The Algerian Democratic and Popular RepublicThe Ministry of Higher Education and Scientific Research

University of Oran

Faculty of SciencesDepartment of Mathematics

ThesisSubmitted in partial fulfillment of the

requirements for the degree of

DOCTOR ES-SCIENCES–MATHEMATICS

by

Mohammed Benharrat

Comparison between the differentdefinitions of the essential spectrum and

Applications.

Thesis directorPr. 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 differentdefinitions of the essential spectrum and

Applications.

Mohammed BenharratEcole Normale Supérieure de l’Enseignement Technologique d’Oran

Département de Mathématiques et InformatiqueB.P. 1525 El M’Nouar. Oran

Email: mohammed.benharrat@gmail.com

Thesis submitted to The University of Oranfor the degree of DOCTOR ES-SCIENCES–MATHEMATICS.

Thesis directorPr. Bekkai MESSIRDI

University of Oran, 2013

Acknowledgments

I would like to thank first and foremost my supervisor, Professor Bekkai Messirdi, whohas invested considerable time and energy into guiding me through my thesis, for his manysuggestions and constant support during this research. Her scholarship and dedication hasbeen an inspiration in my studies.

I am very sensitive to the honor which makes me the Professor C. Bouzar by agreeing tochair this jury; and am him deeply grateful.

I thank Professors B. Bendoukha, M. Terbeche, A. Talhaoui and A. Senoussaoui for theirtime 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 honorsme and I would like to thank him.

Of course, I am grateful to my family for their patience and love. Without them thiswork would never have come into existence.

Abstract

Our main objective in this thesis is to present the most various definitions of the essentialspectrum founded in the mathematical literature, which beginning with the fundamentalwork of Weyl, is becoming more and more a special branch of spectral theory producingresults and problems of its own. On other hand, we gives a remarkable various characteristicstability properties of the essential spectra under appropriate perturbations, as well as someequivalent descriptions of these spectra. Our contributions in this dissertation are:

• We give a survey of results concerning various essential spectra of closed linear opera-tors in a Banach space and we give some relationships between this essential spectraand the SVEP theory.

• We investigate a relationship between the Kato spectrum and another essential spec-trum called the closed-range spectrum of an operator A, defined by Goldberg in [45]by σec(A) = λ ∈ C ; R(λI − A) is not closed, in the case of Banach spaces. Thiswork is extended to the Banach spaces, the result was shown by J.P Labrousse [71] inthe case of Hilbert spaces.

• We show that the symmetric difference between the generalized Kato spectrum andthe the closed-range spectrum is at most countable and we also give some relationshipsbetween the generalized Kato spectrum and the others essential by the use of the localspectral theory.

• We present a survey of results of characterizations and perturbations for various essen-tial spectra and we consider their stability under some classes of perturbations. By theuse of the Fredholm perturbations, we describe the various essential spectra of sometransport operators.

Key Words. Essential spectrum, Semi-Fredholm operators, Fredholm perturbations,Semi-regular operators, Quasi-Fredholm operators, Operators of Kato type, GeneralizedKato spectrum, Closed-range spectrum, Local spectral theory, Transport operators.

AMS Classification: 47A10, 47A53, 47A55, 47A60, 47B07, 47F05, 47G20.

List of Publications

• M. Benharrat, B. Messirdi. Essential spectrum: A Brief survey of concepts and appli-cations. Azerbaijan Journal of Mathematics. V. 2, No 1, 35-61 (2012).

• M. Benharrat, B. Messirdi. On the generalized Kato spectrum. Serdica Math. J. 37(2011), 283-294.

• M. Benharrat, B. Messirdi. Relationship between the Kato essential spectrum and avariant of essential spectrum. To appear in General Mathematics.

List of submitted Papers

• M. Benharrat, B. Messirdi. Quasi-nilpotent perturbations of the generalized Kato spec-trum and Applications. (Submitted ).

• M. Benharrat, B. Messirdi. B-Fredholm spectra of some transport operators. (Submit-ted ).

List of Communications

• M. Benharrat, B. Messirdi. Characterizations of Fredholm operators by quotient oper-ators. Congrés des Mathématiciens Algériens, CMA’2012, Annaba, 07-08 Mars 2012..

