) 0 0 6= 0 6= 0 ∞ (β = β)

σr2(A) 0 0 ∞ 6= 0 ∞

Table 1.2: Note that in the boxes marked by "6= 0" the quantities may be infty. To verify this table, we also use the Table 1.1 and Theorem 1.2. p Example 1.12 Let X = ` , 1 ≤ p ≤ ∞, the space of of all sequences x = (x1, x2, x3,... ) with ﬁnite norm 1 P∞ p p ( n=1 |xn| ) if 1 ≤ p < ∞ kxkp = supn≥1 |xn| if p = ∞ we deﬁne the following operators on `p by

A0x = (0, x1, x2, x3,... )

A1x = (x2, x3, x4,... ) 1 1 A x = (x , x , x ,... ) 2 1 2 2 3 3 1 1 A x = (x , x , x ,... ) 3 2 2 3 3 4 1 1 A x = (0, x , x , x ,... ) 4 1 2 2 3 3

30 A A0 A1 A2 A3 A4 `p 1 < p < ∞ p = 1 or ∞ 1 ≤ p < ∞ p = ∞ 1 ≤ p ≤ ∞ 1 ≤ p ≤ ∞ 1 ≤ p ≤ ∞ 1 1 σp(A) ∅ ∅ D D {1, 2 , 3 ,... } {0} ∅ σc(A) S ∅ S ∅ {0} ∅ ∅ σr(A) D D ∅ ∅ ∅ ∅ {0} 1 1 σ(A) D D D D {1, 2 , 3 ,... } ∪ {0} {0} {0} Table 1.3:

Sometimes it is useful to relate the spectrum of a closed densely deﬁned linear operator to that of its adjoint. Building on classical existence and uniqueness results for linear operator equations in Banach spaces and their adjoints, one may prove the following

Theorem 1.22 ([45]) The spectra and subspectra of an operator A ∈ C(X) and its adjoint A∗ are related by the following relations:

1. σ(A) = σ(A∗).

∗ ∗ 2. σp(A) ⊆ σr(A ) ∪ σp(A ).

∗ 3. σp(A ) ⊆ σr(A) ∪ σp(A).

∗ ∗ 4. σc(A) ⊆ σr(A ) ∪ σc(A ).

∗ 5. σc(A ) ⊆ σc(A).

∗ 6. σr(A) ⊆ σp(A ).

∗ 7. σr(A ) ⊆ σp(A) ∪ σc(A).

∗ ∗ 8. If X is reﬂexive, then σc(A ) = σc(A) and σr(A ) ⊆ σp(A). We now reformulate the Theorem 1.12 in terms of spectral theory

Theorem 1.23 Let dim X = ∞ and K ∈ L(X) be compact. Then the following assertions hold.

1. σ(K) = {0} ∪ {λn : n ∈ J}, where either J = ∅, or J = {1, . . . , n} for some n ∈ N, or J = N. 2. Each λ ∈ σ(K) \{0} is an eigenvalue of K with α(λI − K) = β(λI − K) < ∞.

3. If J = N then λn → 0 as n → ∞.

Note that in Example 1.12 the operators A2, A3 and A4 are compact operators then 0 always belongs to the spectrum if the underlying space is inﬁnite dimensional. But this three operators shows a more precise classiﬁcation of 0 as spectral point is not possible (see Table 1.3).

The Fredholm alternative. Let A ∈ K(X), and y ∈ X. To solve the equation

λx − Ax = y (1.21) we have one of the following alternatives hold:

31 • λx − Ax = 0 has only the trivial solution x = 0. Then for every y ∈ X there is a unique solution x ∈ X of (1.21) given by x = (λI − A)−1y.

• λx − Ax = 0 has an n-dimensional solution space N(λI − A) for some n ∈ N. ∗ ∗ ∗ ∗ Then there are n linearly independent solutions x1, . . . , xn of λx = Ax , and the ∗ equation (1.21) has a solution x ∈ X if and only if hy, xi i = 0 for every i = 1, . . . , n. In this case, every z ∈ X satisfying (1.21) if of the form z = x+x0, where λx−Ax = y and x0 ∈ N(λI − A).

Deﬁnition 1.15 A closed operator A in X has a compact resolvent if there exists a λ ∈ ρ(A) such that R(λ; A) ∈ L(X) is compact.

Remark 1.5 1. If A has a compact resolvent R(λ; A) and µ ∈ ρ(A), then

R(µ; A) = R(λ; A) + (λ − µ)R(λ; A)R(µ; A)

by the resolvent equation, so that R(µ; A) is also compact due to Proposition 1.5.

2. If dim X = ∞, then a closed operator A with compact resolvent R(λ; A) cannot be bounded since otherwise R(λ; A) were bounded and thus I = (λI − A)R(λ; A) were compact by Proposition 1.5.

3. For a closed operator A with λ ∈ ρ(A) the following assertions are equivalent,

i A has a compact resolvent,

ii Each bounded sequence in XA has a subsequence which converges in X,

iii The inclusion map J : XA → X is compact.

Theorem 1.24 ([64]) Let dim X = ∞ and A be a closed operator with compact resolvent. Then σ(A) is either empty or contains only at most countably many eigenvalues λn with α(λnI − A) < ∞. If A has inﬁnitely many eigenvalues λn, then |λn| → ∞ as n → ∞.

Let H be a ﬁnite-dimensional Hilbert space, say H = Cn. It is known from linear algebra that eigenvectors of a self-adjoint operator on H form an orthogonal basis of H. The following theorems generalize this result to inﬁnite dimensional spaces.

Theorem 1.25 ([64]) Let X be a separable Hilbert space and A be densely deﬁned, closed and self-adjoint compact operator. Then

1. σp(A) is a most countable subset of R, 2. If λ 6= µ ∈ σ(A), then N(λI − A) is orthogonal to N(µI − A).

3. If σ(T ) − {0} = {λn; n ∈ N} then there is an orthonormal basis of X consisting of eigenvectors of A and ∞ X = N(A) ⊕n=0 N(λnI − A) Furthermore , the sum ∞ X A = λnPλn n=0

Converge in L(X), where Pλn is the orthogonal projection onto N(λnI − A).

32 Proposition 1.7 ([64]) Let X be a separable Hilbert space and A be densely deﬁned, closed and self-adjoint operator having a compact resolvent. Then

σ(A) = σp(A) = {µn : n ∈ N} ⊂ R with |µn| → ∞ as n → ∞, and there is an orthonormal basis of X consisting of eigenvectors of A. The eigenspaces N(µnI − A) are ﬁnite dimensional, and we have QnD(A) ⊂ D(A) and AQnx = QnAx for all x ∈ D(A) and n ∈ N, where Qn is the orthogonal projection onto N(µnI − A). Finally, the sum X Ax = µnQnx n∈N converges in X for all x ∈ D(A).

1.7 Approximate point spectrum

Another important part of the spectrum is the approximate point spectrum σap(A) that is deﬁned as follows

σap(A) := {λ ∈ C : λI − A is not bounded below}. (1.22)

Next we give an alternative deﬁnition of σap(A) which may come as a motivation for the term approximate point spectrum. The elements of it are sometimes referred to as the approximate eigenvalues of A.

σap(A) = {λ ∈ C : ∀ > 0, ∃x ∈ D(A) with kxk = 1 and k(λI − A)xk < } = {λ ∈ C : ∃(xn)n ⊂ D(A) with kxnk = 1 ∀n and k(λI − A)xnk −→ 0} = {λ ∈ C : ∃(λn)n ⊂ C with λn −→ 0 and ∃(xn)n ⊂ D(A) with kxnk = 1 ∀n such that k(λnI − A)xnk −→ 0}.

Clearly, σp(A) ⊂ σap(A) and

σap(A) = σp(A) ∪ σc(A) ∪ σr2(A) = σ(A) \ σr1(A)

The quantity j(λI − A) := inf k (λI − A)xk kxk=1 is called the injectivity modulus of A at λ, and obviously we have

j(λI − A) = 0 ⇐⇒ λ ∈ σap(A).

Proposition 1.8 We have

1. For all λ, µ ∈ C |j(λ) − j(µ)| ≤ |λ − µ|. 2. If λ ∈ ρ(A) then j(λI − A) = kR(λ, A)k−1.

33 Proof. 1. Let x ∈ D(A) such that kxk = 1. We have

kλx − Axk ≤ |λ − µ| + kµx − Axk hence j(λI − A) ≤ |λ − µ| + j(µI − A) and therefore j(λI − A) − j(µI − A) ≤ |λ − µ| the inequality follows by interchanging λ and µ. 2. Setting S = R(λ, A) , if kxk = 1, then

kxk = kS(λ − A)xk ≤ kSk kλ − Ak kxk hence 1 ≤ kSk j(λI − A) Moreover,

Sx kxk 1 (λ − A) = = kSxk kSxk kxk 1 From this it follows that kSxk ≤ j(λI−A) and consequently 1 kSk ≤ j(λI − A)

This shows the results.

Proposition 1.9 The approximate point spectrum is nonempty, closed in C, and includes the boundary ∂σ(A) of the spectrum.

Proof. Since the function j(λ − A) is continuous in λ and σap(T ) is the inverse image of {0} by j, it follows that σap(A) is closed. −1 Now, let λ ∈ ρ(A), then j(λI − A) = kR(λ, A)k > 0, and λ∈ / σap(A). Hence σap(A) ⊂ σ(A). Since σ(A) is bounded, σap(T ) is bounded and closed in C, hence compact. Let λ ∈ σ(T ) ∩ ρ(T ). Hence there exists a sequence (λn)n in ρ(T ) such that λn −→ λ. Since (λ − T ) is not invertible, then the sequence (λn − A)n is not bounded, so that there exist subsequences of (λn)n, for which limn→+∞ kR(λn,T )k = +∞. Hence

−1 lim j(λn − A) = lim kR(λn,T )k = 0 n→+∞ n→+∞

Since the function j(.) is continuous in λ, we then conclude that j(λ − A) = 0. Hence λ ∈ σap(A). Example 1.13 If A is closed densely deﬁned and symmetric on a Hilbert space X, then σap(A) ⊆ R.

Proposition 1.10 Let A be closed operator on X and λ ∈ ρ(A). Then the following assertions hold.

1 1. σ(R(λ, A)) = { λ−µ ; µ ∈ σ(A)}.

34 1 2. σi(R(λ, A)) = { λ−µ ; µ ∈ σi(A)} for i = p, ap, r, c. 3. If A is unbounded, then 0 ∈ σ(R(λ, A)).

There are some overlapping parts of the spectrum which are commonly used too. For instance, the defect spectrum (or the surjectivity spectrum) σsu(A) and the compression spectrum σco(A) are deﬁned by

σsu(A) := {λ ∈ C : λI − A is not surjective} and σco(A) := {λ ∈ C : R(λI − A) is not dense in X}. By the closed range theorem we easily know that the approximate point spectrum and ∗ the surjectivity spectrum are dual to each other, in the sense that σap(A) = σsu(A ) and ∗ σap(A ) = σsu(A). Moreover, this two sets form a (not necessarily disjoint) subdivision

σ(A) = σap(A) ∪ σsu(A)

of the spectrum and σco(A) ⊂ σsu(A). Moreover, comparing these subspectra with those in Table 1.1 we note that σco(A) = σp2(A) ∪ σp3(A) ∪ σr(A).

1.8 The Riesz projection and the singularities of the resolvent

Let A a closed operator and its spectrum σ(A) decomposes into two non-empty disjoint closed subsets σ and τ. Suppose that σ is bounded, we can surrounding σ by a Jordan contour1 positively oriented Γ. Hence we can associate a Riesz integral with σ I 1 −1 Pσ(A) = (λ − A) dλ, (1.23) 2πi Γ This projection is called the spectral projection associated with the spectral set σ. Since (λ − A)−1 is analytic operator function on the resolvent set of A, a standard argument of complex function theory shows that the deﬁnition of Pσ does not depend on the particular choice of the contour Γ. A fundamental properties of the spectral projection Pσ are given in the next results.

Proposition 1.11 Let A be a closed operator on X such that its spectrum σ(A) is the disjoint union of two non-empty closed subsets σ and τ with σ is bounded. Let Pσ be as deﬁned in (1.23). Then

(i) Pσ is a projection.

(ii) Pσ commutes with A on D(A);

(iii) X = R(Pσ) ⊕ N(Pσ), σ(APσ) = σ and σ(A(I − Pσ)) = σ(A)\σ.

1A subset Γ of C is a Jordan contour if there exists a ﬁnite number of pairwise disjoint closed simple n rectiﬁable Jordan curves positively oriented Γ1, Γ2,..., Γn such that Γ = ∪i=1Γi.

35 Proof. Let Γ1 and Γ2 be two admissible contours around σ separating a from τ = σ(A)\σ. Assume that Γ1 is in the inner domain of Γ2. Then Z 2 1 −1 −1 Pσ = − 2 (λ − A) (µ − A) dλ dµ. 4π Γ1×Γ2 By the ﬁrst resolvent equation we have Z 2 1 1 P σ = − 2 (R(λ, A) − R(µ, A)) dλ dµ. 4π Γ1×Γ2 µ − λ Then Z Z Z Z 2 1 1 1 1 1 1 (Pσ) = R(λ, A) dµ dλ − R(µ, A) dλ dµ. 2πi Γ1 2πi Γ2 µ − λ 2πi Γ2 2πi Γ1 µ − λ

R 1 R 1 The fact that dλ = 0, (µ ∈ Γ2) and dλ = 2πi, (λ ∈ Γ1), we obtain Γ1 µ−λ Γ2 µ−λ 2 Pσ = Pσ. This prove the ﬁrst statement.

The second statement follows from the fact that A and the resolvent of A commute on D(A). To prove the third, denote A1 and A2 the restriction of A in R(Pσ) and N(Pσ) respectively. Note that Pσ is a projection on R(Pσ) along R(I −Pσ) = N(Pσ) and commutes with A, which means that A is decomposed according to X = R(Pσ) ⊕ N(Pσ) and the parts A1 and A2 are deﬁned. Now we have R(λ, A1)x = R(λ, A)Pσx for x ∈ R(Pσ) , λ ∈ ρ(A). But for any λ ∈ ρ(A) outside Γ compute by using the resolvent identity, we have Z 1 −1 −1 R(λ, A)Pσ = (A − λ) (A − µ) dµ 2πi Γ 1 Z Z = [ (λ − µ)−1(A − λ)−1 dµ − (λ − µ)−1(A − µ)−1 dµ]. (1.24) 2πi Γ Γ The ﬁrst integral in (1.24) vanishes since λ outside Γ, this gives Z 1 −1 −1 R(λ, A)Pσ = − (λ − µ) (A − µ) dµ (1.25) 2πi Γ

−1 The integral in (1.25) is analytic in λ outside of Γ, hence R(λ, A)Pσ = (λ−A1) is holomorphic outside Γ. By taking Γ close to the boundary of σ, and using openness of the resolvent −1 set we see that (λ − A1) is analytic on C \ σ. Thus ρ(A1) contains the exterior of Γ and σ(A1) ⊂ σ. In a similar way, it follows from (1.24), if λ inside Γ, that Z 1 −1 −1 R(λ, A)Pσ = R(λ, A) + (λ − µ) (µ − A) dµ. 2πi Γ

This show that R(λ, A)(I −Pσ) has an analytic continuation holomorphic inside Γ. As above, we obtain σ(A2) ⊂ τ. On other hand, a point λ ∈ σ(A) cannot belong to ρ(A1) ∩ ρ(A2), otherwise if would belong to ρ(A), because

−1 −1 −1 (λ − A) = (λ − A) Pσ + (λ − A) (I − Pσ). (1.26)

This shows that σ(Ai) = σi, i = 1, 2.

36 Let A ∈ C(X) and λ0 an isolated point of the spectrum of A. Form a contour

Γλ0 = {λ ∈ C :| λ − λ0 |= ε},

with a bounded region inside Γλ0 intersecting the spectrum of A only at the point λ0. We deﬁne the Riesz projection of A associated to the point λ0 by 1 I P (A) = (λ − A)−1 dλ, (1.27) λ0 2πi Γλ0

Proposition 1.12 Let A be a closed operator on X and λ0 an isolated point of the spectrum of A. Then

(i) N(λ0 − A) ⊆ R(Pλ0 ).

(ii) If X is a Hilbert space and A is self-adjoint, then Pλ0 is the orthogonal projection onto N(λ0 − A).

−1 −1 Proof. (i) Let x ∈ N(λ0 − A), then for λ 6= λ0, we have (λ − A) x = (λ − λ0) x. We

show that Pλ0 x = x, so x ∈ R(Pλ0 ). By the deﬁnition of Pλ0 1 Z 1 Z P x = (λ − A)−1x dλ = (λ − λ )−1x dλ = x λ0 2πi 2πi 0 Γλ0 Γλ0

∗ ∗ (ii) Let X be a Hilbert space and suppose that A = A . We ﬁrst show that Pλ0 = (Pλ0 ) .

Let r > 0 such that Γλ0 = {λ ∈ C such that |λ − λ0| = r} is an admissible contour and iθ λ = λ0 + re . Then I π 1 iθ −1 Pλ0 = (λ0 + re − A) rdθ. 2π −π and I π ∗ 1 iθ −1 ∗ Pλ0 = ((λ0 + re − A) ) rdθ, 2π −π I π 1 iθ ∗ −1 = (λ0 + re − A ) rdθ, 2π −π I π 1 −iθ −1 = (λ0 + re − A) rdθ. (1.28) 2π −π

Reparametrizing (1.28) with θ1 = −θ, then 1 I π ∗ iθ1 −1 Pλ0 = (λ0 + e − A) rdθ1 = Pλ0 . 2π −π

Finally we show that N(λ0 −A) = R(Pλ0 ),which, by part (i) requires that we show R(Pλ0 ) ⊂ N(λ0 − A). We have

1 I (λ − A)P = (λ − A)(λ − A)−1 dλ, 0 λ0 2πi 0 Γλ0 1 I = (λ − λ )(A − λ)−1 dλ. 2πi 0 Γλ0

37 −1 Let Vλ0 denote the interior of Γλ0 , the operator (λ − λ0)(A − λ) is an analytic, operator valued function on Vλ0 {λ0} and satisﬁes |λ − λ | |λ − λ |||(A − λ)−1|| ≤ 0 . 0 d(λ, σ(A)

Now if we take the diameter of Γλ0 small such that

|λ − λ|||(A − λ)−1|| < 1,

−1 then the function (λ − λ0)(A − λ) is uniformly bounded on Vλ0 {λ0} and extends to an analytic function on Vλ0 and hence, by the Cauchy theorem we obtain 1 I (λ − λ )(A − λ)−1 dλ = 0. 2πi 0 Γλ0

This shows that R(Pλ0 ) ⊂ N(λ0 − A). Remark 1.6 Let A ∈ C(X), λ and µ two diﬀerent isolated points of σ(A) then

(i) PλPµ = PµPλ = 0.

(ii) P{λ,µ} = Pλ + Pµ.

Let A ∈ C(X) and λ0 an isolated point of σ(A). The Laurent series for the resolvent −1 (λI − A) in neighborhood of the isolated singularity λ0 is given by

+∞ −1 X n (λI − A) = (λ − λ0) An, (1.29) n=−∞ where 1 Z 1 An = n+1 R(λ, A) dλ, (1.30) 2πi (λ − λ0) Γλ0 and Γλ0 is a positively oriented small circle enclosing λ0 but no other point of σ(A). The coeﬃcients An deﬁned in (1.30) stisfay some useful identities given in the next proposition.

Proposition 1.13 The coeﬃcients An given by (1.30) are bounded operators and satisﬁes the following proprieties

(i) AAn = AnA on D(A) for all n ∈ Z,

(ii) AnAm = (1 − ηn − ηm)An+m+1 where ηn = 1 if n ≥ 0 and ηn = 0 if n < 0.

(iii) A−1 = I − (λ0I − A)A0,

(iv) An−1 = (A − λ0I)An for each n 6= 0,

n n−1 (v) A0 = (A − λ0I) An and A−n = (A − λ0I) A−1 for all n ≥ 1.

38 Proof. The commutativity of An with A follows from the commutativity of A and the resolvent of A. For simplicity, to prove (i), we may assume λ0 = 0, since 0 is an isolated point of σ(A), then there exists δ > 0 such that B(0, δ) ∩ σ(A) = {0}. Denote γr = {λ ∈ C : |λ| = r} for 0 < r < δ. Let r < r1, we have by using the resolvent identities that

1 Z Z A A = λ−n−1µ−m−1R(λ, A)R(µ, A) dλdµ n m (2πi)2 γr γr1 1 Z Z = λ−n−1µ−m−1(µ − λ)−1[R(λ, A) − R(µ, A)] dλdµ (2πi)2 γr γr1 By computing the double integral on the right in any order and the fact that Z 1 −n−1 −1 −n−1 λ (µ − λ) dλ = ηnµ 2πi γr 1 Z µ−m−1(µ − λ)−1dµ = (1 − η )λ−m−1 2πi m γr1 where 1 if n ≥ 0, η = n 0 if n < 0 We obtain Z 1 − ηn − ηm −n−m−2 AnAm = λ R(λ, A)dλ = (1 − ηn − ηm)An+m+1 (1.31) 2πi γr

Now by deﬁnition of An we have R(An) ⊂ D(A). Multiplying (1.29) on the left by λI −A and using that (λI − A)R(λ, A) = I, we obtain

+∞ X n I = (λI − A) (λ − λ0) An n=−∞ +∞ X n = [(λ − λ0)I + (λ0I − A)] (λ − λ0) An n=−∞ +∞ +∞ X n+1 X n = (λ − λ0) An + (λ − λ0) (λ0I − A)An n=−∞ n=−∞ +∞ X n = (λ − λ0) [An−1 + (λ0I − A)An]. n=−∞

The uniqueness of the Laurent series expansion yields I = A−1 + (λ0I − A)A0 and An−1 + (λ0I − A)An = 0 for all n 6= 0. These are (iii) and (iv). The last two identities are straightforward.

From the standard terminology of the complex theory, we call the operator A−1 in the Laurent series (1.29) the residue operator at λ0. A remarkably, when λ0 is an isolated point, by taking n = m = −1 in (1.31), the residue operator A−1 is a projection coincides with the

Riesz projection Pλ0 associated to λ0. Furthermore, by setting D = A−2 and S = −A0, the

39 k−1 n+1 relation (1.31) gives A−k = D for k ≥ 2 and An = −S for n ≥ 0. From this notations the Laurent series (1.29) around λ0 is equivalent to

∞ ∞ 1 X 1 X (λI − A)−1 = P + Dn − (λ − λ )nSn+1; (1.32) λ − λ λ0 (λ − λ )n+1 0 0 n=1 0 n=0 By Proposition 1.11 and compare (1.26) with (1.32) we have

 1 P∞ 1 n R(λ, APλ ) = R(λ; A)Pλ = Pλ + n+1 D ,  0 0 λ−λ0 0 n=1 (λ−λ0)

 P∞ n n+1 R(λ, A(I − Pλ0 )) = R(λ; A)(I − Pλ0 ) = − n=0(λ − λ0) S , where

D = (A − λ0I)Pλ0 = (APλ0 − λ0I)Pλ0 , (A − λ0I)S = I − Pλ0 . (1.33) hence

−1 S = (A(I − Pλ0 ) − λ0I) (I − Pλ0 ) = −R(λ0,A(I − Pλ0 ))(I − Pλ0 ) = lim R(λ, A)(I − Pλ0 ) λ→λ0

and SPλ0 = Pλ0 S = 0,DS = SD = 0,D = DPλ0 = Pλ0 D. We conclude the following

Theorem 1.26 If λ0 is an isolated point in the spectrum of a closed operator A, then Laurent series (1.29) around λ0 is equivalent to

∞ ∞ 1 X 1 X (λI − A)−1 = P + Dn − (λ − λ )nSn+1; λ − λ λ0 (λ − λ )n+1 0 0 n=1 0 n=0

with the residue operator A−1 is a projection coincides with the Riesz projection associated

to λ0 and SPλ0 = Pλ0 S = 0, DS = SD = 0, D = DPλ0 = Pλ0 D.

Now we characterize the isolated points of the spectrum that are poles of the resolvent.

Theorem 1.27 Let λ0 be an isolated point in the spectrum of a closed operator A and let (1.32) be the Laurent series around λ0. Then the following statements are equivalents

1. λ0 is a pole of the resolvent of order m.

2. The operator D = (λ0I − A)Pλ0 is a nilpotent operator of order m.

3. a(λ0I − A) = d(λ0I − A) = m < ∞.

Proof. (1) ⇒ (2). We have A−m 6= 0 and An = 0 for all n > m and we know that m−1 A−1 = Pλ0 , then it follows by Proposition 1.11 that (λ0I − A) Pλ0 = A−m 6= 0 and m (λ0I − A) Pλ0 = A−m−1 = 0. m m+1 (2) ⇒ (3). First we prove that N((λ0I − A) ) = N((λ0I − A) ). Since we already m m+1 know that N((λ0I − A) ) ⊆ N((λ0I − A) ), it suﬃces to prove the inverse inclusion. We m+1 m proceed by contradiction. Let x ∈ N((λ0I − A) ) and x∈ / N((λ0I − A) ), that is, the m m+1 vector y = (λ0I − A) x 6= 0. it follows that (λ0I − A)y = (λ0I − A) x = 0. This implies

by Proposition 1.12 part (ii), Pλ0 y = y. Consequently

m m 0 = (λ0I − A) Pλ0 x = Pλ0 (λ0I − A) x = Pλ0 y = y

40 m m+1 which is a contradiction. Hence, N((λ0I − A) ) = N((λ0I − A) ) and a(λ0 − A) ≤ m. m−1 Now, notice that (λ0I − A) Pλ0 6= 0 guarantees the existence of some vector x ∈ R(Pλ0 ) such that m−1 m−1 (λ0I − A) x = (λ0I − A) Pλ0 x 6= 0 m m From (λ0I − A) x = (λ0I − A) Pλ0 x = 0, it follows that

m m−1 N((λ0I − A) ) 6= N((λ0I − A) ) (1.34)

This shows a(λ0 − A) ≥ m. Thus a(λ0 − A) = m. Next, we consider the unique decomposition described in Proposition 1.11 with σ = n {λ0}. The operator (λ0I − A) is also invertible on N(Pλ0 ) for all n ∈ N. The identity m m m (λ0I − A) Pλ0 = 0, implies that (λ0I − A) = 0 on R(Pλ0 ). Consequently R((λ0I − A) ) = m+1 N(Pλ0 ) = R((λ0I − A) ). Thus λ0I − A has ﬁnite descent, and by Theorem 1.1 we have d(λ0I − A) = m. (3) ⇒ (1). Assume that a(λ0I − A) = d(λ0I − A) = m < ∞, By Proposition 1.11 and m n Theorem 1.1 we have N(Pλ0 ) = R((λ0I − A) ) = R((λ0I − A) ) and R(Pλ0 ) = N((λ0I − m n n n A) ) = N((λ0I − A) ) for all n ≥ m. It follows that D = (A − λ0I) Pλ0 = 0 for all n ≥ m, and so λ0 is a pole of the resolvent of order k with k ≤ m. But from (1.34), it necessarily k = m.