Contents

Introduction 1

1 Spectrum of an operator 91.1 Algebraic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.1.1 Ascent and descent of an operator . . . . . . . . . . . . . . . . . . . . 91.1.2 The nullity and the deficiency of an operator . . . . . . . . . . . . . . 13

1.2 Generalities about Closed operators . . . . . . . . . . . . . . . . . . . . . . . 171.2.1 Closable operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.2.2 Adjoint operator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.3 Operators with closed range . . . . . . . . . . . . . . . . . . . . . . . . . . . 211.4 Compact operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221.5 Perturbations of closed operators . . . . . . . . . . . . . . . . . . . . . . . . 241.6 The spectrum of closed operators . . . . . . . . . . . . . . . . . . . . . . . . 261.7 Approximate point spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . 331.8 The Riesz projection and the singularities of the resolvent . . . . . . . . . . 35

2 Essential Fredholm, Weyl and Browder spectra 422.1 Essential Fredholm spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422.2 Fredholm perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452.3 Browder and Weyl spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

2.3.1 The Browder resolvent . . . . . . . . . . . . . . . . . . . . . . . . . . 572.3.2 The essential spectral radius . . . . . . . . . . . . . . . . . . . . . . . 58

2.4 Characterizations of the essential spectra . . . . . . . . . . . . . . . . . . . . 592.5 Left-right Fredholm and Left-right Browder spectra . . . . . . . . . . . . . . 632.6 Invariance of the essential spectra . . . . . . . . . . . . . . . . . . . . . . . . 67

3 Generalized Kato spectrum 703.1 The semi-regular spectrum and its essential version . . . . . . . . . . . . . . 703.2 Closed-range spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 753.3 Quasi-Fredholm spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 763.4 Generalized Kato spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . 783.5 Saphar operators, essentially Saphar operators and corresponding spectra . . 83

4 Essential spectra defined by means of restrictions 864.1 Descent spectrum and essential descent spectrum . . . . . . . . . . . . . . . 864.2 Ascent spectrum and essential ascent spectrum . . . . . . . . . . . . . . . . . 894.3 Essential spectrum and Drazin invertible operators . . . . . . . . . . . . . . 914.4 B-Fredholm, B-Browder and B-Weyl spectra . . . . . . . . . . . . . . . . . . 944.5 Essential spectra and The SVEP theory . . . . . . . . . . . . . . . . . . . . 994.6 Weyl’s theorem and Browder’s theorem . . . . . . . . . . . . . . . . . . . . . 103

iv

5 Applications 1055.1 One-dimensional transport equation . . . . . . . . . . . . . . . . . . . . . . . 105

5.1.1 Application of the Fredholm perturbations to transport equations . . 1085.1.2 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 modernsection of the spectral analysis widely used in the mathematical and physical sense whenresolving 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 individualoperator, 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 theset of all points of the spectrum of A except isolated eigenvalues of finite multiplicity and heproved 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 inthe description of the essential spectrum of the Schrodinger operators for a large class oftwo-body potentials.

When A is not self-adjoint bounded operator (or just assumed to be closed and denselydefined in an arbitrary Banach space), in analogy with Weyl’s theorem, one would like thatthe essential spectrum to be invariant under arbitrary compact perturbations. The definitiongiven 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 spectrumwhile the point spectrum of the perturbed operator is the whole complex plane, and some op-erators have point eigenvalues which are not isolated and are carried into the resolvent undera compact perturbation. However, there are applications in which one would like to knowthat certain types of singularities are not introduced under compact perturbations, eventhough such singularities lie outside the essential spectrum. This motivates another severalpossible definitions of the essential spectrum (for an arbitrary operator) as the largest sub-set of the spectrum remaining invariant under arbitrary compact perturbations. From thismoment, essential spectrum and their properties of stability under (additive) perturbationsin appropriate class of operators, have been a research interest of many authors.

The theory has been examined in connection with various classes of linear operatorsdefined by means of kernels and ranges, the most important of this classes Fredholm op-erators, semi-Fredholm operators, quasi-Fredholm operators and more recently B-Fredholmoperators and generalized invertible operators (in particular, the Drazin and Koliha invert-ible operators). With these different classes of operators associated essential spectra, whichare qualitatively different subsets of the spectrum of an operator. However, the search fordifferent subsets of the spectrum, satisfying certain properties, is so far, which confirms therelevance of research presented in this thesis.