Corollary 1.3 Let A be a closed operator. If λ0 is a pole of order m of the resolvent around λ0. Then λ0 is an eigenvalue of A. Moreover

n n X = N((λ0I − A) ) ⊕ R((λ0I − A) ) for all n ≥ m.

and the Laurent series (1.32) around λ0 is equivalent to

m−1 ∞ 1 X 1 X (λI − A)−1 = P + Dn − (λ − λ )nSn+1; (1.35) λ − λ λ0 (λ − λ )n+1 0 0 n=1 0 n=0

Remark 1.7 Let λ0 a pole of order m of the resolvent around λ0. From Proposition 1.11 it follows that X = R(Pλ0 ) ⊕ N(Pλ0 ). There are two numbers measuring, roughly speaking, the number of eigenvectors belonging to N(λ0 − A). They are called multiplicities: The algebraic multiplicity of the eigenvalue λ0 is deﬁned as the dimension of the space R(Pλ0 ) and equal m the order of the pole λ0. The geometric multiplicity of λ0 is deﬁned as the dimension of N(λ0 − A) and equal α(λ0 − A). In general, we have α(λ0 − A) ≤ m.

41 Chapter 2

Essential Fredholm, Weyl and Browder spectra

2.1 Essential Fredholm spectra

We now introduce some important classes of operators in Fredholm 1 theory. Let X and Y be Banach spaces and let A be an operator from X into Y . We denote by D(A) ⊂ X its domain and R(A) ⊂ Y its range. For A ∈ C(X,Y ), the nullity, α(A), of A is defined as the dimension of N(A) and the deficiency, β(A), of A is defined as the codimension of R(A) in Y . Definition 2.1 Let X,Y be Banach spaces. The set of upper semi-Fredholm operators from X into Y is defined by

Φ+(X,Y ) = {A ∈ C(X,Y ): α(A) < ∞ and R(A) is closed in Y }, the set of lower semi-Fredholm operators from X into Y is deﬁned by

Φ−(X,Y ) = {A ∈ C(X,Y ): β(A) < ∞ and R(A) is closed in Y }, the set of semi-Fredholm operators from X into Y is deﬁned by

Φ±(X,Y ) = Φ+(X,Y ) ∪ Φ−(X,Y ), the set of Fredholm operators from X into Y is deﬁned by

Φ(X,Y ) = Φ+(X,Y ) ∩ Φ−(X,Y ), the set of bounded Fredholm operators from X into Y is deﬁned by

Φb(X,Y ) = Φ(X,Y ) ∩ L(X,Y ).

If A ∈ Φ±(X,Y ), the number ind(A) = α(A) − β(A) is called the index of A. Clearly, indA is an integer or ±∞. The subset of all compact operators of L(X,Y ) is denoted by b K(X,Y ). If X = Y then Φ+(X,Y ), Φ−(X,Y ), Φ±(X,Y ), Φ(X,Y ) and Φ (X,Y ) are re- b placed, respectively, by Φ+(X), Φ−(X), Φ±(X), Φ(X) and Φ (X). Observe that in the case X = Y the class Φ(X) is non-empty since the identity trivially is a Fredholm operator. But for certain diﬀerent Banach spaces X, Y no bounded Fredholm operators from X to Y may exist (see [38, Lemma 3.3.]).

1It is named in honor of Erik Ivar Fredholm, Swedish mathematician, April 7, 1866 - August 17, 1927.

42 Theorem 2.1 ([105, 90]) Suppose that X,Y and Z are Banach spaces.

(i) If A ∈ Φ+(X,Y ) and B ∈ Φ+(Y,Z), then BA ∈ Φ+(X,Z) and ind(BA) = ind(A) + ind(B),

(ii) If A ∈ Φ−(X,Y ) and B ∈ Φ−(Y,Z), then BA ∈ Φ−(X,Z) and ind(BA) = ind(A) + ind(B),

(iii) If A ∈ Φ(X,Y ) and B ∈ Φ(Y,Z), then BA ∈ Φ(X,Z) and ind(BA) = ind(A) + ind(B),

(vi) If BA ∈ Φ+(X,Z), then B ∈ Φ+(Y,Z),

(v) If BA ∈ Φ−(X,Z), then B ∈ Φ−(Y,Z),

(iv) If BA ∈ Φ(X,Z), then B ∈ Φ−(Y,Z) and A ∈ Φ+(X,Y ),

The converse of (i)–(iii) in Theorem 2.1 is not true in general. To see this, consider the following operators on `2:

Ax = (0, x1, 0, x2, 0,... )

Bx = (x2, x3, x4,... )

Then A and B are not Fredholm, but BA = I. However, if BA = AB then we have if BA ∈ Φ(X) then A ∈ Φ(X) and B ∈ Φ(X). because N(A) ⊂ N(BA) and R(BA) = R(AB) ⊂ R(A). By Theorem 1.7 ∗ ∗ ∗ A ∈ Φ+(X,Y ) ⇔ A ∈ Φ−(Y ,X ), ∗ ∗ ∗ A ∈ Φ−(X,Y ) ⇔ A ∈ Φ+(Y ,X ), A ∈ Φ(X,Y ) ⇔ A∗ ∈ Φ(Y ∗,X∗). ∗ If A ∈ Φ±(X), then ind(A ) = −ind(A).

Example 2.1 If U is the unilateral shift operator on `2, then

ind(U) = 1 and ind(U ∗) = −1.

With U and U ∗, we can build a Fredholm operator whose index is equal to an arbitrary prescribed integer. Indeed if

U p 0 A = : `2 ⊕ `2 → `2 ⊕ `2, 0 U ∗q then A is Fredholm, α(A) = q, β(A) = p, and hence ind(A) = q − p.

By Theorem 1.16 we have the following important stability property of semi-Fredholm operators.

Theorem 2.2 Suppose that A ∈ Φ±(X,Y ). If B ∈ L(X,Y ) such that D(A) ⊂ D(B) and kBk < γ(A) then A + B ∈ Φ±(X,Y ) and

α(A + B) ≤ α(A), β(A + B) ≤ β(A), ind(A + B) = ind(A).

43 This theorem extended to the unbounded case by Theorem 1.17 in the following way

Theorem 2.3 Theorem 2.2 is true if B is A-bonded with a < (1 − b)γ(A) in (1.13).

The following theorem establishes an important characterization of Fredholm operators.

Theorem 2.4 (Atkinson characterization of Fredholm operators) A ∈ Φ(X,Y ) if and only there exist U1,U2 ∈ L(Y,X) and ﬁnite-dimensional operators K1 ∈ F(X),K2 ∈ F(Y ) such that U1A = I − K1 on D(A) and AU2 = I − K2 on Y. For a proof of this classical result we refer to [105]. It should be noted that in the characterization above the ideal F(X) may be replaced by the ideal K(X) of all compact operators. In particular, A ∈ Φb(X) if and only if A is invertible in L(X) modulo the ideal of ﬁnite-dimensional operators F(X).

The fact that K(X) is a closed two-sided ideal in L(X) enables us to deﬁne the Calkin algebra over X as the quotient algebra C(X) = L(X)/K(X) with the product

AbBb = AB,d where Ab is the coset A + K(X).

The space C(X) with this additional operation is a Banach algebra with the identity Iˆ and following the quotient algebra norm

kAke = infK∈K(X) kA + Kk . (2.1)

In particular, by Theorem 2.4 we have A ∈ Φb(X) if and only if A is invertible in C(X).

The classes of semi-Fredholm operator lead to the deﬁnition of the upper semi-Fredholm spectrum of an operator A on a Banach space X, deﬁned by

σuf (A) := {λ ∈ C : λI − A/∈ Φ+(X)}, and the lower semi-Fredholm spectrum of A deﬁned by

σlf (A) := {λ ∈ C : λI − A/∈ Φ−(X)}. The semi-Fredholm spectrum is deﬁned by

σsf (A) = {λ ∈ C : λI − A 6∈ Φ±(X)}, while the Fredholm spectrum is deﬁned by

σef (A) := {λ ∈ C : λI − A/∈ Φ(X)} Clearly, σsf (A) = σuf (A) ∩ σlf (A) and σef (A) = σuf (A) ∪ σlf (A).

The spectrum σef (A) in the literature is often called the essential spectrum or the Wolf essential spectrum of A [118, 101, 117]. σsf (.) is deﬁned by Kato [64]. The two spectra σuf (.) and σlf (.) are also known as the Gustafson and Weidmann essential spectra [55]. It is easy to ﬁnd an example of operator for which σuf (A) 6= σlf (A).

44 Example 2.2 Let A be deﬁned on `2 by

Ax = (x1, 0, x2, 0, x3, 0,... )

Obviously, A is injective with closed range of inﬁnite-codimension, so that 0 ∈ σlf (A) but 0 ∈/ σuf (A). Let A ∈ C(X). If the perturbation B in Theorem 2.2 is caused by a multiple of the identity we have the punctured neighborhood theorem: if A ∈ Φ+(X) then there exists > 0 such that λI − A ∈ Φ+(X) and α(λI − A) is constant on the punctured neighborhood 0 < |λ| < . Moreover, α(λI − A) ≤ α(A) for all |λ| < (2.2) and ind(λI − A) = ind(A) for all |λ| < .

Analogously, if A ∈ Φ−(X) then there exists > 0 such that λI − A ∈ Φ−(X) and β(λI − A) is constant on the punctured neighborhood 0 < |λ| < . Moreover,

β(λI − A) ≤ β(A) for all |λ| < (2.3) and ind(λI − A) = ind(A) for all |λ| < .

It follows that σlf (A), σuf (A), σsf (A) and σef (A) are closed sets of C ( compact sets if A ∈ L(X)). Moreover, the open set ρsf (A) = C\σsf (A), in general, is the union of countable number of connected open sets Ωn. Moreover, in each Ωn , with the possible exception of isolated points, both α(λI − A) and β(λI − A) are constant values αn and βn respectively. At the isolated points,

α(λnjI − A) = αn + r(λnj) and β(λnjI − A) = βn + r(λnj), 0 < r(λnj) < ∞

If αn = βn = 0, Ωn is a subset of ρ(A) except for the λnj, which are isolated eigenvalues of A with ﬁnite algebraic multiplicities r(λnj). in the general case ( in which one or both of αn, βn are positive, the λnj are also eigenvalues of A and behave like isolated eigenvalues , in the sense that their geometric multiplicities are larger by r(λnj) than other eigenvalues in their immediate neighborhood.

2.2 Fredholm perturbations

One of the most important question is the invariance of the essential spectra under additive perturbations. The ﬁrst result, in this context, is due a H. Weyl when he proved the stability of the essential spectrum of the self-adjoint operator under compact perturbation in Hilbert space. M. Schechter extends this result for bounded Fredholm operator in Banach space. Theorem 2.5 ([105]) If A ∈ Φ(X,Y ) and K ∈ K(X,Y ), then A + K ∈ Φ(X,Y ) and ind(A + K) = ind(A). Analog result for semi-Fredholm operators is Lemma 2.1 ([105]) Let K ∈ K(X,Y ). Then the following statements hold.

(i) If A ∈ Φ+(X,Y ), then A + K ∈ Φ+(X,Y ) and ind(A + K) = ind(A).

45 (ii) If A ∈ Φ−(X,Y ), then A + K ∈ Φ−(X,Y ) and ind(A + K) = ind(A). The next result, as consequence of Kato’s perturbation Theorem 1.18, shows the ﬁrst statement of th preceding lemma is not optimal and may be expressed in terms of strictly singular operators.

Theorem 2.6 If A ∈ Φ+(X,Y ) and K ∈ S(X,Y ), then A + K ∈ Φ+(X,Y ) and ind(A + K) = ind(A).

The purpose of this section is to describe the problem of Fredholm perturbations in a more general context. Let us introduce some notations and deﬁnitions. X Let X be a Banach space. If N is a closed subspace of X, we denote by πN the quotient map X → X/N. The codimension of N, codim(N), is deﬁned as the dimension of the vector space X/N.

Deﬁnition 2.2 Let X and Y be two Banach spaces and S ∈ L(X,Y ). S is said to be strictly cosingular operator from X into Y , if there exists no closed subspace N of Y with Y codim(N) = ∞ such that πN S : X → Y/N is surjective. Let CS(X,Y ) denote the set of strictly cosingular operators from X into Y . This class of operators was introduced by Pelczynski [93]. It forms a closed subspace of L(X,Y ) containing K(X,Y ) and CS(X) := CS(X,X), is a closed two-sided ideal of L(X) if X = Y (cf. [113]).

Deﬁnition 2.3 A Banach space X is said to have the Dunford-Pettis property (for short property DP) if for each Banach space Y every weakly compact operator A : X → Y takes weakly compact sets in X into norm compact sets of Y .

It is well known that any L1 space has the DP property [42]. Also, if Ω is a compact Hausdorff space, C(Ω) has the DP property [53]. For further examples we refer to [40] or [43, p. 494, 497, 508, 511]. Note that the DP property is not preserved under conjugation. However, if X is a Banach space whose dual has the DP property then X has the DP property (see [53]). For more information we refer to the paper by Diestel [40] which contains a survey and exposition of the Dunford-Pettis property and related topics. We say that X is weakly compactly generating (w.c.g.) if the linear span of some weakly compact subset is dense in X. For more details and results we refer to [40]. In particular, all separable and all reflexive Banach spaces are w.c.g. as well as L1(Ω, dµ) if (Ω, µ) is σ-finite. We say that X is subprojective, if given any closed infinite-dimensional subspace M of X, there exists a closed and finite dimensional subspace N ⊂ M and a continuous projection p from X onto N. Clearly any Hilbert space is subprojective. The spaces c0, l ,(1 ≤ p < ∞), and Lp (2 ≤ p < ∞), are also subprojective (cf. [116]). We say that X is superprojective if every subspace V having infinite codimension in X is contained in a closed subspace W having infinite codimension in X and such that there is a bounded projection from X to W . The spaces lp,(1 < p < ∞), and Lp (1 < p ≤ 2), are superprojective (cf. [116]).

Deﬁnition 2.4 Let X and Y be two Banach spaces and and A be any class of operators from X to Y . The perturbation of class A denoted by PA is the set

PA = {J ∈ L(X,Y ); A + J ∈ A for every A ∈ A}

46 The set of Fredholm perturbations is PΦb(X,Y ). This class of operators is introduced and investigated in [44]. In particular, it is shown that PΦb(X,Y ) is a closed subset of L(X,Y ) and if X = Y , then PΦb(X) = PΦb(X,X) is a closed two-sided ideal of L(X). The component PA(X,Y ) of the perturbation class PA has sense only when the set A(X,Y ) is non-empty. In the case A(X,Y ) = ∅. we could deﬁne PA(X,Y ) = L(X,Y ), the set of all operators from X to Y . However, this is not useful. p p p p p Indeed, for 1 < p < ∞, p 6= 2, both sets Φ+(L , ` ) and Φ−(` ,L ) are empty because L contains subspaces isomorphic to `2.

Proposition 2.1 ([44, pp. 69-70]) Let X, Y , Z be Banach spaces. If at least one of the sets Φb(X,Y ) or Φb(Y,Z) is not empty, then

(i) F ∈ PΦb(X,Y ), A ∈ L(Y,Z) imply AF ∈ PΦb(X,Z).

(ii) F ∈ PΦ(Y,Z), A ∈ L(X,Y ) imply FA ∈ PΦ(X,Z).

b The sets of upper semi-Fredholm and lower semi-Fredholm perturbations are PΦ+(X,Y ) b b b and PΦ−(X,Y ), respectively. In [44], it is shown that PΦ+(X,Y ) and PΦ−(X,Y ) are closed b b subsets of L(X,Y ), and if X = Y , then PΦ+(X) := PΦ+(X,X) is a closed two-sided ideal of L(X).

Lemma 2.2 ([59]) Let A ∈ C(X,Y ) and F ∈ L(X,Y ). Then

• (i) If A ∈ Φb(X,Y ) and F ∈ PΦb(X,Y ), then A + F ∈ Φb(X,Y ) and ind(A + F ) = ind(A).

b b b • (ii) If A ∈ Φ+(X,Y ) and F ∈ PΦ+(X,Y ), then A + F ∈ Φ+(X,Y ) and ind(A + F ) = ind(A).

b b b • (iii) If A ∈ Φ−(X,Y ) and F ∈ PΦ−(X,Y ), then A + F ∈ Φ−(X,Y ) and ind(A + F ) = ind(A).

The perturbation classes PΦ+ and PΦ− are unknown, in general. The mentioned results of Kato and Vladimirskii and the stability of the index of a semi-Fredholm operator under small perturbations (see [2, Theorem 3.6]) imply that, for every pair X,Y of Banach spaces for which the corresponding perturbation classes are deﬁned

b 1. K(X,Y ) ⊂ S(X,Y ) ⊂ PΦ+(X,Y ),

b 2. K(X,Y ) ⊂ CS(X,Y ) ⊂ PΦ−(X,Y ).

A counterexample was found in [46]: there exists a complex separable Banach space X b b ∗ ∗ such that PΦ+(X) 6= S(X) and PΦ−(X ) 6= CS(X ). However, the space X is very special: it is a ﬁnite product of hereditarily indecomposable spaces. The existence of hereditarily indecomposable Banach spaces was only recently proved, see [1]. So the problem remains open for many spaces, specially classical Banach spaces. b It is known that PΦ+(X,Y ) = S(X,Y ) in the following cases:

1. Y subprojective [79].

2. X is hereditarily indecomposable [1].

3. X is separable and Y contains a complemented copy of C[0, 1] [2].

47 4. X = Lp, 1 < p < 2 and Y satisﬁes the Orlicz property, when p = 1, Y is weakly sequentially complete, and when 2 ≤ p ≤ ∞, Y containing a subspace isomorphic to Lp [48]. X = Lp, Y = Lq, 1 ≤ q ≤ p < 2[48].

b Moreover, PΦ−(X,Y ) = CS(X,Y ) in the following cases: 1. X subprojective [79].

2. X is quotient indecomposable [1].

3. X is separable and Y contains a complemented copy of `1 and Y is separable [2].

4. Y = Lq, 2 < q < ∞, X containing a quotient isomorphic to Lq and X∗ satisﬁes the Orlicz property. [48].

5. Y = Lq, 1 ≤ q ≤ 2, X containing a a quotient isomorphic to Lq[48].

6. X = Lp, Y = Lq, 2 ≤ q ≤ p ≤ ∞ [48].

Furthermore, By the Milman-Weis theorem [114], we have

b p p b p p b p PΦ+(L ) = S(L ) = PΦ−(L ) = CS(L ) = PΦ (L ) (2.4) for p ∈ [1, ∞).

We observe that the perturbation classes PΦ+ and PΦ− studied in [115] correspond to not necessarily bounded upper and lower semi-Fredholm operators. Weis proved that

•P Φ+(X,Y ) = S(X,Y ) if Y is a w.c.g. and superprojective [115, 3.1 Theorem].

•P Φ−(X,Y ) = CS(X,Y ) if Y is a w.c.g. and subrprojective [115, 3.1 Theorem].

•P Φ+(X) = S(X) if every separable subspace of X is contained in a weakly compactly generated and complemented subspace of X [115, 3.2 Corollary].

•P Φ−(X) = CS(X) if every, inﬁnite dimensional quotient space of the Banach space X has an inﬁnite dimensional separable quotient space [115, 3.7 Corollary].

K. Latrach and A. Dehici proved [74, Proposition 3.4] that, If X is a w.c. g. Banach space, then

• If X is superprojective, then S(X) ⊂ PΦ+(X) ∩ PΦ−(X)

• If X is subprojective, then CS(X) ⊂ PΦ+(X) ∩ PΦ−(X)). Let J be a linear operator on X. If D(A) ⊂ D(J), then J will be called A−deﬁned. If J is A−deﬁned operator, we will denote by Jˆ the restriction of J to D(A). Moreover, ˆ if J ∈ L(XA,X) we say that J is A−bounded. One checks easily that if J is closed (or closable). (cf.[64, Remark 1.5, p. 191]), then J is A−bounded.

Let X be a Banach space and let A ∈ C(X). As mentioned above, D(A) provided with the graph norm is a Banach space denoted by XA. Let J be an arbitrary A-bounded operator.

48 ˆ Hence we can regard A and J as operators from XA into X. They will be denoted by A and ˆ J, respectively. These belong to L(XA,X). Furthermore, we have the obvious relations

 α(Aˆ) = α(A), β(Aˆ) = β(A),R(Aˆ) = R(A)  α(Aˆ + Jˆ) = α(A + J) (2.5)  β(Aˆ + Jˆ) = β(A + J), and R(Aˆ + Jˆ) = R(A + J)

In the next, the following lemmas describe some properties of the sets PΦ(X), PΦ+(X), and PΦ−(X). Lemma 2.3 ([74]) Let X be a Banach space. Then

PΦ(X) = PΦb(X).

Proof. Clearly PΦ(X) ⊆ PΦb(X). To prove the oppsite inclusion, let J ∈ PΦb(X). If A ∈ Φ(X) then by Theorem 2.4 there exists A0 ∈ L(X) and F0 ∈ F(X) such that

AA0 = I + F0 on X. (2.6)

This implies that AA0 is a Fredholm operator. The fact that A ∈ Φ(X) implies that ˆ b b A ∈ Φ (XA,X). Applying again the Theorem 2.4 we obtain that A0 ∈ Φ (X,XA). On the other hand, we have (A + J)A0 = I + F0 + JA0 = I + J0. (2.7) This and the fact that PΦb(X) is a closed two sided ideal of L(X) containing F(X) implies b b ˆ ˆ b that J0 ∈ Φ (X), we conclude that (A + J)A0 ∈ Φ (X), it follows that A + J ∈ Φ (XA,X) b because A0 ∈ Φ (X,XA). Now by (2.5), we have A+J ∈ Φ(X). This shows that J ∈ PΦ(X).

Remark 2.1 1. An immediate consequence of the result of Lemma 2.3 is that PΦ(X) is a closed two-sided ideal of L(X).

2. Let X and Y be two Banach spaces. In contrast to the result of Lemma 2.3, the fact that PΦ(X,Y ) is equal or not to PΦb(X,Y ) seems to be unknown. Moreover, b b whether or not PΦ+(X) (resp.PΦ−(X)) is equal to PΦ+(X) (resp. PΦ−(X)) seems to be unknown.

3. In general, we have the following inclusions:

K(X) ⊂ S(X) ⊂ PΦ+(X) ⊂ PΦ(X),

K(X) ⊂ CS(X) ⊂ PΦ−(X) ⊂ PΦ(X)

4. By Lemma 2.3 and (2.4), we have

p p p p b p PΦ+(L ) = S(L ) = PΦ−(L ) = CS(L ) = PΦ (L ) (2.8)

for p ∈ [1, ∞).

Lemma 2.4 ([74]) Let X be a Banach space. Then PΦ+(X) and PΦ−(X) are closed subsets of L(X).

49 Proof. Let (Jn) be a sequence of operators of PΦ+(X) (resp. PΦ−(X)) such that (Jn) converges to J in L(X). If A ∈ Φ+(X) (resp. Φ−(X)), then for n suﬃciently large, applying Theorem 2.2 we get A − (Jn − J) ∈ Φ+(X) (resp. Φ−(X)). Next, using the relation A+J = A−(Jn +J)−Jn, together with the fact Jn ∈ PΦ+(X) (resp. PΦ−(X)) we conclude that J ∈ PΦ+(X) (resp. PΦ−(X)). Lemma 2.5 ([74]) Let J ∈ L(X). Then the following statements hold.

(i) J ∈ PΦ+(X) if and only if α(A + J) < ∞ for each A ∈ Φ+(X).

(ii) J ∈ PΦ−(X) if and only if β(A + J) < ∞ for each A ∈ Φ−(X). (ii) J ∈ PΦ(X) if and only if either α(A + J) < ∞ or β(A + J) < ∞ for each A ∈ Φ(X).

Proof. (i) Let J ∈ PΦ+(X) and let A ∈ Φ+(X). Then A + J ∈ Φ+(X) and consequently α(A + J) < ∞. Conversely, assume that J/∈ PΦ+(X). Then there exists A ∈ Φ+(X) such ˆ ˆ b that A + J/∈ Φ+(X). Therefore, A + J/∈ Φ+(XA,X). Next, applying [79, Lemma 4.3] we ˆ infer that there exists an operator K such that K ∈ K(XA,X) (i.e., K is A-compact) and α(Aˆ + Jˆ + Kˆ ) = ∞. By using (2.5) we have α(A + J + K) = ∞. On the other hand, by Lemma 2.1 we have A + K ∈ Φ+(X), and therefore α(A + J + K) < ∞. This contradicts the fact that α(A + J + K) = ∞. (ii) by the same way as above; it suﬃces to replace [79, Lemma 4.3] by [79, Lemma 5.1]. (iii) Let J ∈ PΦ(X). Hence, for each A ∈ Φ(X), α(A + J) < ∞ and β(A + J) < ∞. 1 Conversely, suppose that α(A + J) < ∞ for each A ∈ Φ(X). By 2.5, µ (A + K) ∈ Φ(X) for each K ∈ K(X) and µ an arbitrary nonzero complex number. Hence α(A + µJ + K) is ﬁnite for all scalar µ. Thus by lemma 2.1 (i), we see that A + µJ ∈ Φ+(X). Now arguing as in the proof of [45, Theorem 2.1, p.117] and using the compactness of the interval [0, 1] we obtain β(A + J) ≤ β(A). Since β(a) < ∞, we get β(A + J) < ∞. Consequently, A + J ∈ Φ(X). This show that J ∈ PΦ(X). If β(A + J) < ∞ for all A ∈ Φ(X), a similar proof as above using [45, Theorem 2.1, p.117] shows that α(A + J) < ∞ for all A ∈ Φ(X) which implies that J ∈ PΦ(X).

The following results generalizes many known perturbation results in the literature.