In this thesis, we give out at least 55 kind spectra and at least 46 kind essential spectraof an operator. Throughout this monograph, let X and Y be complex infinite dimensionalBanach 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) = dimN(A), β(A) = dimY/R(A). If A ∈ C(X), the ascent a(A) of A is definedto 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 thedescent 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 alower 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 invertibleoperators with bounded inverse, bounded below operators, surjective operators, left invertibleoperators, 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-Fredholmoperators, 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-Browderoperators, 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 ofcomplex plane C (see [55, 62, 64, 69, 94, 95, 97, 101, 103, 108, 109, 118, 117, 122, 123]) andhave 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 , theclosure of M and all isolated points of M .

Recall that the discrete spectrum of self-adjoint operator A acting on Hilbert spaceconsists of isolated eigenvalues of finite multiplicity, the remaining part of the spectrum isthe 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 disjointsets:

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. Thereforeλ ∈ σi(A), i ∈ Λ = ec, lf, uf, ef, ew, uw, lw, eb, ub, lb. Consequently we have

σc(A) ⊂⋂i∈Λ

σi(A).

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 ofA-invariant closed subspaces (M,N) such that X = M ⊕N , where A|M is semi-regular andA|N is quasi-nilpotent. If we assume in the definition above that A|N is nilpotent, thenthere 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 isfinite-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 operatorA ∈ L(X), let us define

the semi-regular spectrum2: 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,andthe 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. Katospectrum).

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

4

For each n ∈ N, we set cn(A) = dimR(An)/R(An+1) and c′n(A) = dimN(An+1)/N(An).the essential descent and the essential ascent of A ∈ L(X) are

de(T ) = infn ∈ N : cn(T ) <∞ and ae(T ) = infn ∈ N : c′

n(T ) <∞.

(the infimum of an empty set is defined to be ∞). Now, we continue to introduce thefollowing spectra which were discussed in [11, 12, 49, 51, 62, 69, 78]:

The descent spectrum: σd(A) = λ ∈ C : d(λI − A) =∞,The essential descent spectrum: σed(A) = λ ∈ C : de(λI − A) =∞,The ascent spectrum : σa(A) = λ ∈ C : a(λI − A) =∞ or R(Aa(A)+1) is not closed ,The essential ascent spectrum: σea(A) = λ ∈ C : ae(λI−A) =∞ or R(Aae(A)+1) 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 bothleft and right Drazin invertible if and only if A is Drazin invertible. If ae(A) < ∞ andR(Aae(A)+1) 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 beessentially Drazin invertible (resp. semi-essentially Drazin invertible) if A is left essentiallyDrazin invertible and (resp. or) right essentially Drazin invertible.

Now, we can define the left Drazin spectrum, the right Drazin spectrum, the Drazinspectrum, the left essentially Drazin spectrum, right essentially Drazin spectrum, essentiallyDrazin 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,σeLD(A) = λ ∈ C : λI − A is not a left essentially Drazin invertible operator,σeRD(A) = λ ∈ C : λI − A is not a right essentially Drazin invertible operator,σeD(A) = λ ∈ C : λI − A is not a essentially Drazin invertible operator,σeSD(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].Given n ∈ N, we denote by An the restriction of A ∈ L(X) on the subspace R(An).

According Berkani [18], A is said to be semi B-Fredholm (resp. B-Fredholm, upper semiB-Fredholm, lower semi B-Fredholm), if for some integer n ≥ 0 the range R(An) is closedand An, viewed as a operator from the space R(An) in to itself, is a semi-Fredholm operator(resp. Fredholm, upper semi-Fredholm, lower semi-Fredholm). Analogously, A ∈ L(X) issaid to be B-Browder (resp. upper semi B-Browder, lower semi B-Browder, B-Weyl, uppersemi B-Weyl, lower semi B-Weyl ), if for some integer n ≥ 0 the range R(An) is closedand An is a Browder operator (resp. upper semi-Browder, lower semi-Browder, Weyl, uppersemi-weyl, lower semi-Weyl). If A ∈ L(X) is upper or lower semi-B-Weyl (resp. upper orlower 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 uppersemi-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 spectrumof 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 elegantequalities:

σLD(A) = σubb(A), σRD(A) = σlbb(A);

σeLD(A) = σubf (A), σeRD(A) = σlbf (A);

σD(A) = σbb(A).

Now, we continue to introduce the following essential spectrum which recently discussedin [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 thatR(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 essentialspectra which exists in the form of research papers scattered throughout the literature. Toachieve this goal addressed the following tasks:

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

• Identifies conditions and additional constraints under wich the spectrum splits intoFredholm spectra and eigenvalues or to divide into Weyl spectrum and eigenvalueswhich are both topologically isolated in the spectrum, and geometrically of finite mul-tiplicity, with finite dimensional eigenspaces.

• Proves that both the symmetric difference, between the closed range spectrum and theKato spectrum and between the closed range spectrum and the generalize Kato spec-trum are at most countable. We study the relationships between the generalized Katospectrum and some other spectra originated from Fredholm theory and B-Fredholmtheory. 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 boundedoperator A, or its adjoint A∗, or both, admits the SVEP 3. The second is that the ap-proximate spectrum, or the surjective spectrum, coincide with the connected boundaryspectrum.

• Give a special attention for the properties of stability of the essential spectra of closedlinear operators under (additive) perturbations , such as operators of finite rank, com-pact operators, small in norm and quasinilpotent operators, because they have differentuseful applications and which can be used to obtain information on the location in thecomplex plane of the essential spectrum for large classes of linear operators arising inapplications. For example, differential, integral, integro-differential operators, differ-ence, and pseudo-differential operators, in particular operators of all these types onunbounded domains.

The thesis is organized in five chapters. The first chapter discusses the elements ofspectral theory of operators. The topics discussed include spectral decomposition theorems,Riesz projections, functional calculus, the singularities of the resolvent and eigenvalues offinite type. The beginning of this chapter contains an important proprieties of some classicalalgebraic quantities associated with an operator, such as the ascent, the descent, the nullityand the deficiency of an operator. This quantities are the basic bricks in the construction ofthe most important classes of linear operators.

In the second chapter we gives a survey of results concerning various types of essentialspectra, Fredholm and Browder operators etc. A section of this chapter is also devoted tostudy of some perturbation ideals which occur in Fredholm theory. In particular we study theideal of inessential operators, the ideal of strictly singular operators and the ideal of strictlycosingular operators, which are a generalization of the class of the compact perturbations.

The third chapter provides definitions of semi-regular and essentially semi-regular opera-tors 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 par-ticular we study the generalized spectrum which is characterized by the generalized Katodecomposition. This decomposition first appeared in the classic work of Kato [63] perturba-tion of linear operators, and its has greatly benefited from the work of many authors in thelast 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 classof quasi-Fredholm, semi-regular, Kato type, semi-Fredholm and B-Fredholm operators. Thisconcept leads in a natural way to the generalized Kato spectrum σgk(A), an important subsetof the ordinary spectrum which is defined as the set of all λ ∈ C for which λI −A does notadmit a generalized Kato decomposition. It is shown in [71], in the Hilbert space case, thatthe symmetric difference between the essential Goldberg spectrum defined in [45] and theessential quasi-Fredholm spectrum is at most countable, which is of course, in this case, aquasi-Fredholm operators equivalent to A is of Kato type, but in the case of Banach spacesthe Kato type operator is also quasi-Fredholm, the inverse is not true. We generalize thisresults for the Kato spectrum and the generalized Kato spectrum.

The next chapter deals with spectral theory, we focus on the study of several spectrathat 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 theoremsand the corresponding perturbation theory. In this chapter we also introduce the elegantinteraction between the localized SVEP and Fredholm theory. This interaction is studied inthe more general context of operators admits a generalized Kato decomposition. We givea summary of the Weyl-Browder type theorems and properties, in their classical and morerecently 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

AHψ(x, ξ) = −ξ ∂ψ∂x

(x, ξ)− σ(ξ)ψ(x, ξ) +

∫ 1

−1

κ(x, ξ, ξ′)ψ(x, ξ′) dξ′

= THψ(x, ξ) +Kψ(x, ξ)

with the boundary conditionsψi = H(ψo)

whereH 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 (forinstance gas molecules, photons, or neutrons) in a homogeneous slab of thickness 2a. Thefunctions σ(.) and κ(., ., .) are called, respectively, the collision frequency and the scatteringkernel. Our analysis is based essentially on the perturbations results and the knowledge ofthe essential spectra of T0 where T0 (i.e., H = 0) denotes the streaming operator with vac-uum boundary conditions. We prove that, if the classes of boundary and collision operatorsoperators is in appropriate class of Fredholm perturbation, then the Fredholm spectra of theoperators T0 and AH coincide.Also, we apply the same technic to study the essential spectra of the following singularneutron transport operator