Proposition 2.2 ([74]) Let A ∈ C(X) and let I(X) be any nonzero ideal of L(X) satisfying

I(X) ⊆ PΦ(X). (2.9)

If J ∈ I(X), then

(i) if A ∈ Φ(X), then A + J ∈ Φ(X) and ind(A + J) = ind(A). Moreover,

(ii) if A ∈ Φ−(X) and I(X) ⊆ PΦ+(X), then A + J ∈ Φ+(X);

∗ ∗ (iii) if I(X) ⊂ PΦ−(X) or [I(X)] ⊂ PΦ+(X ), then A + J ∈ Φ−(X) for all A ∈ Φ−(X);

(iv) if A ∈ Φ±(X) and I(X) ⊆ PΦ+(X) ∩ PΦ−(X), then A + J ∈ Φ±(X).

Proof. (i). Let A ∈ Φ(X) and J ∈ I(X). Immediately, we have A + J ∈ Φ(X), then there exists A0 ∈ L(X) and F ∈ F(X) such that

AA0 = I + F on X. (2.10)

50 Thus (A + J)A0 = I + F + JA0 = I + K. (2.11) Since I(X) is a closed two sided ideal containing F(X), we have K ∈ I(X) ⊆ PΦ(X). Then b (2.11) implies that AA0 and (A + J)A0 are in Φ (X) and

ind((A + J)A0) = ind(AA0). (2.12)

b On the other hand, proceeding as in the proof of Lemma 2.3 we see that A0 ∈ Φ (X,XA) ˆ ˆ b and A + J ∈ Φ (XA,X). Next, applying Atkinson theorem to both AA0 and (A + J)A0 and ˆ ˆ ˆ using (2.12 we obtain ind(A) = −ind(A0) and ind(A + J) = −ind(A0) which implies that ind(Aˆ) = ind(Aˆ + Jˆ). Now by (2.5), we have A + J ∈ Φ(X) and ind(A) = ind(A + J). The statement (ii), the ﬁrst part of (iii) and (iv) are trivial. The second part of (iii) ∗ ∗ may be checked as follows. Let A ∈ Φ−(X), then A ∈ Φ+(X ). Moreover, the inclusion ∗ ∗ ∗ ∗ ∗ [I(X)] ⊂ PΦ+(X ) shows that A + J ∈ Φ+(X ). Next, this together with the fact that ∗ ∗ α(A + J ) = β(A + J), implies that A + J ∈ Φ−(X). In the following we give some examples of I(X) satisﬁes the hypothesis (2.9) for which the results of Proposition 2.2 are valid:

1. If I(X) satisﬁes the hypothesis (2.9), then F(X) ⊆ I(X). Hence the ideal of ﬁnite rank operators is the minimal subset of L(X) in the sense of the inclusion.

2. Let A ∈ C(X) and assume that X has the property D P. In this case we take I(X) = Θ(X), where Θ(X) is the ideal of weakly compact operators [73].

3. if X is a w.c.g Banach space, then I(X) = S(X) (resp. I(X) = CS(X)) then only the assertions (i) and (iv) (resp. (ii)) and (iv)) of Proposition 2.2 are valid.

4. if X is w.c.g and superprojective (resp. subprojective) then, for I(X) = S(X) (resp. I(X) = CS(X)), the statements of Proposition 2.2 hold true.

5. Let (Ω, Σ, µ) be a positive measure space. Since p ∈ [1, ∞) the spaces Lp(Ω, dµ) are p p p p p w.c.g., consequently we have PΦ+(L ) = S(L ) and PΦ−(L ) = CS(L ). In L (Ω, dµ) and in C(E) (the Banach space of continuous scalar-valued function on E with the supremum norm) provided that E is a compact Hausdorﬀ space we have a stronger result, namely that S(C(E)) = CS(C(E)) = PΦ(C(E)) (See. [76]).

6. In [116], Whitley proved that if X is an h-space2, then S(X) is the greatest proper ideal of L(X). This together with Remark 2.1 implies that

K(X) ⊆ PΦ+(X) = PΦ(X) = S(X)

K(X) ⊆ PΦ−(X) = PΦ(X) = CS(X)

7. In the following case I(X) = K(X) is the unique proper nonzero closed two-sided ideal of L(X) and K(X) = PΦ+(X) = PΦ−(X) = PΦ(X)

(a) X is a separable Hilbert space. See Calkin [32]. p (b) X = ` , 1 ≤ p < ∞, and X = c0. See Gohberg and al [44].

2A Banach space X is said to be an h-space if each closed inﬁnite dimensional subspace of X contains a complemented subspace isomorphic to X. Any Banach space isomorphic to an h-space is an h-space

51 (c) In [57] Herman established this result for a large class of Banach spaces which have perfectly homogeneous block bases and satisfy certain conditions, for the deﬁnition and more information about these spaces we refer to [57]. (For example the spaces in (b) belong to this class).

As consequence of the Proposition 2.2, the following important stability theorem of the essential Fredholm spectra. Theorem 2.7 Let A ∈ C(X) and let I(X) be any nonzero ideal of L(X) satisfying

I(X) ⊆ PΦ(X).

(i) If J ∈ I(X), then σef (A) = σef (A + J). Further,

(ii) if I(X) ⊆ PΦ+(X), then

σuf (A) = σuf (A + J) for all J ∈ I(X)

∗ ∗ (iii) if I(X) ⊂ PΦ−(X) or [I(X)] ⊂ PΦ+(X ), then

σlf (A) = σlf (A + J) for all J ∈ I(X);

(iv) if I(X) ⊆ PΦ+(X) ∩ PΦ−(X), then

σsf (A) = σsf (A + J) for all J ∈ I(X).

Example 2.3 ,Consider the space X = `p, 1 ≤ p ≤ ∞, and the operators deﬁned in Example 1.12, as follows

A0x = (0, x1, x2, x3,... )

A1x = (x2, x3, x4,... ) 1 1 A x = (x , x , x ,... ) 2 1 2 2 3 3 1 1 A x = (x , x , x ,... ) 3 2 2 3 3 4 1 1 A x = (0, x , x , x ,... ) 4 1 2 2 3 3

We can see that N(A1) consists of thos elements of the form

(x1, 0,...).

p and hence, α(A1) = 1. Moreovere, R(A1) = ` = N(A1)⊕X0, where X0 is the closed suspace consisting of the elements of the form

(0, x2, x3,...).

so that ind(A1) = 1. The operaor A0 is a Fredholm operator with ind(A0) = −1 because A0 is injective and R(A0) = X0. Since the operators A2, A3 and A4 are compact, it follows therefore, that A0 +Ai and A1 +Ai are Fredholm operators with ind(A0 +Ai) = ind(A0) = −1 and ind(A1 +Ai) = ind(A1) = 1, i = 2, 3, 4.

52 The following deﬁnition gives the concept of inessential operators to operators acting between diﬀerent spaces.

Deﬁnition 2.5 An operator A ∈ L(X,Y ) is said to be an inessential operator if IX − SA ∈ Φ(X) for all S ∈ L(X,Y ). The class of all inessential operators is denoted by I(X,Y ).

Theorem 2.8 ([1, pp. 371]) I(X,Y ) is a closed subspace of L(X,Y ) which contains K(X,Y ). Moreover, if T ∈ I(X,Y ), R1 ∈ L(Y,Z), and R2 ∈ L(W, X), where X, Y , W and Z are Banach spaces, then R1TR2 ∈ I(W, Z).

b b In general, for every pair X,Y of Banach spaces for which PΦ−(X,Y ) and PΦ−(X,Y ) are deﬁned, we have

b 1. K(X,Y ) ⊂ S(X,Y ) ⊂ PΦ+(X,Y ) ⊂ I(X,Y ),

b 2. K(X,Y ) ⊂ CS(X,Y ) ⊂ PΦ−(X,Y ) ⊂ I(X,Y ).

Theorem 2.9 ([1, pp. 380]) If I(X,Y ) is not empty, then PΦ(X,Y ) = I(X,Y ).

In the following case we have I(X,Y ) = L(X,Y ) (for more details see Aiena [1, pp. 372-373]): (a) X is reﬂexive and Y has the Dunford-Pettis property. Recall that an operator A is said to be completely continuous if A transforms relatively weakly compact sets into relatively compact sets. Note that if X or Y is reﬂexive then every A ∈ L(X,Y ) is weakly compact, see Goldberg [45]. A Banach space X has the Dunford-Pettis property if any weakly compact operator A from X into another Banach spaces Y is completely continuous.

(b) X has the reciprocal Dunford-Pettis property and Y has the Schur property. Recall that X is said to have the reciprocal Dunford-Pettis property if every completely continuous operator from X into any Banach spaces is weakly compact, whilst Y has the Schur property if the identity IY is completely continuous. (c) X contains no copies of `∞ and Y = `∞, H∞(D), or a C(K), with K σ-Stonian.

(d) X contains no copies of c0 and Y = C(K).

(e) X contains no complemented copies of c0 and Y = C(K), or X contains no complemented copies of `1 and Y = L1.

1 (f) X or Y are ` with 1 ≤ p ≤ ∞ or c0, and X,Y are diﬀerent. Another important class in connection with the Fredholm perturbation and not necessary an ideal is the family of Riesz operators (see [34]).

Deﬁnition 2.6 (Riesz operators) Let X be a Banach space and R ∈ L(X). R is said to be a Riesz operator if λ − R ∈ Φ(X) for all scalars λ ∈ C \{0}. We denote by R(X) the class of all Riesz operators. We have the following characterization of Riesz operators. Theorem 2.10 A ∈ R(X) if and only if each λ ∈ σ(A) \{0} is an isolated point of σ(A) and Pλ ∈ F(X).

53 Riesz operators are a generalization of compact and strictly singular operators and exhibit many of their properties. In [104], it is proved that PΦb(X) is the largest ideal of L(X) contained in the class of Riesz operators. Hence by Lemma 2.3 PΦ(X) is the largest ideal contained in R(X). Moreover, In [34], Cardus proved that every inessential operator lies in R(X). For further information on the family of Riesz operators we refer to [34, 62] and the references therein. Now, we state the result of the Riesz perturbation of the Fredholm operators. Theorem 2.11 ([121]) Let A ∈ L(X) and R ∈ R(X).

(i) If A ∈ Φ+(X) and AR − RA ∈ PΦ+(X), then A + R ∈ Φ+(X) and ind(A + R) = ind(A).

(ii) If A ∈ Φ−(X) and AR − RA ∈ PΦ−(X), then A + R ∈ Φ−(X) and ind(A + R) = ind(A).

2.3 Browder and Weyl spectra

An important classes of operators in Fredholm theory are given by the classes of semi- Fredholm operators which possess finite ascent or finite descent or positive finite index or negative finite index. We shall distinguish the following classes of operators:

The class of all upper semi-Browder operators on a Banach space X that is deﬁned by

B+(X) := {A ∈ Φ+(X): a(A) < ∞}, the class of all lower semi-Browder operators that is deﬁned by

B−(X) := {A ∈ Φ−(X): d(A) < ∞}, the class of all Browder operators3 is deﬁned by

B(X) := B+(X) ∩ B−(X) = {A ∈ Φ(X): a(A), d(A) < ∞}, the set of upper semi-Weyl operators is deﬁned by

W+(X) := {A ∈ Φ+(X): ind(A) ≤ 0}, the set of lower semi-Weyl operators is deﬁned by

W−(X) := {A ∈ Φ−(X): ind(A) ≥ 0}, and the set of Weyl operators is deﬁned by

W(X) := W+(X) ∩ W−(X) = {A ∈ Φ(X): ind(A) = 0}, There exists a Weyl operator which is not Browder. Example 2.4 Put U 0 A = : `2 ⊕ `2 → `2 ⊕ `2, 0 U ∗ where U is the unilateral shift. Evidently, A is Fredholm and ind(A) = ind(U)+ind(U ∗) = 0, which says that A is Weyl. However, σ(A) = {λ ∈ C : |λ| ≤ 1}; so that 0 is not isolated in σ(A), which implies that A is not Browder.

3known in the literature also as Riesz Schauder operators

54 Lemma 2.6 Let A ∈ Φ+(X) . Then the following statements are equivalent 1. ind(A) ≤ 0 2. A can be expressed in the form A = U +K where K ∈ K(X) and U ∈ C(X) an operator bounded below. The various classes of operators defined above motivate the definition of several essential spectra: • The upper semi-Browder spectrum is defined by

σub(A) := {λ ∈ C : λI − A/∈ B+(X)}. • The lower semi-Browder spectrum is deﬁned by

σlb(A) := {λ ∈ C : λI − A/∈ B−(X)} • The Browder spectrum is deﬁned by

σeb(A) := {λ ∈ C : λI − A/∈ B(X)} = σub(A) ∪ σlb(A) • The upper semi-Weyl spectrum is deﬁned by

σuw(A) := {λ ∈ C : λI − A 6∈ W+(X)} • The lower semi-Weyl spectrum is deﬁned by

σlw(A) := {λ ∈ C : λI − A 6∈ W−(X)} • The Weyl spectrum is deﬁned by

σew(A) := {λ ∈ C : λI − A 6∈ W(X)} = σuw(A) ∪ σlw(A)

The set σew(.) known in the literature also as the Schechter essential spectrum [55, 101, 103], 4 and σeb(.) the Browder essential spectrum [31, 55, 62, 101]. σuw(.) and σlw(.) are the essential approximate point spectrum and the essential defect spectrum [94] respectively . The subsets σub(.) and σlb(.) was introduced by Rakočević in [97, 95] and are also another essential version of the approximate spectrum and the defect spectrum respectively. Note that all these sets of essential spectra are closed and in general satisfy the following inclusions

σsf (A) ⊆ σef (A) ⊆ σew(A) ⊆ σeb(A) = σef (A) ∪ accσ(A); (2.13)

σsf (A) ⊆ σuf (A) ⊆ σuw(A) ⊆ σub(A) ⊆ σeb(A); (2.14) and σsf (A) ⊆ σlf (A) ⊆ σlw(A) ⊆ σlb(A) ⊆ σeb(A). (2.15) In particular, if A is a self-adjoint operator acting on Hilbert space, then

σec(A) = σsf (A) = σef (A) = σew(A) = σeb(A), (2.16) if A is closed densely deﬁned and symmetric acting on Hilbert space, then

σec(A) ⊆ σuw(A) ⊆ R, (2.17) and if A is a compact operator in Banach space, then

σec(A) = σsf (A) = σef (A) = σew(A) = σeb(A) = {0}. (2.18) Also the following results follows by duality. 4Felix E. Browder ( July 31 1927) is a United States mathematician.

55 Theorem 2.12 Let A ∈ C(X). Then we have: ∗ ∗ 1. σeb(A) = σeb(A ) and σew(A) = σew(A )

∗ ∗ 2. σub(A) = σlb(A ) and σuw(A) = σlw(A )

∗ ∗ 3. σlb(A) = σub(A ) and σlw(A) = σuw(A ) All essential spectra introduced so far are closed subsets of σ(A), hence compact if A is bounded. They are empty if the underlying Banach space is ﬁnite dimensional. Moreover, for a self-adjoint operator they all coincide. In the following we recall some relations between these subsets. Theorem 2.13 Let A ∈ C(X) with ρ(A) 6= ∅, we have

1. If ρef (A) is connected set then

σef (A) = σew(A), σuf (A) = σuw(A), and σlf (A) = σlw(A);

2. If ρew(A) is connected set then

σeb(A) = σew(A).

Proof. (1). To shows that σef (A) = σew(A), it remains to show σef (A) ⊆ σew(A). Suppose

that ρef (A) ∩ σew (A) 6= ∅, then there exists λ0 ∈ ρef (A) ∩ σew (A), since ρ(A) 6= ∅, there exists λ1 ∈ ρ(A) with λ1 − A ∈ Φ(X) and i(λ1 − A) = 0. On other hand ρef (A) is connected, it follows by Proposition that ind(λ0 − A) = ind(λ1 − A) = 0, hence λ0 ∈/ σew(A). This contradict the assumption. So ρef (A) ∩ σew(A) = ∅ and the first equality hold. For the two last equalities it suffices to shows that σuw(A) ⊆ σuf (A). (resp. σlf (A) ⊆ σlw(A)). We have tow cases. First if λ ∈ (ρuf (A) \ ρef (A)), (resp. λ ∈ (ρlf (A) \ ρef (A)), ) then α(A − λ) < ∞ and β(A − λ) = +∞, (resp. α(A − λ) = +∞ and β(A − λ) < ∞, ), hence ind(A − λ) = −∞ < 0. (resp. ind(A − λ) = +∞ > 0.) and λ ∈ ρuw(A), (resp. λ ∈ ρlw(A)). secondly if λ ∈ ρef (A), since ρef (A) we can find λ0 ∈ ρ(A) such that ind(λ0 − A) = ind(λ − A) = 0. so λ ∈ ρuw(A) ∩ ρlw(A).

2. We have σew(A) ⊂ σeb(A), it suﬃces to shows σeb (A) ⊂ σew (A). Suppose that ρew (A) ∩

σeb (A) 6= ∅, then there exists λ0 ∈ ρew (A) ∩ σeb (A). Since ρew (A) is connected then ind(λ0 − A) = ind(λ − A) = 0 for some λ ∈ ρ(A), hence λ0 ∈/ σeb(A). Proposition 2.3 Let A ∈ C(X) and let I(X) be any nonzero ideal of L(X) satisfying I(X) ⊆ PΦ(X).

Then σew(A) = σew(A + J), for all J ∈ I(X). Moreover, if ρew(A) is connected and neither ρ(A) nor ρ(A + J) is empty, then

σeb(A) = σeb(A + J) for all J ∈ I(X) Proof. Is immediate by using the Proposition 2.2. Example 2.5 Let X = `p, 1 ≤ p ≤ ∞, and consider the operator

Ax = (0, x3, x2, x5, x4, x7, x6,... )

We have N(A) = {(x1, 0,... )} and R(A) = {(0, x1, x2, x3, x4,... )} and ind(A) = 0. So A is a Weyl operator. we know that the operator K given by 1 1 Kx = (x , x , x ,... ) 1 2 2 3 3 is compact, then the operaor A + K is a Weyl operator and

σew(A + K) = σew(A).

56 2.3.1 The Browder resolvent

The discrete spectrum of A, denoted σdis(A) the set of isolated points λ ∈ C of the spectrum such that the corresponding Riesz projectors Pλ(A) are finite dimensional. By Corollary 1.3 the points of σdis(A) being poles of finite rank, i.e., around each of these points there is a punctured disk in which the resolvent has a Laurent expansion (1.35) whose principal part has only finitely many nonzero terms, the coefficients in these being of finite rank. It follows that from the definition the Browder essential spectrum and Theorem 1.27 that

σeb(A) = σ(A) \ σdis(A).

Historically, F. E. Browder, see Definition 11 on page 107 in [31], has defined σeb(A), to be the set of complex numbers λ such that at least one of the following conditions is satisfied: 1. R(λI − A) is not closed. 2. N ∞(A) is of infinite dimension. 3. The point λ is a limit point of the spectrum of A. This set have been investigated extensively by many authors. In the following we give an equivalent definitions to the Browder essential spectrum.

Theorem 2.14 Let A ∈ C(X) and λ0 be a point of the spectrum of A such that R(λ0I − A) is closed. Then the following statements are equivalents

1. λ0 ∈/ σeb(A), (hence λ0 ∈ σdis(A)).

2. λ0 is a pole of the resolvent of order m.

3. a(λ0I − A) = d(λ0I − A) = m < ∞.

4. α(λ0I − A) = β(λ0I − A) < ∞ and d(λ0I − A) < ∞.

5. α(λ0I − A) = β(λ0I − A) < ∞ and a(λ0I − A) < ∞.

6. The operator D = (λ0I − A)Pλ0 is a nilpotent operator of order m, with Pλ0 is the Riesz projection associated to the point λ0.

n n 7. X = N((λ0I − A) ) ⊕ R((λ0I − A) ), for all n ≥ m.

8. There is a punctured disk around λ0 in which the resolvent has a the Laurent expansion:

m−1 ∞ 1 X 1 X (λI − A)−1 = P + Dn − (λ − λ )nSn+1; (2.19) λ − λ λ0 (λ − λ )n+1 0 0 n=1 0 n=0

1 R −1 −1 with D = (λ0I − A)Pλ0 , S = − (λ − λ0) (λI − A) dλ and Γλ0 is a positively 2πi Γλ0 oriented small circle enclosing λ0 but no other point of σ(A).

Denotes by ρB(A) := C \ σeb(A) the Browder resolvent set is the largest open set on which the resolvent is ﬁnitely meromorphic.

For λ ∈ ρB(A), let Pλ be the corresponding ﬁnite rank Riesz projector. From the fact that D(A) is Pλ−invariant, we may deﬁne the operator

Aλ = (λ − A)(I − Pλ) + Pλ

57 with domain D(A) or, with respect to the decomposition X = N(Pλ) ⊕ R(Pλ);

Aλ = (λ − A |N(Pλ)) ⊕ I

Since σ(Aλ) = σ((λ − A)(I − Pλ)) = σ(λ − A) \{0}, Aλ has a bounded inverse which we denote by RB(λ, A) and called the Browder resolvent, i.e.,

−1 RB(λ, A) = ((λ − A) |N(Pλ)) (I − Pλ) + Pλ.

−1 Clearly RB(λ, A) = (λ − A) , for λ ∈ ρ(A) and RB(A, λ) may be viewed as an extension of the usual resolvent from ρ(A) to ρB(A) and retains many of its important properties. For example, because PλAλ = Pλ on D(A) and AλPλ = Pλ on X it follows that PλRB(λ, A) = Pλ = RB(λ, A)Pλ, and we also have the following version of the resolvent identity for RB(λ, A).

Lemma 2.7 ([80]) Let A ∈ C(X). For λ, µ ∈ ρb(A),

RB(λ, A) − RB(µ, A) = (µ − λ)RB(λ, A)RB(µ, A) + MA(λ, µ). (2.20) where MA(λ, µ) = RB(λ, A) [(λ − 1 − A)Pλ − (µ − 1 − A)Pµ] RB(µ, A) is a ﬁnite rank operator with dim R(MA(λ, µ)) = dim(R(Pλ)) + dim(R(Pµ)). Furthermore, in case λ 6= µ the Browder resolvents commute; hence, the function MA(λ, µ) is skew-symmetric, i.e., MA(λ, µ) = −MA(µ, λ).

• If A ∈ R(X), then σef (A) = σew(A) = σeb(A) = {0} and RB(λ, A) exists for all λ 6= 0.

2.3.2 The essential spectral radius In analogy to the radius (1.19) of the whole spectrum, let us consider the radii of the essential spectrum re(A) := sup {|λ| ; λ ∈ σef (A)} . (2.21) The following theorem shows that, although the various essential spectra may be diﬀerent, they all have the same size.

Theorem 2.15 If A ∈ L(X), then

re(A) := sup {|λ| ; λ ∈ σk(A)} , k ∈ {sf, lf, uf, ef, ew, uw, lw, eb, ub, lb}.

Proof. By (2.13), (2.14) and (2.15) it suﬃces to show that reb ≤ rsf (A). Let C be the unbounded connected component of ρeb(A). Since C \ C is compact, we ﬁnd λ0 ∈ C \ C such that |λ0| = max {|λ| ; λ ∈ C \ C} .

But λ0 ∈ ∂ [C \ C] implies that λ0 ∈/ σsf (A), and therefore |λ0| ≤ rsf .

Interestingly, the radii (2.21) of the various essential spectra do not only coincide, but also satisfy a Gel’fand-type formula with the norm kAk replaced by the norm (2.1). In fact, if π denote the natural homomorphism of L(X) onto C(X); π(A) = A + K(X), A ∈ L(X). Then the essential spectral radius of A is given by

1 1 n n n n re(A) = r(π(A)) = lim kA ke = inf kA ke . n→+∞ n∈N

58 2.4 Characterizations of the essential spectra

In the following we give some characterizations of the essential spectra based in the Fredholm perturbations (not necessary an ideals) and considered as a useful equivalent deﬁnitions of this essential spectra. Theorem 2.16 ([17]) Let A ∈ C(X). Then \ σew(A) = σ(A + K) K∈K(X) \ = σ(A + F ) F ∈F(X)

Proof. Let λ∈ / σew(A), without loss of generality, we assume λ = 0. Then A ∈ W(X) and α(A) = β(A) = n. Let P denote the projection of X onto the finite-dimensional space N(A). We have, N(A) ∩ N(P ) = {0} and we can represent the finite rank operator P in the form i=1 X P x = fi(x)xi, n ∗ where the vectors x1, . . . , xn from X, the vectors f1, . . . , fn from X are linearly independent. The set {x1, . . . , xn} forms a basis of R(P ) and P xi = xi for every i = 1, . . . , n, from which we obtain that fi(xj) = δi,j, where δi,j denote the delta of Kronecker. Denote M the topological complement of R(A). Then dim M = n, so we can choose a basis {y1, . . . , yn} of M. Set i=1 X F x = fi(x)yi. n Then F is an operator of finite rank, by Theorem 2.7, A + F ∈ W(X). Now, let x ∈ N(A + F ), then Ax = F x = 0, and this easily implies that fi(x) = 0 for all i = 1, . . . , n. From this it follows that P x = 0 and therefore x ∈ N(A)∩N(P ) = {0}, so A+F is injective. Thus α(A+F ) = 0, and hence β(A+F ) = 0, so R(A+F ) = X. Therefore A+F is invertible. T this show F ∈F(X) σ(A + F ) ⊆ σew(A) and the fact that F(X) ⊆ K(X) we have \ \ σ(A + K) ⊆ σ(A + F ) ⊆ σew(A) K∈K(X) F ∈F(X) Now, suppose that A + K is invertible with K is compact, obviously A + K ∈ W(X), and hence by Theorem 2.7 we conclude that A ∈ W(X). Theorem 2.17 ([74]) Let A ∈ C(X). Then \ σew(A) = σ(A + K) (2.22) K∈M(X) where M(X) be any subset of L(X) satisfying K(X) ⊆ M(X) ⊆ PΦ(X).

T Proof. Set Σ = K∈M(X) σ(A + K). We ﬁrst claim that σew(A) ⊂ Σ. Indeed, if λ∈ / Σ, then there exists K ∈ M(X) such that λ ∈ ρ(A + K). Hence λI − A − K ∈ Φ(X) and ind(λI − A − K) = 0. Since (λI − A − K)−1 ∈ L(X) we have (λI − A − K)−1K ∈ M(X). Therefore Proposition 2.2 proves that I + (λI − A − K)−1K ∈ W(X). Next, using the relation λI − A = (λI − A − K)(I + (λI − A − K)−1K together with Atkinsons theorem we get λI − A is a Weyl operator. This shows that λ∈ / σew(A). The opposite inclusion follows from K(X) ⊆ M(X).

59 Theorem 2.18 ([59]) Let A ∈ C(X). Then \ σuw(A) = σap(A + K). (2.23) K∈K(X)

Proof. Let λ∈ / σuw(A), then λI − A ∈ Φ+(X) with ind(A) ≤ 0. Then by Lemma 2.6, λI − A can be expressed in the form λI − A = U + K where K ∈ K(X) and U ∈ C(X) an operator bounded below. Hence by (1.12) there exists c > 0 such that kUxk ≥ ckxk for all T x ∈ D(A). Thus λ∈ / σap(A + K) and therefore λ∈ / K∈K(X) σap(A + K). T Conversely, if λ∈ / K∈K(X) σap(A + K), then there exists K ∈ K(X) such that (λI − A − K) is injective with closed range, hence (λI − A − K) ∈ Φ+(X) and it follows from Proposition 2.2 that λI − A ∈ Φ+(X) with ind(A) ≤ 0. This completes the proof. Theorem 2.19 ([59]) Let A ∈ C(X). Then \ σlw(A) = σsu(A + K). (2.24) K∈K(X)

Theorem 2.20 ([59]) Let A ∈ C(X) with nonempty resolvent set. Then \ σuw(A) = σap(A + K). (2.25)

K∈∆A(X)

−1 where ∆A = {K ∈ C(X),K is A − bounded and K(µI − A) ∈ PΦ+(X) for some µ ∈ ρ(A)}.

Proof. Since K(X) ⊂ ∆ (X), we refer that T σ (A + K) ⊂ σ (A). Conversely, A K∈∆A(X) ap uw suppose that there exists K ∈ ∆A(X) such that λI − A − K is bounded below, hence λI − A − K ∈ Φ+(X). Since Y = R(λI − A − K) is closed subspace of X, then Y itself is a ˆ ˆ −1 Banach space with the same norm. Therefore (λ − A − K) ∈ L(Y,XA). let µ ∈ ρ(A) such −1 b that K(µ − A) ∈ PΦ+(X), then we have

Kˆ (λ − Aˆ − Kˆ )−1 = [J + (µ − λ + Kˆ )(λ − Aˆ − Kˆ )−1] (2.26)

where J denotes the embedding operator which maps every x ∈ Y on to the same element ˆ ˆ ˆ −1 b in X. Since (λ − A − K) ∈ L(XA,X) and K(λ − A) ∈ PΦ+(X), then it follows from [44, p. 70] and Eq. (2.26) that

ˆ ˆ ˆ −1 b K(λ − A − K) ∈ PΦ+(Y,X) (2.27)

b clearly, J is injective and R(J) = Y . So, J ∈ Φ+(Y,X) and ind(J) ≤ 0. Therefore, we can deduce from (2.27) and Lemma 2.2 that

ˆ ˆ ˆ −1 b ˆ ˆ ˆ −1 J + K(λ − A − K) ∈ Φ+(Y,X) and ind(J + K(λ − A − K) ) ≤ 0 (2.28)

The fact that λ − Aˆ = (J + Kˆ (λ − Aˆ − Kˆ )−1)(λ − Aˆ − Kˆ ) and by using (2.28) together with ˆ b ˆ Theorem 2.1 we get λ − A ∈ Φ+(XA,X) and ind(λ − A) ≤ 0. Now using (2.5) we infer that λ∈ / σuw(A). Theorem 2.21 ([59]) Let A ∈ C(X). Then \ σlw(A) = σsu(A + K). (2.29)

K∈PΦ−(X)

60 Proof. Suppose that there exists K ∈ PΦ−(X) such that λI − A − K is surjective, hence λI − A − K ∈ Φ−(X) and ind(λI − A − K) = α(λI − A − K) ≥ 0. Therefore, by Propo- sition 2.2 we deduce that λI − A ∈ Φ−(X) and ind(λI − A) = ind(λI − A − K) ≥ 0.Thus λ∈ / σlw(A). Conversely, since K(X) ⊂ PΦ−(X), the last inclusion follows.

It follows, immediately, from Theorem 2.20 and Theorem 2.21 that Corollary 2.1 Let M(X) be any subset of L(X). Then

1. σuw(A) = σuw(A + K) for all K ∈ M(X) such that K(X) ⊂ M(X) ⊂ ∆A(X).

2. σlw(A) = σlw(A + K) for all K ∈ M(X) such that K(X) ⊂ M(X) ⊂ PΦ−(X). Now, in the following theorem, we collect some characterizations of the Browder spectra. Theorem 2.22 ([95, 97]) Let A ∈ L(X). Then \ \ \ σeb(A) = σ(A + K) = σ(A + F ) = σ(A + R), K∈K(X),KA=AK F ∈F(X),F A=AF R∈R(X),RA=AR \ \ \ σub(A) = σap(A + K) = σap(A + F ) = σap(A + R), K∈K(X),KA=AK F ∈F(X),F A=AF R∈R(X),RA=AR and \ \ \ σlb(A) = σsu(A + K) = σsu(A + F ) = σsu(A + R). K∈K(X),KA=AK F ∈F(X),F A=AF R∈R(X),RA=AR

Proposition 2.4 The following properties hold:

1. σsf (A) ∪ σp(A) = σap(A).

2. ∂σeb(A) ⊆ ∂σew(A) ⊆ ∂σef (A) ⊆ ∂σsf (A).

Proof. The assertion (1) is obvious. To prove (2), suppose ﬁrst that λ ∈ ∂σeb(A). If λ is isolated, then λ ∈ σew(A) by deﬁnition. Now assume that λ is not isolated and λ∈ / ∂σew(A). Let C be the connected component of ρew(A) containing λ. By Theorem 2.16, there exists a compact operator K such that λ ∈ ρ(A + K). Denoting by E the connected component of ρ(A + K) containing λ. we have C ∩ E 6= ∅ and

R(µ; A) = R(µ; A + K)(I + KR(µ; A + K))−1 (µ ∈ C ∩ E).

This implies that λ can be at most an isolated singularity of I +KR(µ; A+K), and therefore also of R(µ; A). This contradicts our assumption, and so λ ∈ ∂σew(A). To prove the second inclusion in (2), suppose we can ﬁnd λ ∈ ∂σew(A) \ ∂σef (A). Then λI − A is Fredholm, and so there exists δ > 0 such that also µI − A is Fredholm for |λ − µ| < δ, by the stability of the Fredholm property. Moreover, ind(λI − A) = ind(µI − A) for such µ. But we can take λ ∈ ρew(A), this implies that ind(µI −A) = 0, and so ind(λI −A) = 0, a contradiction. Now suppose λ ∈ ∂σef (A) \ ∂σsf (A). Then λI − A is semi-Fredholm, and so there exists δ > 0 such that also µI −A is semi-Fredholm for |λ − µ| < δ, by the stability of the semi-Fredholm property. Moreover, ind(λI − A) = ind(µI − A) for such µ. We can take λ ∈ ρef (A), so that ind(µI − A) = ind(λI − A) < ∞. But this contradicts our assumption.

61 Note that in applications (transport operators, operators arising in dynamic populations, etc., we deal with operators A and B such that B = A + K where A ∈ C(X) and K is, in general, a closed (or closable) A-deﬁned linear operator. The operator K does not necessarily satisfy the hypotheses of the previous results. For some physical conditions on K, we have information about the operator (λI − A)−1 − (λI − B)−1 (λ ∈ ρ(A) ∩ ρ(B)). So the following useful stability result.

Theorem 2.23 ([59, 74]) Let A, B ∈ C(X) such that ρ(A) ∩ ρ(B) 6= ∅. Let I(X) be any nonzero ideal of L(X) satisfying I(X) ⊆ PΦ(X). If for some λ ∈ ρ(A) ∩ ρ(B) the operator (λI − A)−1 − (λI − B)−1 ∈ I(X), then

(i) σef (A) = σef (B) and σew(A) = σew(B). Moreover,

(ii) if I(X) ⊆ PΦ+(X), then

σuw(A) = σuw(B) and σuf (A) = σuf (B;)

∗ ∗ (iii) if I(X) ⊂ PΦ−(X) or [I(X)] ⊂ PΦ+(X ), then

σlw(A) = σlw(B) and σlf (A) = σlf (B);

(iv) if I(X) ⊆ PΦ+(X) ∩ PΦ−(X), then

σsf (A) = σsf (B).

Proof. Without loss of generality, we suppose that λ = 0. Hence 0 ∈ ρ(A) ∩ ρ(B). Therefore, we can write for µ 6= 0

µ − A = −µ(µ−1 − A−1)A.

Since, A is one to one and onto, then

α(µ − A) = α(µ−1 − A−1) and R(µ − A) = R(µ−1 − A−1).

−1 −1 This shows that µ − A ∈ Φ+(X)(resp.µ − A ∈ Φ−(X) ) if and only if µ − A ∈ Φ+(X) −1 −1 −1 −1 (resp. µ − A ∈ Φ−(X) ), in this case we have ind(µ − A) = ind(µ − A ). Similarly, we have µ − A ∈ Φ(X) if and only if µ−1 − A−1 ∈ Φ(X). Assume that A−1 − B−1 ∈ I(X). Hence using Proposition 2.2(i) we conclude that µ − A ∈ Φ(X) if and only if µ − B ∈ Φ(X) and ind(µ − A) = ind(µ − B) for each µ∈ / σef (A). This proves (i). ∗ ∗ If further I(X) ⊆ PΦ+(X) (resp. I(X) ⊂ F−(X) or [I(X)] ⊂ F+(X )), the use of Propo- sition 2.2(ii) (resp. Proposition 2.2(iii)) shows that µ − A ∈ Φ+(X)(resp.µ − A ∈ Φ−(X) ) if and only if µ − B ∈ Φ+(X) (resp. µ − B ∈ Φ−(X) ), and ind(µ − A) = ind(µ − B) for each µ∈ / σuf (A) (resp. µ∈ / σlf (A)). This concludes the proof of (ii) (resp.(iii). Finally, by combining (ii) and (iii), then by Proposition 2.2 (iv) we have (iv).

62 2.5 Left-right Fredholm and Left-right Browder spectra

We use Gl(X) and Gr(X), respectively, to denote the set of all left and right invertible operators on X. It is well-known that A ∈ Gl(X) if and only if A is injective and R(A) is a closed and complemented subspace of X. Also, A ∈ Gr(X) if and only if A is onto and N(A) is a complemented subspace of X. The set of all invertible operators on X is denoted by G(X). An operator A ∈ L(X) is relatively regular if there exists B ∈ L(X) such that ABA = A. We then say that B is a generalized inverse (pseudo inverse) of A. It is easy to see that if ABA = A, then the operator C = BAB satisﬁes the equations ACA = A and CAC = C. It is well known that A is relatively regular if and only if N(A) and R(A) are closed, complemented subspaces of X. In this case AB is a projection onto R(A) and I − BA is a projection onto N(A). In particular, a Fredholm operator is relatively regular and we have,

Theorem 2.24 ([111, Theorem 1.1 and Theorem 2.1]) Let A ∈ L(X). Then

(a) A ∈ W(X) if and only if there exists B ∈ G(X) such that ABA = A.

(b) A ∈ W−(X) if and only if there exists B ∈ Φ(X), α(B) = 0 (hence B ∈ Gl(X)) such that ABA = A.

Generalized invers are useful in solving linear equations. Suppose that B is a generalized inverse of A. If Ax = y is solvable for given y ∈ X; then By is a solution (not necessary only one). Indeed,

Ax = y is solvable ⇒ ∃x0 such that Ax0 = y

⇒ ABy = ABAx0 = Ax0 = y.

Sets of left and right Fredholm operators, respectively, are deﬁned as

Φl(X) = {A ∈ L(X): R(A) is a closed and complemented subspace of X

and α(A) < ∞}, and Φl(X) = {A ∈ L(X): N(A) is a complemented subspace of X and β(A) < ∞}.

It is well-known that the sets Φl(X) and Φr(X) are open , and PΦl(X) = PΦ(X) = PΦr(X). An operator A ∈ L(X) is left (right) Weyl if A is left (right) Fredholm operator and ind(A) ≤ 0(ind(A) ≥ 0). We denote by Wl(X) (Wr(X)) the set of all left (right) Weyl operators. The operator A ∈ L(X) is left Browder if it is left Feredholm of ﬁnite ascent, and A is right Browder if it is right Fredholm of ﬁnite ascent. Let Bl(X) (Br(X)) denote the set of all left (right) Browder operators. The following theorem gives a characterization of left and right Browder operators

63 Theorem 2.25 ([123]) Let A ∈ L(X). Then A is left (right) Browder operator if and only if there exist closed subspaces X1 and X2 invariant with respect to A such that X = X1 ⊕X2,

dim X1 < ∞, the reduction A1 = A|X1 : X1 −→ X1 is nilpotent and the reduction A2 =

A|X2 : X2 −→ X2 is left (right) invertible. The corresponding spectra of A of the classes of operators deﬁned in this section are • The left spectrum: σl(A) := {λ ∈ C : λI − A/∈ Gl(X)}, • The right spectrum:

σri(A) := {λ ∈ C : λI − A/∈ Gr(X)},

• The left Fredholm spectrum:

σlef (A) := {λ ∈ C : λI − A/∈ Φl(X))},

• The right Fredholm spectrum:

σrf (A) := {λ ∈ C : λI − A/∈ Φr(X)},

• The third Kato spectrum:

σK3 (A) = {λ ∈ C : λI − A 6∈ Φl(X) ∪ Φr(X)}.

• The left Browder spectrum:

σleb(A) := {λ ∈ C : λI − A/∈ Bl(X)},

• The right Browder spectrum:

σrb(A) := {λ ∈ C : λI − A/∈ Br(X)},

• The left Weyl spectrum:

σlew(A) := {λ ∈ C : λI − A/∈ Wl(X)},

• The right Weyl spectrum:

σrw(A) := {λ ∈ C : λI − A/∈ Wr(X)},

Note that all these sets of essential spectra are closed and in general satisfy the following inclusions σif (A) ⊆ σiw(A) ⊆ σib(A) = σif (A) ∪ accσi(A); for i = le, r. (2.30)

The following example shows that in general σlew(A) 6= σleb(A) and σrw(A) 6= σrb(A) Example 2.6 Let H be a separable Hilbert space, let V be the right shift on H and let N ∗ be quasi-nilpotent. If A = V ⊕ V ⊕ N, then σleb(A) = σrb(A) = D and σlef (A) = σrf (A) = σlew(A) = σrw(A) = ∂D ∪ {0}, where D is the closed unit ball. Since σeb(A) = D and σlb(A) = σub(A) = D, from σub(A) ⊂ σleb(A) ⊂ σeb(A) and σlb(A) ⊂ σrb(A) ⊂ σeb(A) we get σleb(A) = σrb(A) = D. From σew(A) = ∂D ∪ {0}, ∂σew(A) ⊂ σef (A) ⊂ σew(A) and ∂σew(A) ⊂ σlf (A) ⊂ σew(A) we obtain σlf (A) = σuf (A) = σef (A) = ∂D ∪ {0}. Since σuf (A) ⊂ σlef (A) ⊂ σef (A) and σlf (A) ⊂ σrf (A) ⊂ σef (A), it follows that σlef (A) = σrf (A) = ∂D ∪ {0}. As σlef (A) ⊂ σlew(A) ⊂ σew(A) and σrf (A) ⊂ σrw(A) ⊂ σew(A) we get σlew(A) = σrw(A) = ∂D ∪ {0}.

64 Each of the left Fredholm and right Fredholm spectra are stable under commuting Riesz perturbations:

Theorem 2.26 ([122]) If A ∈ L(X) and R ∈ L(X) is a Riesz operators which commutes with A, then σif (A + R) = σif (A), for i = le, r

Proof. First we claim that if λ ∈ C is arbitrary then

σlef (A + λR) = σlef (A) (2.31)

We combine the two variable spectral mapping theorem for the left spectrum [90, Theorem 8.8] with that the fact in the Calkin algebra C(X) the coset r = R+K(X) is quasi-nilpotent:

σl(a + λr) = {α + λβ :(α, β) ∈ σl(a, r)} ⊆ σl + λσl(r) = σl(a)

since σl(r) = {0}, and the reverse inclusion follows from a = (a + r) − r. This gives σlef (A + R) = σlef (A). For the right Fredholm spectrum, we replace in (2.31) the left spectrum by the right.

Note that, if A ∈ L(X) and R ∈ L(X) is a Riesz operators which commutes with A, by a simple properties of accumulation points, we have

accσi(A + λR) ⊆ acc(σi(A) + λσi(R)) ⊆ accσi(A) + λaccσi(R) = accσi(A) for i = l, ri.

This together with Theorem 2.26 and the relation (2.30) gives

Theorem 2.27 If A ∈ L(X) and R ∈ L(X) is a Riesz operators which commutes with A, then σib(A + R) = σib(A), for i = le, r The commutivity assumption in Theorem 2.27, which cannot more generally be relaxed, even for nilpotent operators

Example 2.7 If A; B and N in L(X) are deﬁned by taking

T 0 S 0 0 I X = `2 × `2,A = ,B = ,N = 0 S 0 T 0 0 where T ∈ L(`2) and S ∈ L(`2) are the left and the right shifts on `2, then A and B are both Weyl, while A + N is not right Weyl and B − N is not left Weyl. In fact

0 0 I − ST 0 (B − N)(A + N) = I − , (A + N)(B − N) = I − 0 I − ST 0 0 both products are Fredholm of index zero, that is Weyl, and we can check that A + N is one one and B − N onto, with

ind(B − N) = 1 = −ind(A + N).

It is familiar that the Weyl and the Browder spectrum of an operator can be written as the intersection of the spectrums of its compact, and its commuting compact, perturbations. This extends to left and right spectra, and Riesz perturbations:

65 Theorem 2.28 ([123]) Let A ∈ L(X). Then \ σleb(A) = σl(A + R), R∈R(X),RA=AR and \ σrb(A) = σri(A + R). R∈R(X),RA=AR

Theorem 2.29 ([123]) Let A ∈ L(X) and J(X) any non zero ideal of Riesz operators. Then \ σlew(A) = σl(A + R), R∈R(X),RA−AR∈J(X) and \ σrw(A) = σri(A + R). R∈R(X),RA−AR∈J(X)

The boundary of the Browder spectrum is a subset of the essential spectrum:

Proposition 2.5 ([123]) If A ∈ L(X) then for each i = u, l, r, le there is inclusion:

∂σeb(A) ⊆ ∂σib(A) ⊆ ∂σiw(A) ⊆ ∂σif (A) ⊆ σib(A) ⊆ σeb(A). (2.32) and hence also σif (A) ⊆ σiw(A) ⊆ σib(A) ⊆ σeb(A) ⊆ ησif (A). (2.33) Here ηK is the connected hull of a compact set K ⊆ C.

Proof. Recall that for compact subsets H; K ⊆ C, ∂H ⊆ K ⊆ H ⇒ ∂H ⊆ ∂K ⊆ K ⊆ H ⊆ ηK = ηH.

By (2.30) we have intσi(A) = intσib(A) and hence

∂σib(A) ⊆ ∂σi(A), and also σib(A) ∩ isoσi(A) ⊆ ∂σif (A), which together with theorem

∂σi(A) ⊆ σif (A) ∪ isoσi(A), give ∂σib(A) ⊆ ∂σi(A) ∩ σib(A) ⊆ ∂σif (A) ∪ (σib(A) ∩ isoσib(A)) = σif (A).

By index continuity the sets σiw(A) \ σif (A) are all open, so that also

∂σiw(A) ⊆ ∂σif (A) ⊆ σiw(A)

Together the two last relation give (2.32), and also (2.33).

66 2.6 Invariance of the essential spectra

Now we want to study the inﬂuence of perturbations on the spectrum. Our hope is that at least some parts of the spectrum remain invariant under additive perturbations, such as operators of ﬁnite rank, compact operators, small in norm and quasi-nilpotent operators.

Remember that Q ∈ L(X) is said to be quasi-nilpotent operator if

1 kQnk n −→ 0.

and is said to be nilpotent if there exists d ∈ N such as Ad−1 6= 0 and An = 0 for all n ≥ d. the natural d is called the degree of nilpotency. An example for quasi-nilpotent but not nilpotent is the operator A4 deﬁned in Example 1.12. An example for quasi-nilpotent but neither nilpotent nor compact:

2 2 2 2 Q = Q1 ⊕ Q2 : ` ⊕ ` −→ ` ⊕ ` , where

Q1x = (0, x1, 0, x3, 0, x5,... ) 1 1 Q x = (0, x , x , x ,... ). 2 1 2 2 3 3 Recall the following properties of the quasi-nilpotent and nilpotent operators: 1. If Q is quasi-nilpotent, then σ(Q) = {0} and Q is neither bounded below nor open. 2. If Q is nilpotent, then Q is neither one to one nor its range is dense. 3. The set of quasi-nilpotent operators is contains in the boundary of the set of invertible operators. 4. Quasi-nilpotent operators of ﬁnite rank or coﬁnite rank are nilpotent operators.

5. An operator R ∈ L(X) is a Riesz operator if and only if the coset Rb is quasi-nilpotent in the Calkin algebra C(X). Now, we will consider, in the following and in the next chapters, for every A ∈ L(X), X inﬁnite dimensional Banach space, the following properties:

(P1) σi(A) 6= ∅.

(P2) σi(A) is closed.

(P3) σi(A + U) = σi(A) whenever AU = UA and kUk < for some > 0.

(P4) σi(A + F ) = σi(A) for every F ∈ F(X) commuting with A.

(P5) σi(A + K) = σi(A) for every K ∈ K(X) commuting with A.

(P6) σi(A + Q) = σi(A) for every quasi-nilpotent operator Q commuting with A.

(P7) σi(A) verifies the spectral mapping theorem: f(σi(A)) = σi(f(A)) where f is an analytic function defined on a neighborhood of σ(A). The properties (P1)-(P7) for these sets σi(A), i ∈ {ap, su, lb, ub, lw, uw, lf, uf, ef, eb, ew} of essential spectra defined above are summarized in the following table:

67 (P1) (P2) (P3) (P4) (P5) (P6) (P7) σi 6= ∅ σi closed Small com. com. ﬁn. com. comp. com. quasi sp. map. pert. rank pert. pert. nilp. pert. theorem

σap(A) yes yes yes no no yes yes

σsu(A) yes yes yes no no yes yes

σuf (A) yes yes yes! yes ! yes ! yes ! yes

σlf (A) yes yes yes! yes ! yes! yes ! yes

σsf (A) yes yes yes! yes! yes! yes! ⊆

σef (A) yes yes yes! yes! yes! yes! yes

σew(A) yes yes yes! yes! yes! yes! ⊇

σuw(A) yes yes yes no yes! yes ⊇

σlw(A) yes yes yes no yes! yes ⊇

σeb(A) yes yes yes yes yes yes yes

σub(A) yes yes yes no yes yes yes

σlb(A) yes yes yes no yes yes yes

Table 2.1:

68 Comments.

1. The boxes marked by "yes!" means that the commutation is not necessary and the boxes marked by ⊆ (resp.⊇) means that we have also f(σi(A)) ⊆ σi(f(A)) (resp. f(σi(A)) ⊇ σi(f(A))).

2. The properties (P2) and (P3) holds for all σi, by Theorem 2.2 and Theorem 2.3.

3. The properties (P1)-(P7) when are valid for σi, i ∈ {lb, ub, lw, uw, lf, uf, ef, eb, ew} see [1], [105], [64] and [90].

4. The property (P7) is valid for σi, i ∈ {lb, ub, lf, uf, ef, eb} by [52], [91], [95], [92]. The interested reader may ﬁnd further results on the spectral mapping Theorem also in Schmoeger [107]. In the same paper Schmoeger has described the set of all A ∈ L(X) such that property (P7) holds for σi, i ∈ {lw, uw, sf, ew}.

5. For interest investigation of the stability of σeb(A) under commuting compact perturbation and more generaly by Riesz perturbation see also [62].

6. The table 2.1 is valid for σi(A), i ∈ {lb, ub, lw, uw, lf, uf, ef, eb, ew} for all closed densely deﬁned linear operators on X (see the results of this Chapter, and for more details see also [105], [64] and [59]).

7. Consider the identity operator in a Hilbert space and let P be a 1−dimensional orthogonal projection. Then I − P is not onto and (P4) and (P5) fail for σsu(A).

∞ 8. Consider the bilateral shift A in Hilbert space H with an orthonormal basis {ei}i=−∞ deﬁned by Aei = ei+1 and F x = hx, e0i e1. Then d(A − F ) = ∞ so that σub(A) and σlb(A) ( by taking adjoint) do not have property (P4).

9. The inclusion of (P7) for σew(A) may be proper. For example, if U is the unilateral shift, consider U + I 0 A = : `2 ⊕ `2 → `2 ⊕ `2, 0 U ∗ − I

Then σew(A) = σ(A) = {λ ∈ C : |λ + 1| ≤ 1} ∪ {λ ∈ C : |λ − 1| ≤ 1}. Let . p(λ) = (λ + 1)(λ − 1)

Therefore 0 ∈ p(σew(A)), however,

U + 2I 0 U 0 p(A) = (A + I)(A − I) = 0 U ∗ 0 U ∗ − 2I

∗ so that ind(p(A)) = ind(U ) + ind(U) = 0, which implies 0 ∈/ σew(p(A)).

69 Chapter 3

Generalized Kato spectrum

3.1 The semi-regular spectrum and its essential version

The semi-regular spectrum was first introduced by Apostol [8] for operators on Hilbert spaces and successively studied by several authors Muller[89]and Rakocevic [98], Mbekhta and Ouahab [86]and Mbekhta [82] in the more general context of operators acting on Banach spaces. Definition 3.1 Let A ∈ L(X). A is said to be semi-regular if R(A) is closed and N(An) ⊆ R(A), for all n ≥ 0. Equivalently, A is a semi-regular if and only if R(A) is closed and A verifies one of the equivalent conditions of lemma 1.2 or Corollary 1.1. Trivial examples of semi-regular operators are surjective operators as well as injective operators with closed range.

Let > 0 as in (2.2) or (2.3). If A is semi-Fredholm operator, the jump, jump(A), of A is deﬁned by:

if A ∈ Φ+(X), jump(A) = α(A) − α(λI − A) for all 0 < |λ| < .

if A ∈ Φ−(X), jump(A) = β(A) − β(λI − A) for all 0 < |λ| < .

Observe that jump(A) ≥ 0 and the continuity of the index ensures that both deﬁnitions of jump(A) coincide whenever A ∈ Φ(X), it is know by [1, Theorem 1.58] that if A ∈ Φ±(X), then jump(A) = 0 if and only if A is semi-regular operator.