Aψ(x, ξ) = −ξ · ∇xψ(x, ξ)− σ(ξ)ψ(x, ξ) +

∫Rnκ(x, ξ′)ψ(x, ξ′) dξ′ (x, ξ) ∈ Ω× V,

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 boundedsubset 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 wherewe give further information and discuss some of the more recent developments in the the-ory previously developed. In general, all the results established in these final sections arepresented without proofs. However, we always give appropriate references to the originalsources, 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 unboundedlinear operator in a Banach spaces. Let us start by setting the stage, introducing thebasic notions necessary to study linear operators. Through this monograph, an operatormeans a linear transformation defined on Banach space. Although many of the results inthese monograph are valid for real Banach spaces, we always assume that all Banach spacesare complex infinite-dimensional Banach spaces. First we present some classical quantitiesassociated with an operator. These quantities, such as the ascent, and the descent of anoperator are defined in the first section and are the basic bricks in the construction of oneof the most important branches of spectral theory, the theory of Fredholm operators.

1.1 Algebraic PropertiesLet X and Y be two vector spaces over the real or complex numbers (the scalars) andL(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 itskernel and R(A) = Ax, x ∈ D(A) its range. For all n ∈ N domain, kernel and the rangeof 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

andR(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 definitionsand 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 spaceX, form the following two increasing and decreasing chains (sequences of subspaces), respec-tively:

N(A0) = 0 ⊆ N(A) ⊆ N(A2) ⊆ . . .

9

andR(A0) = Y ⊇ R(A) ⊇ R(A2) ⊇ . . .

Generally all these inclusions are strict. In this subsection we shall consider operators forwhich 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

Am(N(An+m)) = R(Am) ∩N(An), 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 andonly 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 thatR(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), wethen obtain that z ∈ N(Aq)∩R(Aq), and hence the last intersection is 0. Therefore z = 0and y = Aqw ∈ R(Aq)and this shows R(Aq) = R(Ap), from whence we obtain p ≥ q, so thatp = 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, sinceApx = Apu = 0, then A2pv = 0, hence v ∈ N(A2p) = N(Ap) and Apv = 0, consequentlyx = u ∈ N(Aq). This shows that N(Aq) = N(Ap) and q ≥ p, we conclude p = q.

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Lemma 1.2 For a linear operator A on a vector space X the following statements are equiv-alent:

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 obtainN(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 subspaceR(An). Then

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

andR(Amn ) = R(Am+n) = R(Anm) for all m,n ∈ N, (1.5)

Lemma 1.3 Let A be a linear operator on a vector space X. Then the following statementsare equivalent:

(i) a(A) <∞;

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

(iii) there exists k ∈ N such that a(Ak) <∞.

Proof. (i)⇔ (ii). If p = a(A) <∞, by Lemma 1.1, then N(Ap) = R(Ap) ∩N(A) = 0.Conversely , suppose that N(Ak) = 0, for some k ∈ N. If x ∈ N(Ak+1) then A(Akx) = 0,so

Akx ∈ N(A) ∩R(Ak) = N(Ak) = 0.

Hence x ∈ N(Ak). This shows that N(Ak+1) ⊆ N(Ak), thus N(Ak+1) = N(Ak) and conse-quently 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

0 = N(Ak) ∩R(Amk ) = (N(A) ∩R(Ak)) ∩R(Amk ) = N(A) ∩R(Amk )

= N(A) ∩R(Am+k) = N(Am+k)

so that the equivalence is proved.

A similar result holds for the descent:

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Lemma 1.4 Let A be a linear operator on a vector space X. Then the following statementsare 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

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

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

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

thus d(A) ≤ k.The implication (ii) ⇒ (iii) is obvious. we show the opposite implication, let m =

d(Ak) < ∞ for some k ∈ N. Then R(Amk ) = R(Am+1k ), so R(Am+k) = R(Am+k+1), hence

d(A) ≤ k +m.

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

a(A) = infk ∈ N : Ak is injective .

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

d(A) = infk ∈ N : Ak is onto .

The finiteness of the ascent and the descent of a linear operator A is related to a certaindecomposition 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) thena(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 ≤ ∞), theBanach space of of all p-summable sequences (bounded sequences for p = ∞) of complexnumbers 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

yn =

xn+1 if n is evenxn if n is odd

Then N(A) = N(A2) = (xn)n ∈ `p : x2n = 0 for each n ∈ N. Hence a(A) = 1. AlsoR(A) = R(A2) = (yn)n ∈ `p : y2n+1 = y2n for each n ∈ N. Threfore d(A) = 1.