The semi-regularity of an operator may be expressed in terms of the concept of the gap metric. let us deﬁned this concept. Let M,N be two closed linear subspaces of the Banach space X and set

δ(M,N) = sup{dist(x, N): x ∈ M, kxk = 1}, in the case that M 6= {0}, otherwise we deﬁne δ({0},N) = 0 for any subspace N. The gap between M and N is deﬁned by

δb(M,N) = max{δ(M,N), δ(M,N)}

δb is a metric on the set F(X) of all linear closed subspaces of X, and the convergence Mn −→ M in F(X) is obviously deﬁned by δb(Mn,M) −→ 0 as n −→ ∞ in R. Moreover

70 (F(X), δb) is complete metric space (see [64]).

Note that if X is Hilbert space, then the gap metric is deﬁned in terms of the orthogonal projection as follows: δ(M,N) = k(I − PN )PM k and δb(M,N) = kPM − PN k

where PM and PN are the orthogonal projection onto M and N respectively.

In the following theorem we extend to the Banach space, the result was shown by J. P. Labrousse [71] in the case of Hilbert spaces and we shows that the semi-regularity of an operator may be characterized in terms of the continuity of the gap metric. For α a nonzero positive real number, we introduce the following set

R(α) = {λ ∈ C : γ(λI − A) ≥ α}

Theorem 3.1 Let (λn)n ⊂ R(α) nonstationary sequence and λn −→ λ0 in C, then

1 1. δb(N(λnI − A),N(λ0I − A)) ≤ α |λn − λ0|.

2. λ0 ∈ R(α).

3. λ0I − A is semi-regular. To prove this theorem we will need the following two propositions.

Proposition 3.1 ([1]) For every operator A ∈ L(X), and arbitrary λ, µ ∈ C, we have: 1. γ(λI − A).δ(N(µI − A),N(λI − A)) ≤ |µ − λ|.

2. min{γ(µI − A), γ(λI − A)}δb(N(µI − A),N(λI − A)) ≤ |µ − λ|. Proof. (1). The statement is trivial for λ = µ. Suppose that λ 6= µ and consider an element 0 6= x ∈ N(µI − A). Then x∈ / N(λI − A) and hence

γ(λI − A)dist(x, N(λI − A)) ≤ k(λI − A)xk = k(λI − A)x − (µI − A)xk = |µ − λ|

From this estimate we obtain, if B = {x ∈ N(µI − A), kxk ≤ 1}, that

γ(λI − A) sup dist(x, N(λI − A)) ≤ |µ − λ| x∈B and therefore we deduce (1). (2). Clearly, the inequality follows from (1) by interchanging λ and µ.

Proposition 3.2 ([1]) Let M,N ∈ F(X). For every x ∈ X and 0 < < 1 there exists x0 ∈ X such that (x − x0) ∈ M and 1 − δ(M,N) dist(x ,N) ≥ (1 − ) kx k . (3.1) 0 1 + δ(M,N) 0

71 Proof. If x ∈ M it suﬃces to take x0 = 0. Assume therefore that x∈ / M. Let Xb = X/M denote the quotient space and put xb = x + M the equivalence class of x. Evidently, kxk = inf kzk > 0. We claim that there exists an b z∈xb element x0 ∈ X such that

kxbk = dist(x0,M) ≥ (1 − ) kx0k Indeed, when it is not so, then

kxbk = kzk < (1 − ) kzk for every z ∈ xb and therefore kxbk ≤ inf kzk = (1 − ) kxbk z∈xb This is impossible since kxbk > 0. Let µ = dist(x0,N) = infu∈N kx0 − uk. We know that there exists y ∈ N such that kx0 − yk ≤ µ + kx0k. From that we obtain kyk ≤ (1 + ) kx0k + µ. On the other hand, we have dist(y, M) ≤ δ(N,M) kyk and hence

(1 − ) kx0k ≤ dist(x0,M)

≤ kx0 − yk + dist(y, M)

≤ µ + kx0k + δ(N,M) kyk

≤ µ + kx0k + δ(N,M)[(1 + ) kx0k + µ]

From this we obtain that 1 − − δ(N,M) µ ≥ − kx k . 1 + δ(N,M) 0

Since > 0 is arbitrary, this implies the inequality (3.1).

Proof of Theorem 3.1. 1. For n, m ∈ N by proposition 3.1 part (2) we have 1 δb(N(λnI − A),N(λmI − A)) ≤ |λn − λm| min{γ(λnI − A), γ(λmI − A)}

Since by assumption γ(λnI −A) ≥ α, for all n ∈ N we have min{γ(λnI −A), γ(λmI −A)} ≥ α and 1 δb(N(λnI − A),N(λmI − A)) ≤ |λn − λm| (3.2) α The sequence (N(λnI − A))n) is a Cauchy sequence in the complet metric space F(X), thus it converges. Let F = limn→∞ N(λnI − A). Let x ∈ N(λ0I − A), by proposition 3.1 part (1) we ﬁnd 1 δ(N(λ I − A),N(λ I − A)) ≤ |λ − λ | 0 n α n 0

From this estimate we deduce that x ∈ F and N(λ0I − A) ⊂ F . conversely, let x ∈ F , by proposition 3.2 for 0 < < 1, N = F and M = N(λnI − A) there exists xn ∈ X such that (x − xn) ∈ N(λnI − A) and 1 − δ(N(λnI − A),F ) dist(xn,F ) ≥ (1 − ) kxnk . 1 + δ(N(λnI − A),F )

72 From this we obtain 1 − δ(N(λnI − A),F ) δb(F,N(λnI − A)) ≥ dist(xn,F ) ≥ (1 − ) kxnk . 1 + δ(N(λnI − A),F )

which yields, xn → 0 as n → ∞. On other hand

(λ0I − A)xn = (λ0I − A)x − (λ0I − A)(x − xn)

= (λ0I − A)x + (λn − λ0)(x − xn) and hence (λn − λ0)(x − xn) → 0. We obtain that

(λ0I − A)xn → (λ0I − A)x

Hence N(λ0I − A) is closed, and (λ0I − A)x = 0, consequently F ⊂ N(λ0I − A). To end the proof of (1) we take n → ∞ in (3.2). 2. Suppose that λ0 ∈/ R(α), then there exists x ∈ X and 0 < < 1 ; kxk = 1, x∈ / N(λ0I−A) and k(λ0I − A)xk < (1 − )α kxk

We can ﬁnd xn ∈/ N(λnI − A) such that (x − xn) ∈ N(λnI − A) and take n ∈ N such that |λ − λ | < α n 0 2 Then k(λnI − A)xk ≤ k(λ0I − A)xk + |λn − λ0| kxk and therefore k(λ I − A)xk ≤ (1 − )α kxk (3.3) n 2 On other hand, We have (x − xn) ∈ N(λnI − A) and hence

kx − xnk ≤ sup{dist(y, N(λnI − A)); y ∈ N(λ0I − A), kyk = 1}

≤ δ(N(λ0I − A),N(λnI − A)) kxk

≤ δb(N(λ0I − A),N(λnI − A)) kxk 1 ≤ |λ − λ | kxk α n 0 ≤ kxk 2 From the last inequality it ﬂlows that 1 kxk ≤ kxnk (3.4) (1 − 2 ) From (3.3) and (3.4) we obtain

k(λnI − A)xnk < α kxnk

what contradicts the fact that (λn)n ⊂ R(α). k 3. It is clear that N(λnI − A) ⊂ R((λ0I − A) ) for every k ∈ N. For every x ∈ N(λ0I − A), k ∈ N, and λn 6= λ0 we then have k dist(x, R((λ0I − A) )) ≤ dist(x, N(λnI − A)) kxk

≤ δ(N(λ0I − A),N(λnI − A)) kxk

≤ δb(N(λ0I − A),N(λnI − A)) kxk

73 k k This implies that x ∈ R((λ0I − A) ) for every k ∈ N. Hence N(λ0I − A) ⊂ R((λ0I − A) ) for every k ∈ N. k To establish 3 it suﬃces to prove that R((λ0I − A) ) is closed for k ∈ N. We proceed by induction. The case k = 1 is obvious from 2. k k k Assume that R((λ0I − A) ) is closed. Then N(λ0I − A) ⊂ R((λ0I − A) ) = R((λ0I − A) ) k and hence N(λ0I − A) + R((λ0I − A) ) is closed. By proposition 2.3 we then conclude that k k+1 A(R((λ0I − A) )) = R((λ0I − A) ) is closed.

A semi-regular operator A has a closed range. It is evident that the reduced minimum modulus of A is useful to ﬁnd conditions which ensure that R(A) is closed. The following theorem gives several equivalent conditions for the continuity of the function λ → γ(λI −A).

Theorem 3.2 ([89]) For A ∈ L(X) and λ0 ∈ C, the following statements are equivalent:

1. λ0I − A is semi-regular.

2. γ(λ0I − A) > 0 and the mapping λ → γ(λI − A) is continuous at λ0

3. γ(λ0I − A) > 0 and the mapping λ → N(λI − A) is continuous at λ0 in the gap topology.

4. R(λ0I − A) is closed in a neighborhood of λ0 and the mapping λ → R(λI − A) is continuous at λ0 in the gap topology. For an essential version of semi-regular operators we use the following notation. For subspaces M,L ⊂ X write M ⊂e L if there exists a ﬁnite-dimensional subspace F of X for which M ⊂ L + F . Obviously

M ⊂e L ⇔ dim(M/M ∩ L) < ∞

Deﬁnition 3.2 An operator A ∈ L(X) is called essentially semi-regular if R(A) is closed n and N(A ) ⊂e R(A), for all n ≥ 0. Now, set V0(X) = {A ∈ L(X): A is semi-regular }, V(X) = {A ∈ L(X): A is essentially semi regular }.

It is well known that Φ+(A) ∪ Φ−(A) ⊂ V(X), V0(X) and V(X) are neither semi-groups nor open or closed subset of L(X) and

int(V(X)) = Φ+(X) ∪ Φ−(X),

int(V0(X)) = {T ∈ Φ±(X): α(A) = 0 or β(A) = 0}.

Lemma 3.1 ([68]) A ∈ L(X) is semi-regular (resp. essentially semi-regular) operator if and only if there exists a closed subspace V of X such that TV = V and the operator Aˆ : X/V → X/V induced by A is bounded below (resp. upper semi-Fredholm).

74 The semi-regular spectrum of a bounded operator A on X is deﬁned by

σse(A) := {λ ∈ C : λI − A is not semi-regular} and its essential version by

σes(A) := {λ ∈ C : λI − A is not essentially semi-regular}

The sets σse(A) and σes(A) are always non-empty compact subsets of the complex plane, σse(f(A)) = f(σse(A)) and σes(f(A)) = f(σes(A)) for any analytic function f in a neighborhood of σ(A) [98]. Now we recall some results about σse(A) and σes(A) Theorem 3.3 ([98]) Let A ∈ L(X).

∗ ∗ 1. σse(A) = σse(A ) and σes(A) = σes(A ).

2. ∂σ(A) ⊆ σse(A); where ∂σ(A) is the boundary of the spectrum of A.

3. λ ∈ σse(A) \ σes(A) if and only if λ is an isolated point of σse(A), n supn∈N(dim N(λI − A) + N((λI − A) ))/N(λI − A) < ∞ and R(λI − A) is closed. Theorem 3.4 ([98]) Let A ∈ L(X). Then \ \ σes(A) = σse(A + K) = σse(A + F ) K∈K(X),KA=AK F ∈F(X),F A=AF

Let us mention that the mappings A → σse(A) and A → σes(A) are not upper semi- continuous at A in general [98, Remark 4.4].

Theorem 3.5 ([98]) Let A, An ∈ L(X). and AAn = AnA for each positive integer n. Then

lim sup σse(An) ⊂ σse(A) and lim sup σes(An) ⊂ σes(A) n∈N n∈N 3.2 Closed-range spectrum

Most of the classes of operators considered before require that the operators have closed ranges. Thus it is natural to consider the closed-range spectrum or the Goldberg spectrum of an operator A ∈ C(X),

σec(A) = {λ ∈ C ; R(λI − A) is not closed}. However, the closed-range spectrum has not good properties:

1. σec(A) is not necessarily non-empty. For example, A = 0.

2. σec(A) may be not closed. There exists an operator A such that R(A) is closed but R(λI − A) is not closed for all λ ∈ D(0, 1) \{0}. For example, the right shift operator A deﬁned on `2 by A(x1, x2, x3,... ) = (0, x1, x2, x3,... ). 3. It is possible that R(A2) is closed but R(A) is not. Let A be deﬁned on `2 by 1 1 A(x , x , x ,... ) = (0, x , 0, x , 0, x , 0,... ) 1 2 3 1 3 2 5 3 The operator A is compact and R(A) is not closed, A2 = 0 and R(A2) is closed.

75 VI 4. Conversely, it is also possible that R(A) is closed but R(A2) is not. let A = 0 0 be an operator deﬁned on `2 ⊕ `2, where V has the following proprieties that V 2 = 0 and R(V ) is not closed. Then R(A) is closed, R(A2) is not closed, A3 = 0.

5. σec(A) is unstable under nilpotent perturbations. For example, A = 0 and N the nilpotent operator defined in (3.). Then 0 ∈ σec(A + N) but 0 ∈/ σec(A). Note that the essentially semi-regular spectrum, which has very nice spectral properties, is not too far from the closed-range spectrum. Clearly σec(A) ⊂ σes(A) and is at most countable. Thus the essentially semi-regular spectrum can be considered as a nice completion of the closed-range spectrum. However, the spectrum σec(A) can be used (see Remark 3.1 and Corollary 3.1 below) to obtain information on the location in the complex plane of the various types of essential spectra, Fredholm, Weyl and Browder spectra etc... , for large classes of linear operators arising in applications. For example , integral, pseudo-differential, difference, and pseudo-differential operators (see [13, 45, 72, 73, 74]).

Remark 3.1 If λ in the continuous spectrum σc(A) of A then R(λ − A) is not closed. Therefore λ ∈ σi(A), i ∈ Λ = {ec, es, se, lf, uf, ef, ew, uw, lw, eb, ub, lb}. Consequently we have \ σc(A) ⊂ σi(A). i∈Λ

Corollary 3.1 For an operator A, if σ(A) = σc(A) then

σ(A) = σi(A) for all i ∈ {ec, es, se, lf, uf, ef, ew, uw, lw, eb, ub, lb}.

3.3 Quasi-Fredholm spectrum

Quasi-Fredholm operators have been introduced by J.P Labrousse [70] as a generalization of the semi-Fredholm operators. Deﬁne

n m ∆(A) = {n ∈ N : ∀m ∈ N : m ≥ n ⇒ R(A ) ∩ N(A) ⊆ R(A ) ∩ N(A)}. The degree of stable iteration is deﬁned as dis(A) = inf ∆(A) if ∆ 6= ∅, while dis(A) = ∞ if ∆ = ∅.

Deﬁnition 3.3 A is said to be quasi-Fredholm if there exists d ∈ N such that 1. dis(A) = d

2. R(An) is closed for all n ≥ d.

3. R(A) + N(Ad) is a closed subspace of X.

An operator is quasi-Fredholm if it is quasi-Fredholm of some degree d.

We denote by qΦ(X) the set of all quasi-Fredholm operators. Examples of quasi-Fredholm operators are semi-regular operators (quasi-Fredholm of degree 0), essentially semi-regular operators, Fredholm operators and semi-Fredholm operators. Some other examples of quasi- Fredholm operators operators may be found in Mbekhta [86], Labrousse [70] and in Chapter 4. We give now a fundamental characterizations of quasi-Fredholm operators.

76 Theorem 3.6 ([70, Theorem 3.2.2]) Let H be a Hilbert space. An operator A ∈ L(H) is quasi-Frdholm operator if and only if there exists a pair of closed subspaces (M,N) of H such that H = M ⊕ N and

1. A(M) ⊂ M and A/M is a semi-regular operator.

2. A(N) ⊂ N and A/N is a nilpotent operator. The pair (M,N) is said to be a Kato decomposition of A. For an operator the property of being quasi-Fredholm my be described in terms of restrictions, Theorem 3.7 A ∈ L(X) is quasi-Fredholm operator if and only if there exists n ∈ N such n that R(A ) is closed and An is semi-regular operator. In this case Tn is semi-regular for all m ≥ n.

n Proof. Suppose that there exists n ∈ N such that R(A ) is closed and An is semi- m p regular operator. Then An is semi-regular for all m ≥ 1, it follows that R(A ) is closed for n m all p ≥ n. Since An is semi-regular we have N(An) = N(A) ∩ R(A ) = N(A) ∩ R(A ) for all m ≥ n. Hence d = dis(A) ∈ N and N(Am) + R(A) = N(Ad) + R(A) for all m ≥ d. Moreover, R(Am) is closed for all m ≥ d because R(Am) is closed for each m ≥ n. So A is quasi-Fredholm operator. Conversely, suppose that A is a quasi-Fredholm operator and let d = dis(A). Thus R(Ad) d d d+1 is closed. Consider the operator Ad : R(A ) → R(A ), then R(Ad) = R(A ) is closed and d m m N(Ad) = N(A) ∩ R(A ) = N(A) ∩ R(A ) ⊂ R(Ad ) for all m ≥ d. So Ad is a semi-regular operator. Moreover, for all m ≥ d n d+1+n N(Am) ⊆ N(Ad) ⊆ R(Ad ) = R(A )) for all n ∈ N, In particular d+(m−d)+1+n m+1+n n N(Am) ⊆ R(A )) = R(A ) = R(Am) for all n ∈ N. n Moreover, since Ad is semi-regular for all n ∈ N, it then follows R(Am) is closed for all m ≥ d. Hence Am is semi-regular. Deﬁnition 3.4 Let A ∈ L(X), the essential quasi-Fredholm spectrum is deﬁned by

σqf (A) := {λ ∈ C : λ − A 6∈ qΦ(X)} Note that the set qΦ(X) is open (see [70, 18, 64]), consequently the essential quasi- Fredholm spectrum is a compact set of the spectra σ(A) of A. σqf (A) may be empty, this is the case where the spectrum σ(A) is a ﬁnite set of poles of the resolvent. Note that all these sets of spectra deﬁned above in general satisfy the following inclusions

σqf (A) ⊆ σes(A) ⊆ σse(A) and σec(A) ⊆ σes(A) ⊆ σse(A). By Theorem 3.1, we easily obtain that

Proposition 3.3 Let A ∈ L(X). The sets σse(A) \ σqf (A) and σes(A) \ σqf (A) are at most countable.

The comparison between σqf (A) and σec(A), gives Theorem 3.8 ([71]) Let H be a Hilbert space and A ∈ L(H). Then the symmetric diﬀer- ence σqf (A)∆σec(A) is at most countable.

77 3.4 Generalized Kato spectrum

Now, we introduce an important class of bounded operators which involves the concept of semi-regularity. Deﬁnition 3.5 An operator A ∈ L(X), is said to admit a generalized Kato1 decomposition, if there exists a pair of closed subspaces (M,N) of X such that : 1. X = M ⊕ N.

2. A(M) ⊂ M and A/M is semi-regular.

3. A(N) ⊂ N and A/N is quasi-nilpotent. (M,N) is said to be a generalized Kato decomposition of A, abbreviated as GKD(M,N).

If we assume in the deﬁnition above that A/N is nilpotent, then there exists d ∈ N for d which (A/N ) = 0. In this case T is said to be of Kato type of order d. An operator is of Kato type if it is of Kato type of some degree d. Operators which admit a generalized decomposition was originally introduced by M. Mbekhta[81, 83, 85] in the Hilbert spaces, were called pseudo-Fredholm operators. Recently appeared in the Book of Aiena [1] and the work of Q. Jiang-H. Zhong [60]. The operators which satisfy this property form a class which includes the most class of operators deﬁned in this thesis.

Examples of operators admits a generalized Kato decomposition: 1. Kato type operators.

2. Semi-regular operator is of Kato type with M = X and N = {0}.

3. Quasi-nilpotent operator has a GKD with M = {0} and N = X.

4. Essentially semi-regular with N is ﬁnite-dimensional and A/N is nilpotent. 5. If 0 is an isolated point in σ(A), then admits a GKD , see Theorem 1.26 or Theorem 3.14. In particular, if 0 is a pole of the resolvent of A, see Theorem 1.27 or Theorem 3.15.

6. Riesz operators. If A is a Riesz operator, then A = A1 ⊕ A2 with A1 is compact and A2 is quasi-nilpotent operator. 7. Quasi-polar and polar operators. If A ∈ L(X) is said to be quasi-polar (resp., polar) if there is a projection P commuting with A for which A has a matrix representation A 0 A = 1 : R(P ) ⊕ N(P ) −→ R(P ) ⊕ N(P ) 0 A2

where A1 is invertible and A2 is quasi-nilpotent ( resp., nilpotent).

8. Browder operators. If A is a Browder operator, then A = A1 ⊕ A2 with A1 is invertible and A2 is nilpotent operator.

9. Right Browder operators. If A is a right Browder operator, then A = A1 ⊕ A2 with A1 is right invertible and A2 is nilpotent operator, see Theorem 2.25. 1Tosio Kato, August 25, 1917 - October 2, 1999. Japanese mathematician

78 10. Left Browder operators. If A is a Browder operator, then A = A1 ⊕ A2 with A1 is left invertible and A2 is nilpotent operator, see Theorem 2.25. 11. Semi-Fredholm operators. Kato proved that a closed semi-Fredholm operator is of Kato type [63, Theorem 4].

12. Quasi-Fredholm operators. In Hilbert spaces, this class coincide with Kato type operators, Theorem 3.6. The same decomposition exists also for quasi-Fredholm operators on Banach spaces under the additional assumption that the subspaces R(T d) ∩ N(T ) and R(T ) + N(T d) are complemented, see Remark after Theorem 3.2.2 in [70].

For other examples of pseudo-Fredholm operators see Chapter 4.

Theorem 3.9 ([1]) Let A ∈ L(X), and assume that A is of Kato type of order d Then:

1. M ∩ N(A) = R(An) ∩ N(A) = R(Ad) ∩ N(A) for every n ∈ N, n ≥ d. 2. R(A) + N(An) = A(M) ⊕ N for every natural n ≥ d. Moreover R(A) + N(An) is closed in X.

Note that by Theorem 3.6, in the case of Hilbert spaces, the set of quasi-Fredholm operators coincides with the set of Kato type operators. But in the case of Banach spaces the Kato type operator is also quasi-Fredholm (see Theorem 3.9), according to the remark following Theorem 3.2.2 in [70] the converse is true when R(T d) ∩ N(T ) and R(T ) + N(T d) are complemented in the Banach space X. For every operator A ∈ L(X), let us deﬁne the Kato type spectrum and the generalized Kato spectrum as follows respectively:

σk(A) := {λ ∈ C : λI − A is not of Kato type}

σgk(A) := {λ ∈ C : λI − A does not admit a generalized Kato decomposition} σgk(A) ( resp. σk(A)) is not necessarily non-empty. For example, each quasi-nilpotent (resp. nilpotent) operator A has empty generalized Kato spectrum (resp. kato spectrum).

The following result shows that the generalized Kato spectrum of a bounded operator is a closed subset of the spectra σ(A) of A. The next theorem is due to Q. Jiang , H. Zhong ([60], Theorem 2.2) :

Theorem 3.10 Suppose that A ∈ L(X), admits a GKD(M,N). Then there exists an open disc D(0, ) for which λI − A is semi-regular for all λ ∈ D(0, ) \{0}

This theorem extend works of P. Aiena and E. Rosas for an operator of Kato type (see [1]):

Theorem 3.11 ([1]) Suppose that A ∈ L(X), is of Kato type. Then there exists an open disc D(0, ) for which λI − A is semi-regular for all λ ∈ D(0, ) \{0} Note that the set of all Kato type operators is open by Theorem 3.11 (see, for example, [64], [1]), consequently the Kato spectrum is a closed set of the spectrum σ(A) of A. However, since σgk(A) ⊆ σk(A) ⊆ σes(A) ⊆ σse(A), as a straightforward consequence of Theorem 3.10 and Theorem 3.11 , we easily obtain that these spectra diﬀer from each other on at most countably many isolated points.

79 Proposition 3.4 ([1, 60]) The sets σse(A) \ σgk(A), σes(A) \ σk(A), σes(A) \ σgk(A) and σk(A) \ σgk(A) are at most countable.

Note that all these spectra and the semi-Fredholm spectra can by ordered as follows,

σgk(A) ⊆ σk(A) ⊆ σes(A) ⊆ σse(A),

σgk(A) ⊆ σk(A) ⊆ σes(A) ⊆ σsf (A) ⊆ σef (A),

σec(A) ⊆ σes(A) ⊆ σse(A), and σec(A) ⊆ σes(A) ⊆ σsf (A) ⊆ σef (A) ⊆ σew(A) ⊆ σeb(A).

Proposition 3.5 ([1]) For A ∈ L(X) the following properties hold:

1. If λ ∈ ∂σef (A) is non-isolated point of σef (A) then λ ∈ σk(A).

2. ∂σef (A) ⊆ σes(A).

3. σse(A) \ σk(A) is at most countable.

Moreover, similar statements hold if, instead of boundary points of σef (A), we consider boundary points of σlf (A), σuf (A) and σsf (A).

Remark 3.2 Let A ∈ Φ±(X) such that jump(A) > 0, since A is semi-Fredholm operator there exists 1 > 0 for which λI − A ∈ Φ±(X) for all 0 ≤ |λ| < 1. On other hand, A is of Kato type, then by Theorem 3.11, there exists 2 > 0 such that λI − A is semi-regular for all 0 < |λ| < 2. Consequently, jump(λI − A) = 0 for all 0 < |λ| < with = min{1, 2}. We conclude that if A ∈ Φ±(X) such that jump(A) > 0, then there exists > 0, such that λI −A is semi-regular operator for all 0 < |λ| < , Hence there exists a sequence of semi-regular operators (An)n≥1 such that An −→ A.

For A ∈ L(X), there are two linear subspaces of X deﬁned in [83], the quasi-nilpotent part H0(A) of A:

n n 1 o H0(A) = x ∈ X : lim kA xk n = 0 n→∞ and the analytical core K(A) of A:

K(A) = {x ∈ X : there exist a sequence (xn) in X and a constant δ > 0 such that

n Ax1 = x, Axn+1 = xn and kxnk ≤ δ kxk for all n ∈ N}

It easily follows, from the deﬁnitions, that H0(A) and K(A) are generally not closed and A(K(A)) = K(A). Observe that if Y is a closed subspace of X such that A(Y ) = Y , then Y ⊂ K(A) [106, Proposition 2]. Furthermore, if A is quasi-nilpotent then H0(A) = X. Theorem 3.12 ([1]) Suppose that (M,N) is a GKD for A ∈ L(X). Then we have:

1. K(A) = K(A|M) and K(A) is closed;

2. K(A) ∩ N(A) = N(A|M).

80 Theorem 3.13 ([1]) Assume that A ∈ L(X), admits a GKD (M,N). Then

H0(A) = H0(A|M) ⊕ H0(A|N) = H0(A|M) ⊕ N (3.5)

Theorem 3.14 ([37]) Assume that A ∈ L(X), X a Banach space. The following assertions are equivalent:

1. 0 is an isolated point in σ(A);

2. K(A) is closed and X = K(A) ⊕ H0(A)

3. H0(A) is closed and X = K(A) ⊕ H0(A)

4. there is a bounded projection P on X such that R(P ) = K(A) and N(P ) = H0(A). Here ⊕ denotes the algebraic direct sum.

Theorem 3.15 ([1]) Assume that A ∈ L(X), X a Banach space. The following assertions are equivalent:

1. 0 is a pole of the resolvent of A;

p p 2. There exists p ∈ N such that K(A) = R(A ) and H0(A) = N(A ).

Motivated by the relation of the essential quasi-Fredholm spectrum and the closed range spectrum given in Theorem 3.8, we study this relation in the case of the generalized Kato spectrum. We begin with the following result

Proposition 3.6 If λ ∈ σec(A) is non-isolated point then λ ∈ σgk(A).

Proof. Let λ ∈ σec(A) be a non-isolated point. Assume that λI −A admits a GKD(M,N). Then by Theorem 3.10 there exists an open disc D(λ, ) such that µI − A is semi-regular in D(λ, )\{λ}, so that R(µI −A) is closed if µ ∈ D(λ, )\{λ}. This contradicts our assumption that λ is a non-isolated point.

Theorem 3.16 The symmetric diﬀerence σgk(A)∆σec(A) is at most countable.

Proof. We have

σgk(A)∆σec(A) = (σgk(A) ∩ (C \ σec(A))) ∪ (σec(A) ∩ (C \ σgk(A)))

From Proposition 3.6 the set σec(A) \ σgk(A) is at most countable, we have C \ σec(A) = S∞ 1 m=1 R( m ) and ∞ [ 1 σ (A) ∩ ( \ σ (A)) = (σ (A) ∩ R( )). gk C ec gk m m=1 1 To ﬁnish the proof we prove that the set σgk(A) ∩ R( m ) is at most countable. Let λ0 be 1 1 a non-isolated point of σgk(A) ∩ R( m ). Then there exists (λn)n ⊂ σgk(A) ∩ R( m ) such that λn → λ0, by Theorem 3.1 λ0I − A is semi-regular operator, hence λ0 ∈/ σgk(A). This contradicts the closedness of σgk(A).

Proposition 3.7 σse(A) \ (σgk(A) ∩ σec(A)) is at most countable.

81 Proof. We have

σse(A) \ (σgk(A) ∩ σec(A)) = (σgk(A)∆σec(A)) ∪ σse(A) \ (σgk(A) ∪ σec(A))

Since the sets σse(A) \ σgk(A), σse(A) \ σec(A) are at most countable, Theorem 3.16 implies that σgk(A)∆σec(A) is at most countable, establishing the result.

The fact that σk(A) ⊆ σes(A) ⊆ σse(A) then we have

Corollary 3.2 σes(A)\(σgk(A)∩σec(A)) and σk(A)\(σgk(A)∩σec(A)) are at most countable.

Proposition 3.8 If λ ∈ ∂σ(A) is a non-isolated point, then λ ∈ σgk(A).

Proof. Let λ ∈ ∂σ(A) a non-isolated point. Since ∂σ(A) ⊆ σse(A), then λ ∈ σse(A) is non-isolated point, hence λ ∈ σgk(A). Example 3.1 Let X = l2 the space of complex square-summable sequences and the linear operator A deﬁned by 1 1 Ax = (0, x , 0, x , 0, x , 0,... ), x = (x ) ∈ `2 1 3 2 5 3 n

The operator A is compact and R(A) is not closed, then 0 ∈ σec(A). It easy to see that 2 A = 0, so 0 ∈/ σgk(A), σec(A) = {0} and σgk(A) = ∅.

Example 3.2 Let Pg(X) be the class of operators on a Banach space X which satisfy a polynomial growth condition. An operator A satisfies this condition if there exists K > 0, and δ > 0 for which δ kexp(iλA)k ≤ K(1 + |λ| ) for all λ ∈ R, Examples of operators which satisfy a polynomial growth condition are Hermitian operators on Hilbert spaces, nilpotent and projection operators, algebraic operators with real spectra. It is shown that Pg(X) coincides with the class of all generalized scalar operators having real spectra. We first note that the polynomial growth condition may be reformulated as follows (see [1]) : A ∈ Pg(X) if and only if σ(A) ⊆ R and there is a constant K > 0, and δ > 0 such that −1 −δ (λI − A) ≤ K(1 + |Imλ| ) for all λ ∈ C with Imλ 6= 0, (3.6) The following proposition establishes the finiteness of the ascent of a linear operator A ∈ Pg(X).

Proposition 3.9 ([1]) Assume that A ∈ Pg(X), for every λ ∈ σ(T ) we have: 1. a(λI − A) < ∞ .

2. R((λI − A)p) = R((λI − A)p+k); k ∈ N. and p = a(λI − A).

Proposition 3.10 Let A ∈ Pg(X), we have:

1. If λ ∈ σ(A) \ σec(A), then λ is an isolated point in σ(A).

p 2. If λ ∈ σec(A) and R((λI − A) ) is closed for some p ∈ N, then λ is a pole of the resolvent of A.

82 Proof. 1. If we assume that A ∈ Pg(X) and R((λI −A)) is closed for some λ ∈ C then also p = a(λI−A) is ﬁnite, R((λI−A))+N((λI−A)p) is closed and R((λI−A))+N((λI−A)p) = R((λI − A)) + N((λI − A)n) for all n ≥ p. Since a(λI − A) < ∞, it follows by [18, Theorem 2.5] that λ is an isolated point in σ(A). 2. If R((λI − A)p) is closed, then R((λI − A)p) = R((λI − A)p+k); k ∈ N, so d(λI − A) < ∞, it follows that λ is a pole of the resolvent of A.

Corollary 3.3 Let A ∈ Pg(X), then σgk(A)∆σec(A) is at most countable.

Proof. From Proposition 3.10, if λ∈ / σec(A), then λ is a an isolated point in σ(A). This implies By [37, Theorem 6.7] that A admits GKD and λ∈ / σgk(A) and the set σgk(A)\σec(A)) is empty. Now if λ ∈ σec(A), we have two cases. First if there exists p ∈ N such that p R((λI−A) ) is closed, by Proposition 3.10 part 2, λ is a pole of the resolvent and λ∈ / σgk(A)), p thus σec(A)\σgk(A) is at most countable . Now if R((λI −A) ) is not closed for every p ∈ N, p then R(((λI − A)/M ) ) is not closed for every A-invariant closed subset M and p ∈ N, so λI − A does not admits GKD and λ ∈ σgk(A). The set σec(A) \ σgk(A) is then empty.

Remark 3.3 Similar results of Proposition 3.6, Theorem 3.16, Proposition 3.7 and Corol- lary 3.3 holds if, instead of the generalized Kato spectrum σgk(A), we consider the Kato spectrum σk(A),see [15].

3.5 Saphar operators, essentially Saphar operators and corresponding spectra

In this section we study a class of operators studied by P. Saphar [100], as the "complemented" version of the semi-regular operators. Deﬁnition 3.6 An operator A ∈ L(X) is called Saphar operator if A has a generalized inverse and N(An) ⊆ R∞(A). n ∞ A is called essentially Saphar operator if A is has a generalized inverse and N(A ) ⊆e R (A). Equivalently, A is Saphar (resp. essentially Saphar) operator if and only if A is semi- regular (resp. essentially semi regular) operator and has a generalized inverse. Obviously, in Hilbert spaces the Saphar (resp. essentially Saphar) operators coincide with the semi-regular (resp. essentially semi regular) operators. Let

Sa(X) = {A ∈ L(X): A is Saphar operator } and Se(X) = {A ∈ L(X): A is essentially Saphar operator } Saphar operators have an important property

Theorem 3.17 An operator A ∈ Sa(X) if and only if there is a neighborhood U ⊆ C of 0 and a holomorphic function F : U −→ L(X) such that

(λI − A)F (λ)(λI − A) = (λI − A) and F (λ)(λI − A)F (λ) = F (λ), for all λ ∈ U.

Let us remark that for F it is possible to take

∞ X F (λ) = λnBn+1 n=0

83 −1 where B ∈ L(X) is a generalized inverse of A, and U = {λ ∈ C : |λ| < kBk }. Further F (λ) − F (µ) = (λ − µ)F (λ)F (µ) for all λ, µ ∈ U,

i.e. F (λ) satisﬁes the resolvent identity on U. Theorem 3.17 shows that the set

ρsr(A) = {λ ∈ C :(λI − A) is Saphar operator } is open. The next theorem shows that if it is possible to ﬁnd a global analytic genaral inverse of (λI − A).

Theorem 3.18 Let A ∈ L(X), there exists an holomorphic function S : ρsr(A) −→ L(X) such that

(λI − A)S(λ)(λI − A) = (λI − A) and S(λ)(λI − A)S(λ) = S(λ), for all λ ∈ ρsr(A).

Remark 3.4 The existence of global analytic general resolvent of λI−A is an open question.

The Saphar spectrum of a bounded operator A on X is deﬁned by

σsa(A) := {λ ∈ C : λI − A/∈ Sa(X)} and its essential version by

σesa(A) := {λ ∈ C : λI − A/∈ Se(X)}.

The sets σsa(A) and σesa(A) were studied (under various names and notations) by many authors, see [69, 108, 109, 89] , are always non-empty compact subsets of the complex plane. Clearly, σse(A) ⊆ σsa(A) and σes(A) ⊆ σesa(A)

If H is a Hilbert space, then σse(A) = σsa(A) and σes(A) = σesa(A) Now we recall some results about σsa(A) and σesa(A)

∗ Theorem 3.19 ([109, Proposition 1]) Let A ∈ L(X). σsa(A) ⊂ σsa(A ) and in general ∗ σsa(A) 6= σsa(A ).

Theorem 3.20 ( [69, Corollary 2.14]) Let A ∈ L(X). For any analytic function f in a neighborhood of σ(A), we have

σsa(f(A)) = f(σsa(A)) and σesa(f(A)) = f(σesa(A)).

The properties of σi (i = se, es, ec, qf, k, gk,) are summarized in the following table:

84 (P1) (P2) (P3) (P4) (P5) (P6) (P7) σi 6= ∅ σi closed Small com. com. ﬁn. com. comp. com. quasi sp. map. pert. rank pert. pert. nilp. pert. theorem

σse(A) yes yes yes no no yes yes

σes(A) yes yes yes yes yes yes yes

σec(A) no no no no no no no

σqf (A) no yes no yes no no yes

σk(A) no yes no ? no no ?

σgk(A) no yes no ? no yes ?

Table 3.1:

Comments.

1. It well-known that ∂σ(A) ⊆ σse(A) and ∂σef (A) ⊆ σes(A), so both are non-empty (for inﬁnite dimentional Banach spaces).

2. For property (P3) for σse and σes see [90]. 3. For semi-regular and essentially semi-regular operators the property (P6) was proved in [68], for σgk is proved in [16]. 4. The stability of essentially semi-regular spectrum under commuting compact perturbation was shown in [98], and under not necessary commuting ﬁnite rank perturbation in [67].

5. Observation 6 after Table 2.1 sows that (P4) and (P5) fail for semi-regular operators.

6. Since every operators commutes with the zero operator, σqf (A) cannot have properties (P1), (P3), (P5) and (P6). The property (P4) for σqf (A) is proved in [66].

7. The spectral mapping theorem for σqf in a Hilbert space was proved in [24] ( and hence for σk). For Banach spaces the theorem hold for every function f non constant on each component of its domain of deﬁnition (see[69]).

8. The boxes marked by "?" represent open problems.

85 Chapter 4

Essential spectra deﬁned by means of restrictions

4.1 Descent spectrum and essential descent spectrum

n n+1 Let A ∈ L(X), and consider the decreasing sequence cn(A) = dim R(A )/R(A ), n ∈ N, see [62]. the esential descent of A is

de(A) = inf{n ∈ N : cn(A) < ∞} (the infimum of an empty set is defined to be ∞). We say that A has finite essential descent if de(A) < ∞. Clearly, every lower semi-Fredholm operators has finite essential descent and we have de(A) = 0. This class of operators contain also every operator of finite descent. The notion of essential descent was studied by many authors, for instance, we cite [11, 49, 50, 51, 78]. The descent and the essential descent spectrum are defined respectively by σd(A) = {λ ∈ C : d(λI − A) = ∞}; e σd(A) = {λ ∈ C : de(λI − A) = ∞}; We begin by the following result which shows that an operator with finite essential descent is either semi-regular or 0 is an isolated point of its semi-regular spectrum.

Theorem 4.1 Let A ∈ L(X) be an operator with ﬁnite essential descent, then there exists an open disc D(0, ) and a positive integer d such that for all λ ∈ D(0, ) \{0}, we have the following assertions:

1. λI − A is both semi-regular and lower semi-Fredholm operator,

2. β((λI − A)n) = n dim(R(Ad)/R(Ad+1)),

3. α((λI − A)n) = n dim(N(Ad+1)/N(Ad)).

The proof of this theorem requires the following lemma.

Lemma 4.1 Let A ∈ L(X) be an operator is both lower semi-Fredholm and semi-regular, then β(An) = nβ(A) for all n ∈ N. Proof. Let n ≥ 2 and S : X −→ X/R(A) be the operator given by Sx = An−1x + R(An). Since A is semi-regular operator, we have N(S) = R(A) + N(An−1) = R(A), and consequently X/R(A) is isomorph to R(An−1)/R(An). On other hand, it is well-known that

86 X/R(An−1) × R(An−1)/R(An) ∼= X/R(An). Therefore X/R(An−1) × X/R(A) ∼= X/R(An), and hence β(An) = β(An−1) + β(A). By induction, β(An) = nβ(A).

Proof of Theorem 4.1. If de(A) < ∞ , we put d = inf{n ∈ N : cn(A) = cp(A) for all p ≥ n}. Clearly, de(A) ≤ d, and if d(A) < ∞ then we have d(A) = d. We d denote by Ad the restriction of A ∈ L(X) on the subspace R(A ). We deﬁne a new norm on R(Ad) by |y| = kyk + inf{kxk : x ∈ X and y = Ax}, for all y ∈ R(Ad) d It is easy to verify that R(A )) equipped with this norm is a Banach space, and that Ad is d a bounded operator on (R(A ), |.|). Hence it follows that Ad is both lower semi-Fredholm d+1 and semi-regular. In fact, Ad is lower semi-Fredholm because R(Ad) = R(A ) is of ﬁnite d codimension in R(A ). Moreover, since de(A) < ∞ , [50, Theorem 3.1] ensures that

d n+d n+d n N(Ad) = N(A) ∩ R(A ) = N(A) ∩ R(A ) ⊆ R(A ) = R(Ad ) for all n ∈ N.

Let > 0 be such that for every 0 < |λ| < , λI − Ad both semi-Fredholm and semi-regular, k k it follows then that, since N(λI − A) = N(λI − Ad) ⊆ R((λI − Ad) ) ⊆ R((λI − A) ) for all k ∈ N, λI − A is also semi-regular. On the other hand, by Corollary 1.2 and Lemma 4.1 we obtain n n d d+1 β((λI − A) ) = β((λI − Ad) ) = nβ(Ad) = n dim(R(A )/R(A ) In prticular, λI − A is lower semi-Fredholm. For the last statement, we have

n n α((λI − A) ) = α(λI − Ad) ) n n = ind((λI − Ad) ) + β((λI − Ad) )

= n[ind(λI − Ad) + β(λI − Ad)]

= n[ind(Ad) + β(Ad)] = nα(Ad) = n dim(R(Ad) ∩ N(A))

But, since Ad induces an isomorphism from N(Ad+1)/N(Ad) onto R(Ad) ∩ N(A), we obtain the desired result.

As a direct consequence of the preceding theorem and Lemma 1.4, the following result shows that an operator with ﬁnite descent is either surjective or 0 is an isolated point of its surjective spectrum.

Corollary 4.1 Let A ∈ L(X) be an operator with ﬁnite descent d = d(A), then there exists an open disc D(0, ) for which λI − A is onto and α(λI − A) = dim(N(A) ∩ R(Ad)) for all λ ∈ D(0, ) \{0} Also as immediate consequence of Theorem 4.1 and Corollary 4.1, we have

e Corollary 4.2 If A ∈ L(X), then σd(A) and σd(A) are a compact subset of C. Moreover e σd(A) \ σd(A) is an open set. Proof. The ﬁrst assertion follows directly from Theorem 4.1 and Corollary 4.1. For the e second, let λ ∈ σd(A) \ σd(A) and d be as in the proof of Theorem 4.1. Then there exist a e punctured open neighborhood V of λ such that V ⊆ ρd(A) and for all µ ∈ V and n ∈ N, β((µI − A)n) = n dim(R(µI − A)d/R(µI − A)d+1)

87 Since µI − A has inﬁnite descent, dim(R(µI − A)d/R(µI − A)d+1) is non-zero, and conse- n quently {β((µI−A) )}n is a strictly increasing sequence for each µ ∈ V . Thus V ⊂ σd(A).

The spectral mapping theorem holds for the descent and the essential descent spectrum (see [87]):

Theorem 4.2 Let A ∈ L(X) and f be an analytic function an an open neighborhood of σ(A), not identically constant on each connected component of its domain, then

e e σd(f(A)) = f(σd(A)) and σd(f(A)) = f(σd(A)).

It is clear that the descent spectrum, and therefore the essential descent spectrum, of an operator can be empty. In the next theorem we show that this occurs for algebraic operators, that is, there exists a non-zero complex polynomial p for which p(A) = 0.

Theorem 4.3 Let A ∈ L(X). Then

e ρd(A) ∩ ∂σ(A) = ρd(A) ∩ ∂σ(A) = σdis(A).

Moreover, the following assertions are equivalent:

1. σd(A) = ∅,

e 2. σd(A) = ∅,

3. ∂σ(A) ⊆ ρd(A),

e 4. ∂σ(A) ⊆ ρd(A), 5. A is algebraic.

By [11, Theorem 2.9] and [11, Corollary 2.10], we have

e Theorem 4.4 Let A ∈ L(X) and Ω be a connected component of ρd(A). Then

Ω ⊂ σ(A) or Ω \ σdis(A) ⊆ ρ(A).

Moreover, the following assertions are equivalent:

1. σ(A) is at most countable,

2. σd(A) is at most countable,

e 3. σd(A) is at most countable,

e In this case, σd(A) = σd(A) and σ(A) = σd(A) ∪ σdis(A). Form this theorem, it follows in particular that A ∈ L(X) is meromorphic (i.e σ(A)\{0} ⊆ e σdis(A)) if and only if σd(A) ⊆ {0}, if and only if σd(A) ⊆ {0}.

88 4.2 Ascent spectrum and essential ascent spectrum

Associated to an operator A on X we consider the non-increasing sequence

0 n+1 n cn(A) = dim N(A )/N(A ).

It follows from [62] that, for every n ∈ N,

0 n cn(A) = dim N(A) ∩ R(A ).

The esential ascent of A is

0 ae(A) = inf{n ∈ N : cn(A) < ∞}. (the infimum of an empty set is defined to be ∞). We say that A has finite essential ascent if ae(A) < ∞. Clearly, every upper semi- Fredholm operators has finite essential ascent and we have ae(A) = 0. This class of operators contain also every operator of finite ascent. Operators with finite essential ascent was studied by many authors, for instance, we cite [12, 49, 50, 51, 87]. In [51], it was established that if A ∈ L(X) has finite essential ascent then

n n R(A ) is closed for some n > ae(A) if and only if R(A ) is closed for all n ≥ ae(A) (4.1)

The ascent resolvent set and the essential ascent resolvent set of an operator A ∈ L(X) are deﬁned respectively by

a(A)+1 ρa(A) = {λ ∈ C : a(λI − A) < ∞ and R(A ) is closed };

e ae(A)+1 ρa(A) = {λ ∈ C : ae(λI − A) < ∞ and R(A ) is closed}; e e The complementary sets σa(A) = C \ ρa(A) and σa(A) = C \ ρa(A) are the ascent spectrum e and essential ascent spectrum of A, respectively. It is clear that σa(A) ⊆ σa(A) ⊆ σ(A). The next theorem shows that an operator with ﬁnite essential ascent is either semi-regular or 0 is an isolated point of its semi-regular spectrum. The proof of this theorem requires the following lemma.

Lemma 4.2 Let A ∈ L(X) be an operator is both upper semi-Fredholm and semi-regular, then α(An) = nα(A) for all n ∈ N.

Proof. Let n ∈ N. Since N(An−1) ⊆ R(A), A is a surjection from N(An) to N(An−1), and consequently α(An) = α(An−1) + α(A). Thus, a successive repetition of this argument leads to α(An) = nα(A).

e Theorem 4.5 Let A ∈ L(X) be an operator with 0 ∈ ρa(A), then there exists an open disc D(0, ) and a positive integer d such that for all λ ∈ D(0, ) \{0}, we have the following assertions:

1. λI − A is both semi-regular and upper semi-Fredholm operator,

2. α((λI − A)n) = n dim(N(Ad+1)/N(Ad)),

3. β((λI − A)n) = n dim(R(Ad)/R(Ad+1)).

89 0 0 Proof. If ae(A) < ∞ , we put d = inf{n ∈ N : cn(A) = cp(A) for all p ≥ n}. Clearly, ae(A) ≤ d, and if a(A) < ∞ then we have a(A) = d. We denote by Ad the restriction of d A ∈ L(X) on the subspace R(A ). Hence it follows that Ad is both upper semi-Fredholm and semi-regular. In fact, Ad is upper semi-Fredholm because N(Ad) has ﬁnite dimension, d −d d+1 d and R(Ad) = (R(A)+N(A ))/N(Ad) = A (N(A ))/N(A ) is closed. On the other hand, we have

d n+d n+d n N(Ad) = N(A) ∩ R(A ) = N(A) ∩ R(A ) ⊆ R(A ) = R(Ad ) for all n ∈ N, which proves that Ad is semi-regular. Hence there exists > 0 such that for every 0 < |λ| < , λI − Ad is both semi-regular and upper semi-Fredholm with α(λI − Ad) = α(Ad). For λ ∈ C with 0 < |λ| < , we have

n n d n d d N((λI − Ad) ) = N((λI − A) )/N(A ) = (N((λI − A) ) ⊕ N((A )))/N((A )) (4.2)

d d d R(λI − Ad) = (R(λI − A) + N((A ))/N((A ) = R((λI − A))/N((A ) (4.3) n Consequently, R(λI−A) is closed and contains the ﬁnite dimensional subspace N((λI−Ad) ) for all n ∈ N. This implies the ﬁrst statement. For the second, by (4.2) and the previous Lemma,

n n α((λI − A) ) = α(λI − Ad) )

= nα(λI − Ad))

= nα(Ad)) = n dim(N(Ad+1)/N(Ad)).

Now by the continuity of the index we get

β((λI − A)n) = codimR((λI − A)n)/N(Ad) n = β((λI − Ad) ) n n = α(λI − Ad) ) − ind((λI − Ad) )

= n[α(Ad) − ind(Ad)] = nβ(Ad) = n dim X/(R(A) + N(Ad)) = n dim(R(Ad)/R(Ad+1)

As a corollary of the previous theorem and Lemma 1.3, the following result shows that an operator with ﬁnite ascent is either bounded below or 0 is an isolated point of its approximate spectrum.

e Corollary 4.3 Let A ∈ L(X) be an operator with 0 ∈ ρa(A), then there exists an open disc D(0, ) for which λI − A is bounded below and β((λI − A)n) = n dim(R(Aa(A))/R(Aa(A)+1)) for all λ ∈ D(0, ) \{0} Also as immediate consequence of Theorem 4.5 and Corollary 4.3, we have

e Corollary 4.4 If A ∈ L(X), then σa(A) and σa(A) are a compact subset of C. Moreover e σa(A) \ σa(A) is an open set. The spectral mapping theorem holds for the ascent and the essential ascent spectrum (see [87]):

90 Theorem 4.6 Let A ∈ L(X) and f be an analytic function an an open neighborhood of σ(A), not identically constant on each connected component pf its domain, then

e e σa(f(A)) = f(σa(A)) and σa(f(A)) = f(σa(A)). The ascent spectrum and the essential ascent spectrum of an operator is not necessarily non-empty, this occurs for algebraic operators. Theorem 4.7 Let A ∈ L(X). Then

e ρa(A) ∩ ∂σ(A) = ρa(A) ∩ ∂σ(A) = σdis(A). Moreover, the following assertions are equivalent:

1. σa(A) = ∅,

e 2. σa(A) = ∅,

3. ∂σ(A) ⊆ ρa(A),

e 4. ∂σ(A) ⊆ ρa(A), 5. A is algebraic. By [12, Theorem 2.9] and [12, Corollary 2.10], we have

e Theorem 4.8 Let A ∈ L(X) and Ω be a connected component of ρa(A) . Then

Ω ⊂ σ(A) or Ω \ σdis(A) ⊆ ρ(A).

Moreover, the following assertions are equivalent: 1. σ(A) is at most countable,

2. σa(A) is at most countable,

e 3. σa(A) is at most countable,

e In this case, σa(A) = σa(A), and σ(A) = σa(A) ∪ σdis(A). Form this theorem, it follows in particular that A ∈ L(X) is meromorphic (i.e σ(A)\{0} ⊆ e σdis(A)) if and only if σa(A) ⊆ {0}, if and only if σa(A) ⊆ {0}.

4.3 Essential spectrum and Drazin invertible operators

An operator A ∈ L(X) is said to be Drazin invertible if there exists an operator AD ∈ L(X) such that AAD = ADA, ADAAD = AD,Ak+1AD = Ak for some nonnegative integer k. The operator AD is said to be a Drazin inverse of A. It follows from [65] that AD is unique. The smallest k in the previous deﬁnition is called as the Drazin index of A and denoted by i(A). For A ∈ L(X), if a(A) < ∞ and R(Aa(A)+1) is closed, then A is said to be left Drazin invertible. If d(A) < ∞ and R(Ad(A)) is closed, then A is said to be right Drazin invertible. If a(A) = d(A) < ∞, then A is said to be Drazin invertible. Clearly,

91 A ∈ L(X) is both left and right Drazin invertible if and only if A is Drazin invertible. If ae(A)+1 ae(A) < ∞ and R(A ) is closed, then A is said to be left essentially Drazin invertible. de(A) If de(A) < ∞ and R(A ) is closed, then A is said to be right essentially Drazin invertible. A is said to be essentially Drazin invertible (resp. semi-essentially Drazin invertible) if A is left essentially Drazin invertible and (resp. or) right essentially Drazin invertible. For A ∈ L(X), let us deﬁne the left Drazin spectrum, the right Drazin spectrum, the Drazin spectrum, the left essentially Drazin spectrum, right essentially Drazin spectrum, essentially Drazin spectrum and semi-essentially Drazin spectrum the of A as follows, respectively:

σLD(A) = {λ ∈ C : λI − A is not a left Drazin invertible operator};

σRD(A) = {λ ∈ C : λI − A is not a right Drazin invertible operator};

σD(A) = {λ ∈ C : λI − A is not a Drazin invertible operator}; e σLD(A) = {λ ∈ C : λI − A is not a left essentially Drazin invertible operator}; e σRD(A) = {λ ∈ C : λI − A is not a right essentially Drazin invertible operator}, e σD(A) = {λ ∈ C : λI − A is not a essentially Drazin invertible operator}, e σSD(A) = {λ ∈ C : λI − A is not a semi-essentially Drazin invertible operator}. These spectra have been extensively studied by several authors, see e.g [6, 18]. We have

e e σD(A) = σLD(A) ∪ σRD(A), σLD(A) = σa(A) ⊂ σap(A), σLD(A) = σa(A) ⊂ σap(A). and e e e e e e σSD(A) = σRD(A) ∩ σLD(A), σD(A) = σRD(A) ∪ σLD(A). It is well know that A is Drazin invertible if and only if A is ﬁnite ascent and descent, which is also equivalent to the fact that A = R ⊕ N where R is invertible and N is nilpotent (see [78, Corollary 2.2]).

Corollary 4.5 If A ∈ L(X) then σgk(A) ⊆ σD(A)

Theorem 4.9 ([6]) Let A ∈ L(X). If N ∈ L(X) is a nilpotent operator such that AN = NA. Then σLD(A) = σLD(A + N)

d Let Ad the restriction of A ∈ L(X) on the subspace R(A ). An immediate consequence of Theorem 4.5 and Theorem 4.1, we have Theorem 4.10 Let A ∈ L(X) be an operator semi-essentially Drazin invertible operator, then there exists an open disc D(0, ) and a positive integer d such that for all λ ∈ D(0, )\{0}, we have the following assertions: 1. λI − A is both semi-regular and semi-Fredholm operator,

n 2. β((λI − A) ) = nβ(Ad),

n 3. α((λI − A) ) = nα(Ad).

Theorem 4.11 Let A ∈ L(X) be an operator essentially Drazin invertible operator, then there exists an open disc D(0, ) and a positive integer d such that for all λ ∈ D(0, ) \{0}, we have the following assertions:

92 1. λI − A is both semi-regular and Fredholm operator,

n 2. β((λI − A) ) = nβ(Ad),

n 3. α((λI − A) ) = nα(Ad),

n 4. ind((λI − A) ) = nind(Ad). and by Corollary 4.1 and corollary 4.3 ,we have

Corollary 4.6 Let A ∈ L(X) Drazin invertible, then there exists an open disc D(0, ) for which λI − A is invertible for all λ ∈ D(0, ) \{0} Recently, Koliha introduced the concept of a generalized Drazin inverse [65]. The generalized Drazin inverse of A ∈ L(X) exists if and only if 0 is not accumulation point of the spectrum of A and is described as follows. If 0 is not accumulation point of the spectrum of A, then the spectral projection of A at 0 is the unique idempotent P ∈ L(X) such that (see section 1.8) AP = PA is quasi-nilpotent and A + P is invertible . Then there exist r > 0 such that λI − (A + P ) and λI − AP are invertible. From

λI − A = (λI − AP )P + (λI − (A + P ))(I − P )

it follows for any λ satisfying 0 < |λ| < r,

(λI − A)−1 = (λI − AP )−1P + (λI − (A + P ))−1(I − P ) +∞ +∞ X X = λ−n−1AnP − λn((A + P )−1)n+1(I − P ) n=0 n=0 +∞ +∞ X X = λ−n−1AnP − λn(AD)n+1. n=0 n=0 where AD = (A + P )−1(I − P ) is called the generalized Drazin inverse (g-Drazin inverse) of A. In this case A is called g-Drazin invertible. In this case, we can deﬁne the Drazin index i(A) of A as follows   0 if A is invertible, i(A) = k if AP is nilpotent of degree k ≥ 1,  ∞ if AP is quasi-nilpotent but not nilpotent,

Observe that AAD = ADA = (A + P )(A + P )−1(I − P ) = (I − P ), that is P = I − AAD. A g-Drazin invertible A ∈ L(X) has a unique decomposition A = S + Q, where S ∈ L(X) is g-Drazin invertible of Drazin index 1 or 0, Q ∈ L(X) is quasi-nilpotent, and SQ = 0 = QS. The operator S is called the core part of A.

Note that A has the Drazin inverse if and only if the point 0 is a pole of the resolvent of A. The order of this pole is equal to i(A). Particularly, it follows that 0 is not the point

93 of accumulation of the spectrum of A. Furthermore, the comparison between the Browder resolvent deﬁned in the subsection 2.3.1 and the the Drazin inverse gives

D RB(λ, A) = (λI − A) + Pλ

where Pλ is the Riesz projection associated to the point λ, and we have the following results

Corollary 4.7 Let A ∈ L(X) and λ0 be a point of the spectrum of A such that R(λ0I − A) is closed. Then the statements (1)−(8) of Theorem 2.14 are equivalents to λ0I −A is Drazin invertible with index m.

4.4 B-Fredholm, B-Browder and B-Weyl spectra

In 1958 Kato proved that a closed semi-Fredholm operator is of Kato type. In 1987 J.P Labrousse [70] studied and characterized a new calss of operators named quasi-Fredholm operators, in the case of Hilbert spaces, and he proved that this class coincides with the set of Kato type operators. In 1999 M. Berkani [19] studied a class of bounded linear quasi-Fredholm operators acting on a Banach space X called B-Fredholm operators and characterized a B-Fredholm operators as the direct sum of a nilpotent operator and a Fred- holm operator. In 2000 M. Berkani and M. Sarih, [26], studied the class of semi-B-Fredholm operators and proved, in Hilbert spaces, that every semi-B-Fredholm is a direct sum of a nilpotent operator and a Fredholm operator. Recently (2008) in [29] Berkani extended this characterization of B-Fredholm bounded operators to the class of B-Fredholm closed linear operators acting on a Hilbert space and study its properties. n For each integer n, deﬁne An to be the restriction of A to R(A ) viewed as the map n n n from R(A ) into R(A ) (in particular A0 = A). If there exists n ∈ N such that R(A ) is closed and An is Fredholm (resp. upper semi-Fredholm, lower semi-Fredholm, Brow- der, upper semi-Browder, lower semi-Browder), then A is called B-Fredholm (resp. upper semi-B-Fredholm, lower semi-B-Fredholm, B-Browder, upper semi-B-Browder, lower semi-B- Browder). If A ∈ L(X) is upper or lower semi-B-Browder, then A is called semi-B-Browder. If A ∈ L(X) is upper or lower semi-B-Fredholm, then A is called semi-B-Fredholm. It follows n from [26, Proposition 2.1] that if there exists n ∈ N such that R(A ) is closed and An is m semi-Fredholm, then R(A ) is closed, Am is semi-Fredholm and ind(Am) = ind(An) for all m ≥ n. This enables us to deﬁne the index of a semi-B-Fredholm operator A as the index n of the semi-Fredholm operator An, where n is an integer satisfying R(A ) is closed and An is semi-Fredholm. An operator A ∈ L(X) is called B-Weyl (resp. upper semi-B-Weyl, lower semi-B-Weyl) if A is B-Fredholm and ind(An) = 0 (resp. A is upper semi-B-Fredholm and ind(An) ≤ 0, A is lower semi-B-Fredholm and ind(An) ≥ 0). If A ∈ L(X) is upper or lower semi-B-Weyl, then A is called semi-B-Weyl.

The following theorem shows that every semi B-Fredholm operator is quasi-Fredholm operator Theorem 4.12 ([26]) Let A ∈ L(X). Then A is an upper semi B-Fredholm (resp. a lower semi B-Fredholm) operator if and only if there exists an integer d ∈ N such that A is a quasi-Fredholm and such that N(A) ∩ R(Ad) is of ﬁnite dimension (resp. N(Ad) + R(A) is of ﬁnite codimension. We give now a fundamental characterizations of semi B-Fredholm operators which proves that admits a Kato decomposition.

94 Theorem 4.13 ([19]) An operator A ∈ L(X) is B-Frdholm operator if and only if there exists a pair of closed subspaces (M,N) of X such that X = M ⊕ N and

1. A(M) ⊂ M and A/M is a Fredholm operator.

2. A(N) ⊂ N and A/N is a nilpotent operator.

Theorem 4.14 ([26]) Let H be a Hilbert space. An operator A ∈ L(H) is semi B-Frdholm operator if and only if A = A1 ⊕ A2 such that A1 is a semi-Fredholm operator and A2 is a nilpotent operator.

Theorem 4.15 ([25]) Let H be a Hilbert space. An operator A ∈ L(H) is upper semi B- Frdholm operator if and only if A = A1 ⊕ A2 such that A1 is a upper semi-Fredholm operator and A2 is a nilpotent operator. Moreover, in this case we have ind(A) = ind(A1).

Corollary 4.8 ([25]) Let H be a Hilbert space. An operator A ∈ L(H) is upper semi B- Weyl operator (resp lower semi B-Weyl) if and only if A = A1 ⊕ A2 such that A1 is a upper semi-Weyl operator (resp. A1 is a lower semi-Weyl operator) and A2 is a nilpotent operator.

This classes of operators defined above motive the definition of several spectra: • The B-Ferdholm spectrum is defined by

σbf (A) := {λ ∈ C : λI − A is not B-Ferdholm}

• the semi B-Fredholm spectrum is deﬁned by

σsbf (A) := {λ ∈ C : λI − A is not semi B-Fredholm}

• the upper semi B-Fredholm spectrum is deﬁned by

σubf (A) := {λ ∈ C : λI − A is not upper semi B-Fredholm}

• the lower semi B-Fredholm spectrum is deﬁned by

σlbf (A) := {λ ∈ C : λI − A is not lower semi B-Fredholm}

• the B-Browder spectrum is deﬁned by

σbb(A) := {λ ∈ C : λI − A is not B-Browder}

• the upper semi B-Browder spectrum is deﬁned by

σubb(A) := {λ ∈ C : λI − A is not upper semi B-Browder}

• the lower semi B-Browder spectrum is deﬁned by

σlbb(A) := {λ ∈ C : λI − A is not lower semi B-Browder}

• the semi B-Browder spectrum is deﬁned by

σsbb(A) := {λ ∈ C : λI − A is not semi B-Browder}

95 • the B-Weyl spectrum is deﬁned by

σbw(A) := {λ ∈ C : λI − A is not B-Weyl}

• the upper semi B-Weyl spectrum is deﬁned by

σubw(A) := {λ ∈ C : λI − A is not upper semi B-Weyl}

• the lower semi B-Weyl spectrum is deﬁned by

σlbw(A) := {λ ∈ C : λI − A is not lower semi B-Weyl}

• while the semi B-Weyl spectrum is deﬁned by

σlbw(A) := {λ ∈ C : λI − A is not semi B-Weyl}

We have σbf (A) = σubf (A) ∪ σlbf (A)

σbw(A)) = σubw(A) ∪ σlbw(A) and σqf (A) ⊆ σbf (A) ⊆ σbw(A) ⊆ σbb(A) = σubb(A) ∪ σlbb(A). Note that all the B-spectra are compact subsets of C (see [18], [70]) , and may be empty. This is the case where the spectrum σ(A) of A is a ﬁnite set of poles of the resolvent. Furthermore σgk(A) ⊆ σk(A) ⊆ σbf (A) ⊆ σbb(A) ⊆ σbw(A). For any A ∈ L(X), Berkani have found in [18, Theorem 3.6] , the following elegant equalities: σLD(A) = σubb(A), σRD(A) = σlbb(A); e e σLD(A) = σubf (A), σRD(A) = σlbf (A);

σD(A) = σbb(A), and by [25, Lemma 2.12] we have

σubw(A) ⊂ σLD(A) ⊂ σap(A).

Theorem 4.16 ([20]) Let A ∈ L(X). Then \ σbw(A) = σD(A + F ) F ∈F (X)

The following essential spectrum named the topological uniform descent spectrum, as we know, and in our opinion, deserve further attention, recently investigated in [18, 61]:

σud(A) = {A ∈ L(X): λI − A does not have eventual topological uniform descent},

where A ∈ L(X) is said to have a topological uniform descent if there exists d ∈ N such that R(A) + N(Ad) is closed and R(A) + N(An) = R(A) + N(Ad) for all n ≥ d. Operators

96 with eventual topological uniform descent are introduced by Grabiner in [50]. It includes all classes of operators introduced above. It also includes many other classes of operators such as operators of Kato type, quasi-Fredholm operators, operators with finite descent and operators with finite essential descent, and so on. Especially, operators which have topological uniform descent for n ≥ 0 are precisely the semi-regular operators. Discussions of operators with eventual topological uniform descent may be found in [50, 18, 87, 33, 120]. From the definition, if A is a quasi Fredholm operator of degree d then A is an operator of topological uniform descent for n ≥ d. But the converse is not true(see [18, Example, p173]. Not that σud(A) may be empty, precisely when A is algebraic. we have the following inclusion.

σsbw(A) σbw(A) σud(A) ⊆ σsbf (A) ⊆ { e ⊆ σubw(A) ⊆ { ⊆ σbb(A) = σD(A), σubf (A)=σLD(A) σubb(A)=σLD(A)

σsbw(A) σbw(A) σud(A) ⊆ σsbf (A) ⊆ { e ⊆ σlbw(A) ⊆ { ⊆ σbb(A) = σD(A), σlbf (A)=σRD(A) σlbb(A)=σRD(A)

σud(A) ⊆ σsbb(A) ⊆ σbb(A) = σD(A) and σud(A) ⊆ σbf (A) ⊆ σbb(A) = σD(A). The properties (P1)-(P7) for these sets of essential spectra deﬁned above are summarized in the following table:

97 (P1) (P2) (P3) (P4) (P5) (P6) (P7) σi 6= ∅ σi closed Small com. com. ﬁn. com. comp. com. quasi sp. map. pert. rank pert. pert. nilp. pert. theorem

σbb(A) no yes no yes! no no yes

σbw(A) no yes no yes! no no ⊆

σbf (A) no yes no yes! no no yes

σsbf (A) no yes no yes! no no yes

σubf (A) no yes no yes! no no yes

σlbf (A) no yes no yes! no no yes

σubb(A) no yes no yes no no yes

σlbb(A) no yes no yes no no yes

σubw(A) no yes no yes no no ⊇

σlbw(A) no yes no yes no no ⊇

σd(A) no yes no no no no yes

e σd(A) no yes no yes no no yes

σa(A) no yes no no no no yes

e σa(A) no yes no yes no no yes

σud(A) yes no no yes no no yes

Table 4.1:

98 Comments.

1. All properties (P1)-(P7) for these sets of essential spectra σi(A), i ∈ {bb, bw, bf, sbf, ubf, lbf, ubb, lbb, ubw} are proved by Berkani in [18],[19], [20], [21], [22], [23], [25], [26], [28], [29].

2. If K is a compact operator such that R(Kn) is not closed for every positive integer n, then K is not a B-Fredholm operator. So if F is a ﬁnite rank operator, then F is a B- Fredholm operator, but K +F is not a B-Fredholm operator, otherwise K = K +F −F would be a B-Fredholm operator. Hence the class of B-Fredholm operators is not stable under compact perturbation.

e e 3. We have σa(0) = σd(0) = ∅. Since every operator commutes with 0, σa and σd cannot have properties (P1), (P3), (P5),(P6).

4. If A ∈ L(X) is the zero operator, then σ(A) = {0} = σ(A + Q), σk(A) = ∅ and σk(A) 6= {0} = σk(A+Q), k ∈ {ubb, lbb, bb}. Thus the invariance under perturbation by commuting quasi-nilpotent (more generally, Riesz) operators does not hold for σk(A), k ∈ {ubb, lbb, bb}. Moreover, B. P. Duggal in [39] proved that σbb(A) = σbb(A + N), if N is a nilpotent operator with AN = NA and if either the Banach space is a Hilbert space or the adjoint operator has the SVEP (single-valued extension property), then the nilpotent commuting perturbations preserves the upper semi-B-Browder.

5. σud cannot have properties (P1), (P3), (P5),(P6), see [66, Example 1. , 2, 3].

4.5 Essential spectra and The SVEP theory

Let A ∈ L(X). We say that A has the single-valued extension property at λ0 ∈ C, abbreviated A has the SVEP at λ0, if for every neighborhood Uλ0 of λ0 the only analytic function f : Uλ0 → X which satisﬁes the equation

(λI − A)f(λ) = 0, for all λ ∈ Uλ0 is the constant function f ≡ 0. The operator A is said to have the SVEP if A has the SVEP at every λ ∈ C.

We collect some basic properties of the SVEP (see [1]):

1. Every operator A has the SVEP at an isolated point of the spectrum.

2. If a(λI − A) < ∞, then A has the SVEP at λ.

3. If d(λI − A) < ∞, then A∗ has the SVEP at λ

For an arbitrary operator A ∈ L(X) let us consider the set

Ξ(A) = {λ ∈ C : A does not have the SVEP at λ} The following theorems describe the relationships between an operator which admits a GKD(M,N) and the points where A, or its adjoint A∗ have the SVEP.

Theorem 4.17 ([1]) Suppose that A ∈ L(X) admits a GKD(M,N). Then the following assertions are equivalent:

99 1. A has the SVEP at 0;

2. A|M has the SVEP at 0;

3. A|M is injective;

4. H0(A) = N;

5. H0(A) is closed;

6. H0(A) ∩ K(A) = {0};

7. H0(A) ∩ K(A) is closed.

Theorem 4.18 ([1]) Suppose that A ∈ L(X) admits a GKD(M,N). Then the following assertions are equivalent: 1. A∗ has the SVEP at 0;

2. A|M is surjective;

3. K(A) = M;

4. X = H0(A) + K(A);

5. H0(A) ∩ K(A) = {0};

6. X = H0(A) + K(A) is norm dense in X.

Theorem 4.19 ([1]) Let A ∈ L(X). Then

∗ σeb(A) = σef (A) ∪ Ξ(A) ∪ Ξ(A ) (4.4) and ∗ σeb(A) = σew(A) ∪ Ξ(A) = σew(A) ∪ Ξ(A ) (4.5)

Note that Ξ(A) ⊆ σap(A) and σ(A) = Ξ(A) ∪ σsu(A) ∗ In particular, if A (resp. A ) has the SVEP then σ(A) = σsu(A) (resp. σ(A) = σap(A) ). All results established above have a numerous of interesting applications. In the next theorem we consider a situation which occurs in some concrete cases.

Theorem 4.20 Let A ∈ L(X) be an operator for which σap(A) = ∂σ(A) and every λ ∈ ∂σ(A) is non-isolated in σ(A). Then σec(A) ⊆ σgk(A) = σes(A) = σse(A). Proof. Since λ ∈ ∂σ(A) is non-isolated, according to Proposition 3.8,

σap(A) = ∂σ(A) ⊆ σgk(A) ⊆ σk(A) ⊆ σes(A) ⊆ σse(A) ⊆ σap(A), that is, σgk(A) = σk(A) = σes(A) = σse(A) = σap(A) = σp(A) ∪ σec(A) and σec(A) ⊆ σgk(A) = σes(A) = σse(A).

Dually we have

100 Theorem 4.21 Let A ∈ L(X) an operator for which σsu(A) = ∂σ(A) and every λ ∈ ∂σ(A) is non-isolated in σ(A). Then σec(A) ⊆ σgk(A) = σes(A) = σse(A).

∗ Proof. Since λ ∈ ∂σ(A) is non-isolated, then σsu(A) cluster in λ. Observe that A has the SVEP at λ ∈ ∂σ(A), then λI − A does not admit a generalized Kato decomposition and thus λ ∈ σgk(A). So

σsu(A) = ∂σ(A) ⊆ σgk(A) ⊆ σk(A) ⊆ σes(A) ⊆ σse(A) ⊆ σap(A) and σgk(A) = σk(A) = σes(A) = σse(A) = σsu(A). Thus we have σec(A) ⊆ σgk(A) = σes(A) = σse(A).

Since the ascent implies the SVEP then we have

∗ Ξ(A) ⊆ σLD(A) and Ξ(A ) ⊆ σRD(A) The following theorem proves an equality up to Ξ(A) between the left Drazin spectrum and the left B-Fredholm spectrum and by duality we ﬁnd a similar result holds for the right Drazin spectrum and the right B-Fredholm spectrum. Theorem 4.22 ([6]) Let A ∈ L(X). Then

σLD(A) = σlbf (A) ∪ Ξ(A) (4.6) and ∗ σRD(A) = σubf (A) ∪ Ξ(A ) (4.7) Theorem 4.23 ([35]) Let A ∈ L(X). Then

σubb(A) = σqf (A) ∪ Ξ(A) = σubw(A) ∪ Ξ(A) (4.8) and ∗ ∗ σlbb(A) = σqf (A) ∪ Ξ(A ) = σlbw(A) ∪ Ξ(A ) (4.9) Moreover, ∗ σbb(A) = σbw(A) ∪ Ξ(A) = σbw(A) ∪ Ξ(A ) (4.10) Corollary 4.9 Let A ∈ L(X). Then we have

1. σeb(A) = σef (A) ∪ σgk(A), σD(A) = σbw(A) ∪ σgk(A) and σbb(A) = σqf (A) ∪ σgk(A). 2. If A has the SVEP then σqf (A) = σubw(A) = σubb(A) (4.11) and σbw(A) = σbb(A) = σlbb(A) = σlbw(A) (4.12) 3. If A∗ has the SVEP then

σqf (A) = σlbw(A) = σlbb(A) (4.13) and σbw(A) = σbb(A) = σubb(A) = σubw(A) (4.14)

101 ∗ 4. If both A, A have SVEP then σgk(A) is empty and

σeb(A) = σew(A) = σef (A) (4.15)

σqf (A) = σD(A) = σubb(A) = σlbb(A) = σbb(A) = σlbw(A) = σubw(A) = σbw(A) (4.16)

From the deﬁnition of loacalized SVEP it is easily seen that Ξ(A) ⊆ accσap(A); and ∗ dually Ξ(A ) ⊆ accσsu(A), where accK denote the set oﬀ all accumulation points of K ⊆ C.

Theorem 4.24 ([35]) Let A ∈ L(X) an operator for which σap(A) = ∂σ(A) and every λ ∈ ∂σ(A) is non-isolated in σ(A). Then

σqf (A) = σubb(A) = σubw(A) = σap(A) = σub(A) = σuw(A) = σse(A)

By duality we have

Theorem 4.25 ([35]) Let A ∈ L(X) an operator for which σsu(A) = ∂σ(A) and every λ ∈ ∂σ(A) is non-isolated in σ(A). Then

σqf (A) = σlbb(A) = σlbw(A) = σsu(A) = σlb(A) = σlw(A) = σse(A).

By Theorem 4.20 and Theorem 4.24 we have

Corollary 4.10 Let A ∈ L(X) an operator for which σap(A) = ∂σ(A) and every λ ∈ ∂σ(A) is non-isolated in σ(A). Then

σgk(A) = σqf (A) = σubb(A) = σubw(A) = σap(A) = σub(A) = σuw(A) = σse(A)

= σk(A) = σec(A) = σes(A)

By duality we obtain by Theorem 4.21 and Theorem 4.25

Corollary 4.11 Let A ∈ L(X) an operator for which σsu(A) = ∂σ(A) and every λ ∈ ∂σ(A) is non-isolated in σ(A). Then

σgk(A) = σqf (A) = σlbb(A) = σlbw(A) = σsu(A) = σlb(A) = σlw(A) = σse(A)

= σk(A) = σec(A) = σes(A)

Example 4.1 We consider the Cesaro operator Cp on the classical Hardy space Hp(D), where D is the open unit disc of C and 1 < p < ∞. Cp is deﬁned by

1 Z λ f(λ) (Cpf)(λ) = dµ, for all f ∈ Hp(D) and λ ∈ D. λ 0 1 − λ

p p The spectrum of the operator Cp is the closed disc Γp centred at 2 with radius 2 , see [1], and σef (Cp) ⊆ σap(Cp) = ∂Γp. From Corollary 4.10 we also have

σgk(Cp) = σqf (Cp) = σlbb(Cp) = σlbw(Cp) = σap(Cp) = σlb(Cp) = σlw(Cp)

= σse(Cp) = σk(Cp) = σec(Cp) = σes(Cp) = σef (Cp = ∂Γp

102 4.6 Weyl’s theorem and Browder’s theorem

H. Weyl examined the spectra of all compact perturbations of a hermitian operator on Hilbert space and found in 1909 that their intersection consisted precisely of those points of the spectrum which were not isolated eigenvalues of ﬁnite multiplicity. This Weyl’s theorem has since been extended to hyponormal and to Toeplitz operators [36], to seminormal and other operators [17] and recently to Banach spaces operators berkani6. Extended and variants of this theorem have been discussed in [4, 25, 27, 41, 96]. In the following, we use the abbreviations gaW , aW , gW , W , (gw), (w), (gaw) and (aw) to signify that an operator A ∈ L(X) obeys generalized a-Weyl’s theorem, a-Weyl’s theorem, generalized Weyl’s theorem, Weyl’s theorem, property (gw), property (w), property (gaw) and property (aw). Similarly, the abbreviations gaB, aB, gB, B, (gb), (b), (gab) and (ab) have analogous meaning with respect to Browder’s theorem or the properties. The following table summarizes the meaning of various theorems.

gaW σap(A)\σubw(A) = Ea(A) gaB σap(A)\σubw(A) = Πa(T ) 0 0 aW σap(A)\σuw(T ) = Ea(T ) aB σa(A)\σuw(T ) = Πa(T ) gW σ(A)\σbw(A) = E(A) gB σ(A)\σbw(A) = Π(A) 0 0 W σ(A)\σew(T ) = E (A) B σ(A)\σew(A) = Π (A) (gw) σap(A)\σubw(A) = E(A) (gb) σap(A)\σubw(T ) = Π(A) 0 0 (w) σap(A)\σuw(A) = E (A) (b) σap(A)\σuw(A) = Π (A) (gaw) σ(A)\σbw(A) = Ea(A) (gab) σ(A)\σbw(A) = Πa(A) 0 0 (aw) σ(A)\σew(A) = Ea(A) (ab) σ(A)\σew(A) = Πa(A) Where, • E(A) = {λ ∈ isoσ(A) : 0 < α(λI −A)}, the set of all eigenvalues of A isolated in σ(A).

0 • E (A) = {λ ∈ E(A): α(λI − A) < ∞} = σdis(A), the set of all eigenvalues of A of ﬁnite multiplicity isolated in σ(A).

• Ea(A) = {λ ∈ isoσap(A) : 0 < α(λI − A)}, the set of all eigenvalues of A isolated in σap(A).

0 • Ea(A) = {λ ∈ Ea(A): α(λI − A) < ∞}, the set of all eigenvalues of A of ﬁnite multiplicity isolated in σap(A). • Π(A) = {λ ∈ σ(A) : 0 < max(a(λI − A), d(λI − A)) < ∞}, the set of the poles of the resolvent of A. • Π0(A) = {λ ∈ Π(A): α(λI − A) < ∞}, the the set of the poles of the resolvent of A of ﬁnite rank.

• Πa(A) = {λ ∈ σap(A) : 0 < max(a(λI − A), d(λI − A)) < ∞}, the set of the left poles of the resolvent of A.

0 • Πa(A) = {λ ∈ Πa(A): α(λI − A) < ∞}, the set of the left poles of the resolvent of A of ﬁnite rank. The following diagram summarizes the implications between various Weyl type theorems, Browder type theorems and the various proprieties. The numbers near the implications are references to the results to the bibliography therein.

103 [28] gB (aw) ⇐= (gaw) gaB gW ⇑ [25] ⇑ [28][27] ⇑ [5] ⇑ [25] [25] [25] [28] [28] [27] gaB ⇐= gaW =⇒ gW =⇒ gB ⇐= (gab) ⇐= (gb) ⇐= (gw) m [7] ⇓ [25] ⇓ [25] m [7] ⇓ [28][27] ⇓ [5] ⇓ [25] [96] [10] [28] [28] [27] aB ⇐= aW =⇒ W =⇒ B ⇐= (ab) ⇐= (b) ⇐= (w) ⇓ [41] ⇑ [28][27] ⇓ [4] ⇓ B (aw) aB W

Weyl-Browder type theorems and the various proprieties, in their and more recently in their generalized form, have been studied by a large of authors and becomes a signiﬁcant sector of the development of spectral theory. Two important studies in this important sector are

• The stability of this Weyl-Browder type theorems and proprieties by additive small, quasi-nilpotent and compact perturbations, see [1, 4, 22, 29, 30].

• Describe the class of operators satisfying Weyl-Browder type theorems or one of this proprieties deﬁned in the table above, see [1, 17, 23, 25, 36, 96].

104 Chapter 5

Applications

5.1 One-dimensional transport equation

In this section, we shall apply the results of perturbations to the one-dimensional transport equation on Lp-spaces, with p ∈ [1, ∞). Let p Xp = L ((−a, a) × (−1, 1), dx dξ), a > 0 and p ∈ [1, ∞). We consider the boundary spaces :

o p p o o Xp := L [{−a} × (−1, 0), |ξ|dξ] × L [{a} × (0, 1), |ξ|dξ] := X1,p × X2,p and

i p p i i Xp := L [{−a} × (0, 1), |ξ|dξ] × L [{a} × (−1, 0)], |ξ|dξ] := X1,p × X2,p respectively equipped with the norms 1 1 Z 0 Z 1 p o o p o p p p p kψ kXo = kψ k o + kψ k o = |ψ(−a, ξ)| |ξ| dξ + |ψ(a, ξ)| |ξ| dξ p 1 X1,p 2 X2,p −1 0 and 1 1 Z 1 Z 0 p i i p i p p p p kψ k i = kψ k + kψ k = |ψ(−a, ξ)| |ξ| dξ + |ψ(a, ξ)| |ξ| dξ . Xp 1 Xi 2 Xi 1,p 2,p 0 −1

Let Wp the space deﬁned by: ∂ψ W = ψ ∈ X : ξ ∈ X . p p ∂x p

o i It is well-known that any function ψ in Wp has traces on −a and a in Xp and Xp. They are denoted, respectively by ψo and ψi, and represent the outgoing and the incoming ﬂuxes, with o i o ψ1 i ψ1 ψ = o and ψ = i , ψ2 ψ2 given by

o ψ1(ξ) = ψ(−a, ξ), ξ ∈ (−1, 0), o ψ2(ξ) = ψ(a, ξ), ξ ∈ (0, 1), i ψ1(ξ) = ψ(−a, ξ), ξ ∈ (0, 1), i ψ2(ξ) = ψ(a, ξ), ξ ∈ (−1, 0).

105 We deﬁne the operator TH by:

 T : D(T ) ⊆ X −→ X  H H p p  ∂ψ  ψ −→ T ψ(x, ξ) = −ξ (x, ξ) − σ(ξ)ψ(x, ξ) H ∂x   i o D(TH ) = ψ ∈ Wp such that ψ = Hψ ∞ o i Where σ(.) ∈ L (−1, 1) and H is bounded from Xp to Xp. The function ψ(x, ξ) represents the number density of gas particles having the position x and the direction cosine of propagation ξ. The variable ξ may be thought of as the cosine of the angle between the velocity of particles and the x-direction. The function σ(.), is called the collision frequency.

The spectrum of the operator T0 (i.e., H = 0) was analyzed in [88]. in particular we have

∗ σ(T0) = σc(T0) = {λ ∈ C : Reλ ≤ −λ }, (5.1)

∗ where σc(T0) is the continuous spectrum of T0 and λ = −lim inf σ(ξ) ,(for more detail see |ξ|→0 [88]). Combining the Corollary 3.1 with (5.1) we obtain

∗ σi(T0) = {λ ∈ C : Reλ ≤ −λ }, i ∈ {ec, es, se, lf, uf, ef, ew, uw, lw, eb, ub, lb} (5.2)

Let us now consider the resolvent equation for TH

(λ − TH )ψ = ϕ (5.3) where ϕ is a given element of Xp and the unknown ψ must be founded in D(TH ). For Reλ + λ∗ > 0, the solution of (5.3) is formally given by:

 Z x 0 − (λ+σ(ξ))|a+x| −(λ+σ(ξ))|x−x |  ψ(−a, ξ) e |ξ| + 1 e |ξ| ϕ(x0, ξ) dx0 if 0 < ξ < 1,  |ξ|  −a ψ(x, ξ) = Z a 0  − (λ+σ(ξ))|a−x| 1 −(λ+σ(ξ))|x−x |  |ξ| |ξ| 0 0  ψ(a, ξ) e + e ϕ(x , ξ) dx if − 1 < ξ < 0. |ξ| x

where  Z a −2a(λ+σ(ξ)) − −(λ+σ(ξ))|a−x|  ψ(a, ξ) = ψ(−a, ξ) e |ξ| + 1 e |ξ| ϕ(x, ξ) dx  |ξ|  −a 

 Z a  − 2a(λ+σ(ξ)) −(λ+σ(ξ))|a+x|  |ξ| 1 |ξ|  ψ(−a, ξ) = ψ(a, ξ) e + |ξ| e ϕ(x, ξ) dx  −a In the sequel we shall consider the following operators:

106  M : Xi −→ Xo,M u := (M +u, M −u) where  λ p p λ λ λ   −2a + |ξ| (λ+σ(ξ)) Mλ u(−a, ξ) = u(−a, ξ) e if 0 < ξ < 1   −2a  − |ξ| (λ+σ(ξ))  Mλ u(a, ξ) = u(a, ξ) e if 0 < ξ < 1  i − + Bλ : X −→ Xp; Bλu = χ(−1,0)(ξ)B u + χ(0,1)(ξ)B u where  p λ λ   −(λ+σ(ξ)) + |ξ| |a+x| (Bλ u)(−a, ξ) = u(−a, ξ) e , if 0 < ξ < 1    − (λ+σ(ξ)) |a−x|  − |ξ| (Bλ u)(−a, ξ) = u(−a, ξ) e , if − 1 < ξ < 0

 G : X −→ Xo,G ϕ := (G+ϕ, G−ϕ) where  λ p p λ λ λ   Z a  1 −(λ+σ(ξ)) |a−x|  + |ξ| Gλ ϕ = e ϕ(x, ξ) dx, if 0 < ξ < 1 |ξ| −a   Z a  1 −(λ+σ(ξ)) |a+x|  G−ϕ = e |ξ| ϕ(x, ξ) dx, if − 1 < ξ < 0  λ |ξ| −a

 C : X −→ X ; C ϕ = χ (ξ)C+ϕ + χ (ξ)C−ϕ where  λ p p λ (−1,0) λ (0,1) λ   Z x  1 −(λ+σ(ξ)) |x0−x|  − |ξ| 0 0 Cλ ϕ = e ϕ(x , ξ) dx if 0 < ξ < 1 |ξ| −a   Z a  1 −(λ+σ(ξ)) |x0−x|  C+ϕ = e |ξ| ϕ(x0, ξ) dx0 if − 1 < ξ < 0  λ |ξ| x

where χ(−1,0) and χ(0,1) denote, respectively the characteristic functions of the intervals (−1, 0) and (0, 1). The operators Mλ,Bλ,Gλ and Cλ are bounded on their respective −1 −1 domains respectively, by e−2a(Reλ+λ∗), [p(Reλ + λ∗)] p , [(Reλ + λ∗)] q and [(Reλ + λ∗)]−1 where q denotes the conjugate of p. We deﬁne the real λ0 by  ∗  −λ , if ||H|| ≤ 1 λ0 =  1 ∗ 2a log ||H|| − λ if ||H|| > 1

It follows from the norm estimate of Mλ that, for Reλ > λ0, ||MλH|| < 1 and consequently

+∞ X n ψ0 = (MλH) Gλϕ (5.4) n=0 On the other hand, we have

ψ = BλHψ0 + Cλϕ +∞ X n = (BλH (MλH) Gλ + Cλ)ϕ n=0

107 Hence, {λ ∈ C such that Reλ > λ0} ⊂ ρ(TH ) and for Reλ > λ0 −1 −1 (λ − TH ) = BλH(I − MλH) Gλ + Cλ (5.5)

−1 On the other hand, observe that the operator Cλ is nothing else but (λ − T0) . Obviously, if the operator I − MλH is boundedly invertible , (for example, if Reλ > λ0), then λ ∈ ρ(TH ) ∩ ρ(T0) and −1 −1 (λ − TH ) − (λ − T0) = Dλ, (5.6) where +∞ X n Dλ = BλH (MλH) Gλ n=0

5.1.1 Application of the Fredholm perturbations to transport equations Next we consider the transport operator

AH = TH + K where K is the collision operator given by

K : Xp −→ Xp Z 1 ψ −→ κ(x, ξ, ξ0)ψ(x, ξ0) dξ0 −1 where κ(., ., .) is a measurable function form [−a, a] × [−1, 1] × [−1, 1] to R. Observe that the operator K acts only on the variable ξ, so x may be viewed merely as a parameter in [−a, a]. Hence we may consider K :[−a, a] −→ K(x) ∈ Z = L(Lp([−1, 1], dξ)). In view of this function, we will make use of the following class of collision operators introduced in [88] and referred to as regular operators. Deﬁnition 5.1 A collision operator K is called regular if satisﬁed the following assumptions: {x ∈ [−a, a] such that K(x) ∈ O} is measurable if O ⊂ Z is open. (5.7) There exists a compact subset E ⊂ Z such that K(x) ∈ E a .e.on [−a, a] . (5.8) K(x) ∈ K(Lp([−1, 1], dξ)) a .e.on [−a, a] . (5.9) Remark 5.1 1. the assumption (5.7 ) means that t K(.) is measurable. 2. It follows form that (5.8) that K(.) ∈ L∞(]−a, a[ , Z). (5.10)

3. The collision operator is bounded operator on Xp. Indeed, let ψinp. it easy to see that (Kψ)(x, ξ) = (K(x)ψ(x, ξ))(ξ) and then by (5.10), we have

Z 1 Z 1 p p p |(Kψ)(x, ξ)| dξ ≤ kK(.)kL∞(]−a,a[,Z) |ψ(x, ξ)| dξ −1 −1 This leas to the estimate

∞ kKkXp ≤ kK(.)kL (]−a,a[,Z). (5.11)

108 4. The assumptions (5.8) and (5.9) mean that K(x, ξ, ξ0 ) is the kernel compact operator on L∞(]−1, 1[ , Z)(for x ﬁxed in [−a, a]) and that this family of operators, indexed by x ∈ [−a, a] , lies in compact subset of Z.

Proposition 5.1 ([75, Proposition 3.2]) Assume that K is a regular operator, then for ∗ −1 any λ ∈ C such that Reλ ≤ −λ , the operator (λ−TH ) K is compact (resp. weakly compact) on Xp, 1 < p < ∞ (resp. X1).

Recall that the key of this proposition is the following lemma

Lemma 5.1 ([88, Lemma 2.3]) The space of collision operators with kernels of the form

n 0 X 0 κn(x, ξ, ξ ) = αi(x)fi(ξ)gi(ξ ) i=1

∞ p q where αi(.) ∈ L (]−a, a[), fi(.) ∈ L (]−1, 1[) and gi(.) ∈ L (]−1, 1[) (q denote the conjugate of p), is dense , in the uniform topology, in the class of regular collision operators.

Now we state the main result of this section.

Theorem 5.1 Let p ∈ [1, ∞[ and suppose that collision operator K is a regular operator, then σi(AH ) = σi(TH ), i = lf, uf, sf, ef, ew, uw, lw. (5.12) Moreover, if H is a strictly singular boundary operator, then

∗ σi(AH ) = σi(T0) = {λ ∈ C : Reλ ≤ −λ }, i = lf, uf, sf, ef, ew, uw, lw. (5.13)

−1 Proof. Let λ be such that r[(λ − TH ) K] < 1 then λ ∈ ρ(AH ) and

+∞ −1 −1 X −1 n −1 (λ − AH ) − (λ − TH ) = [(λ − TH ) K] (λ − TH ) (5.14) n=1

−1 −1 By Proposition 5.1 (λ−AH ) −(λ−TH ) is compact (resp. weakly compact) on Xp, 1 < p < ∞ (resp. X1). Threfore, it follows from the inclusion K(Xp) ⊆ S(Xp) ⊆ PΦ+(Xp) ⊆ PΦ(Xp) for 1 < p < ∞ and the fact that the set of weakly compact operators consides with the set of 1 −1 −1 strictly singular on L spaces that (λ−AH ) −(λ−TH ) ∈ S(Xp) = PΦ+(Xp), 1 ≤ p < ∞. Hence by Theorem 2.23 we obtain (5.12). On other hand, the use of equation (5.6) allows us to write (5.14) in the form

+∞ −1 −1 X −1 n −1 (λ − AH ) − (λ − T0) = Dλ + [(λ − TH ) K] (λ − TH ) n=1

−1 −1 Therefore the strict singularity of H implies that of (λ − AH ) − (λ − T0) . Now, the equality (5.13) follows, arguing as above, from Theorem 2.23 and Proposition 5.1.

109 5.1.2 Application of the quasi-nilpotent perturbations to transport equations Next we consider the transport operator

AH = TH + K where K is the bounded operator given by  K : X −→ X  p p Z ξ 0 0 0  ψ −→ κ(x, ξ, ξ )ψ(x, ξ ) dξ −1 and κ satisﬁes the following assumptions:

κ(., ., .) is a measurable function form [−a, a] × [−1, 1] × [−1, 1] to and (H) R |κ(x, ξ, ξ0)| ≤ c < ∞, a.e. Lemma 5.2 If κ satisﬁes (H) then, for any integer n ≥ 1

1 +n+1 2 q kKnk ≤ cn n! 1 1 where p + q = 1.

Proof. Let ψ ∈ Xp. Holder’s inequalities implies that Z ξ 0 0 0 |Kψ(x, ξ)| = κ(x, ξ, ξ )ψ(x, ξ )dξ −1 1 1 Z ξ q Z 1 p ≤ |κ(x, ξ, ξ0)|q dξ0 |ψ(x, ξ0)|p dξ0 −1 −1 1 Z 1 p 1 0 p 0 ≤ c(ξ + 1) q |ψ(x, ξ )| dξ −1

and Z ξ 2 0 0 0 K ψ(x, ξ) = κ(x, ξ, ξ )Kψ(x, ξ )dξ −1 1 Z ξ Z 1 p 2 0 1 0 0 p 0 ≤ c (ξ + 1) q dξ |ψ(x, ξ )| dξ −1 −1 1 Z 1 p 1 1 p 2 q +1 0 0 ≤ c 1 (ξ + 1) |ψ(x, ξ )| dξ q + 1 −1 we proceed by induction to obtain

1 Z 1 p 1 1 p n n q +n 0 0 |K ψ(x, ξ)| ≤ c 1 1 1 (ξ + 1) |ψ(x, ξ )| dξ ( q + 1)( q + 2) ... ( q + n) −1 then, by Fubini’s theorem we deduce

Z a Z 1 Z ξ p 0 0 0 0 n 1 1 +n p κ(x, ξ, ξ )ψ(x, ξ )dξ dξ dx ≤ 2c (ξ + 1) q kψk Xp −a −1 −1 n! this shows the result.

110 Theorem 5.2 Let p ∈ [1, ∞[ and suppose that the collision operator satisﬁes (H) on Xp and KTH − TH K ∈ PΦ(Xp), then

σi(AH ) = σi(TH ), i = lf, uf, sf, ef, ew. (5.15)

Furthermore, if the boundary operator H is a strictly singular operator, then

∗ σi(AH ) = σi(T0) = {λ ∈ C : λ ≤ −λ }, i = lf, uf, sf, ef, ew. (5.16)

Proof. Let p ∈ [1, ∞[, by virtue of Lemma 5.2 the operator K is quasi-nilpotent. Now, by the relation (5.2) and Theorem 2.11 we obtain (5.15). Furthermore, the last result follows as in the proof of Theorem 5.1.

5.2 Singular transport operators

In this section we are concerned with the essential spectra of singular transport operators Z 0 0 0 Aψ(x, ξ) = −ξ · ∇xψ(x, ξ) − σ(ξ)ψ(x, ξ) + κ(x, ξ )ψ(x, ξ ) dξ (x, ξ) ∈ Ω × V, (5.17a) Rn

φ|Γ− (x, ξ) = 0 (x, ξ) ∈ Γ−. (5.17b) where Ω is a smooth open subset of Rn (n ≥ 1), V is the support of a positive Radon measure n p dµ on R and ψ ∈ L (Ω × V, dxdµ(ξ)) (1 ≤ p < ∞). In (5.17b) Γ− denotes the incoming part of the boundary of the phase space Ω × V

Γ− = {(x, ξ) ∈ ∂Ω × V ; ξ · n(x) < 0}

where n(x) stands for the outward normal unit at x ∈ ∂Ω. The operator A describes the transport of particle (neutrons, photons, molecules of gas, etc.) in the domain Ω. The function ψ represents the number (or probability) density of particles having the position x and the velocity ξ. The functions σ(.) and κ(., .) are called, respectively, the collision frequency and the scattering kernel. Let us ﬁrst introduce the functional setting we shall use in the sequel. Let

p Xp = L (Ω × V, dxdµ(ξ)), σ p Xp = L (Ω × V, σ(ξ)dxdµ(ξ)) 1 ≤ p < ∞, p n p n Lσ(R ) = L (R , σ(ξ)dµ(ξ)). We deﬁne the partial Sobolev space

Wp = {ψ ∈ Xp ; ξ · ∇xψ ∈ Xp}.

For any ψ ∈ Wp, one can deﬁne the space traces ψ|Γ− on Γ−,

Wfp = {ψ ∈ Wp ; ψ|Γ− = 0}.

The streaming operator T associated with the boundary condition (5.17b) is ( T : D(T ) ⊂ Xp → Xp

ψ 7→ T ψ(x, ξ) := −ξ · ∇xψ(x, ξ) − σ(ξ)ψ(x, ξ),

111 with domain σ D(T ) := Wfp ∩ Xp . The transport operator (5.17) can be formulated as follows A = T + K, where K denotes the following collision operator  K : Xp → Xp  Z 0 0 0  ψ 7→ κ(x, ξ )ψ(x, ξ ) dξ Rn We will assume that the scattering kernel κ(., .) is non-negative and there exists a closed n subset E ⊂ R with zero dµ measure and a constant σ0 > 0 such that ∞ n σ(.) ∈ Lloc(R \ E), σ(ξ) > σ0; (5.18a) 1 "Z 0 # q κ(., ξ ) 0 q p n ( 1 )dµ(ξ ) ∈ L (R ). (5.18b) 0 Rn σ(ξ ) p σ Using boundedness of Ω and the assumption (5.18b) we can ﬁned that K ∈ L(Xp ,Xp) with 1 "Z 0 # q κ(., ξ ) q 0 kKkL(Xσ,X ) ≤ ( 1 ) dµ(ξ ) (5.19) p p 0 Rn σ(ξ ) p Lp(Rn σ Note that a simple calculation using the assumption (5.18a) shows that Xp is a subset of σ Xp and the the embedding Xp ,→ Xp is continuous.

Let us now consider the resolvent equation (λI − T )ψ = ϕ (5.20) where ϕ is a given element of Xp and the unknown ψ must be founded in D(T ). For Reλ > −σ0, the solution of (5.20) reads as follows Z t(x,ξ) ψ(x, ξ) = e−(λ+σ(ξ))sϕ(x − sξ, ξ) ds (5.21) 0 where t(x, ξ) = sup{ t > 0 ; x − sξ ∈ Ω, ∀ 0 < s < t } = inf{ s > 0 ; x − sξ∈ / Ω}. Lemma 5.3 The collision operator K is T -bounded.

Proof. Let λ ∈ C be such that Reλ > −σ0 and consider ψ ∈ Xp. It follows from (5.21) that Z Z −1 p 1 p (λI − T ) ψ(x, ξ) dx ≤ p |ψ(x, ξ)| dx Ω (Reλ + σ(ξ)) Ω Therefore, −1 σ(ξ) (λI − T ) ψ σ ≤ sup kψk (5.22) Xp p Xp ξ∈Rn (Reλ + σ(ξ)) −1 σ Hence, (λI − T ) ∈ L(Xp,Xp ). Using now the equation (5.19) to deduce that the operator K is T -bounded. The spectrum of the operator T was analyzed in [88, 77] and we have

σ(T ) = σc(T ) = {λ ∈ C : Reλ ≤ −σ0}, (5.23) Again, by Corollary 3.1 and (5.23) we obtain

σi(T ) = {λ ∈ C : Reλ ≤ −σ0}, i ∈ {ec, es, se, lf, uf, ef, ew, uw, lw, eb, ub, lb}. (5.24)

112 Proposition 5.2 ([77, Proposition 4.1]) Let Ω be a bounded subset of Rn and 1 < p < ∞. If the the hypotheses (5.18a) and (5.18b) are satisﬁed, the measure dµ satisﬁes ( the hyperplanes have zero dì measure, i.e., (5.25) for each, e ∈ Sn−1, dµ{ξ ∈ Rn, ξ.e = 0} = 0,

n−1 n p n p n where S denotes the unit ball of R and the collision operator K : Lσ(R ) −→ L (R ) is −1 compact. Then for any λ ∈ C such that Reλ > −σ0, the operator K(λI − T ) is compact on Xp. Now we are in position to state the main result of this section.

Theorem 5.3 Assume that the hypotheses of Proposition 5.2 are satisﬁed. Then

σi(A) = {λ ∈ C : Reλ ≤ −σ0}, i ∈ {lf, uf, sf, ef, ew, uw, lw}. (5.26)

Proof. Let λ ∈ C be such that Reλ > −σ0. It follows from (5.22) that,

−1 σ(ξ) (λI − T ) σ ≤ sup . L(Xp,Xp ) p ξ∈Rn (Reλ + σ(ξ))

So, since Xσ is continuously embedded in X , we infer that lim kK(λI − T )−1k = p p λ→+∞ L(Xp) 0. Therefore, there exists λ ∈ ρ(T ) such that r(K(λI − T )−1) < 1. For such λ we have

+∞ X (λI − A)−1 − (λI − T )−1 = (λ − T )−1[K(λI − T )−1]n (5.27) n=1

−1 −1 By using Proposition 5.2, (λI−A) −(λI−T ) is compact on Xp, 1 < p < ∞. Therefore, it follows from the inclusion K(Xp) ⊆ S(Xp) = PΦ+(Xp) = PΦ(Xp) for 1 < p < ∞. Hence by Theorem 2.23 we obtain (5.26).

113 Conclusion and perspectives

The main thrust of this thesis is in the spirit of the spectral theory and operator theory; its aim is to give a survey of various characteristic stability properties of diﬀerent notions of essential spectrum under perturbations belonging to a large of class operators ( Fredholm perturbations, compact, quasi-nilpotent, . . . ), as well as some equivalent descriptions of these spectra, and many cases when these essential spectra coincide or diﬀer from each other on at most countably many isolated points. The results obtained are used for describing the essential spectra of some transport operators.

By the preceding chapters, some questions do, however, remain open to further investigation. This is not intended to be a complete survey of such problems but a sample of those that the authors considers to be of greatest interest.

1. According to the Table 3.1, It was natural to ask the following questions:

(a) The Kato spectrum and the generalized Kato spectrum must be stable under additive commuting ﬁnite rank operators? (b) If f is an analytic function deﬁned on a neighborhood of σ(A), must

f(σk(A)) = σk(f(A)) and f(σgk(A)) = σgk(f(A))?

Note that, in the Hilbert space, the spectral mapping theorem for σk(A) was proved in [24] .

2. In view of chapter 5, an interest research is the use of essential spectra and their perturbations in the spectral analysis of the Cauchy abstract problem and associated perturbed Cauchy abstract problem, to derive sufficient or necessary (or both ) conditions for well-posedness, exponential stability and norm continuity of the solutions. As an applications of these topics: transport equations, infinite linear systems, control theory of infinite dimensional linear systems.

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121 Conclusion

The main thrust of this thesis is in the spirit of the spectral theory and operator theory; its aim is to give a survey of various characteristic stability properties of diﬀerent notions of essential spectrum under perturbations belonging to a large of class operators ( Fredholm perturbations, compact, quasi-nilpotent, ...), as well as some equivalent descriptions of these spectra, and many cases when these essential spectra coincide or diﬀer from each other on at most countably many isolated points. The results obtained are used for describing the essential spectra of some transport operators.