12

Example 1.2 let B be defined by

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

As B is surjective, so d(B) = 0. Further we note that for each n ≥ 1, en ∈ N(Bn) buten /∈ N(Bn−1). Note that the sequence en = δjn. 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

yn =

x0 if n = 0, 1xn if n ≥ 1

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

Example 1.4 Consider the operator D be defined by D(x) = y, where x = (xn)n andy = (yn)n are related by

yn =

xn+2 if n is oddx0 if n = 0, 2xn−2 if n is even and n ≥ 4

Then for each n ∈ N, e2n+1 ∈ N(Dn+1) but e2n+1 /∈ N(Dn+2). Hence a(D) = ∞. Furtherwe note that R(D) = (yn)n ∈ `p : y0 = y2, R(D2) = (yn)n ∈ `p : 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) = dimN(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) andβ(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 propertieshold:

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 thatβ(A) <∞, so β(An) <∞, from Lemma 1.1 we have N(A) ∩ R(Ap) = 0 and this impliesthat 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 ≥ maxp, 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 caseq =∞, 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 thatβ(A) = dimY ≤ 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 ≥ maxp, q the quantity nind(A) = α(Ap) − β(Aq) isconstant, hence ind(A) = 0 and α(A) = β(A) = 0. Now if p = ∞, then α(An) → ∞ asn→∞, 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∈N

N(An).

The hyper-range of A is the subspace

R∞(A) =⋂n∈N

R(An).

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

N(A) ∩R(Am) = N(A) ∩R(Am+k), 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 dimF < ∞. Let Dn = N(A) ∩ R(A), we haveDn+1 ⊆ Dn, foe every n ∈ N. Suppose that there exist k ∈ N such that Di 6= Di+1,i = 1 . . . k, Then for every one of these i , we can find an element wi such that Aiwi ∈ Di

and Aiwi /∈ 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,if∑k

i=1 λiui = 0, thenk∑i=1

λiwi =k∑i=1

λivi = 0

and therefore from the equalities Akw1 = . . . = Akwk−1 = 0, we deduce that

Ak(k∑i=1

λiwi) = λkAkwk = Ak(

k∑i=1

λivi) ∈ R(Ak+1)

From Akwk ∈ N(A), we obtain λkAkwk ∈ Dk+1, but λkAk /∈ Dk+1, this is possible only if

λk = 0. Analogously we have λk−1 = . . . = λ1 = 0, so the vectors u1, . . . , uk are linearlyindependent and we deduce that k ≤ dimF , But then for a sufficiently large m we obtainthe 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.

On other handN(An)∩R(Am) ⊆ ∩mi=0R(Ai), henceN(An)∩R(Am) ⊆ ∩∞i=0R(Ai), this provedthe 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), sox ∈ 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

A(R∞(A)). Let y ∈ R∞(A) , then y ∈ R(An), for every n ∈ N, so there exists xk ∈ X suchthat y = Am+kxk, for every k ∈ N. If we set

zk = Amx1 − Am+k−1x, k ∈ N.

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

y = Amx1 = zk + Am+k−1xk ∈ R(Am+k−1), 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 qhave 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 soX = 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 impliesN(An) ⊆ (λI−A)(N(An)). The converseinclusion 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

(µI − A)(N((λI − A)n)) = N((λI − A)n), 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

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

2. α((λI − Am)n) ≤ α((λI − A)n), 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))

= β((λI − Am)n).

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

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1.2 Generalities about Closed operatorsIf X, Y are Banach spaces, we says that an operator A from X into Y is bounded (orcontinuous) if there is a constant c ≥ 0 such that

‖Ax‖ ≤ c ‖x‖ 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

||A|| := supx 6=0

||Ax||||x||

.

(unless further specification is necessary, ‖.‖ will always denote the norm in an appropriatespace). For linear operators the concepts of continuity at a point, uniform continuity andboundedness coincide. But when one deals with differential operators, one discovers the needto consider also unbounded linear operators. For example,

Example 1.5 The differential operator of first order A = i∂x on L2([0, 1]). It readily seen tobe unbounded since one can find a sequence of functions ϕn ∈ L2([0, 1]), given by ϕn = einx

for n ∈ N, satisfying ‖ϕn‖L2([0,1]) = 1 and ‖Aϕn‖L2([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), whereM denote, in the sequel, the closure of subset M of X . When A is bounded and denselydefined, 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 propertyof being closed:

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

Definition 1.4 Let A : X −→ Y be a linear operator with the domain D(A). The graphof 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

‖x‖A = ‖x‖+ ‖Ax‖ .

We write XA if D(A) is equipped with the graph norm. Clearly XA is a normed vectorspace and A ∈ L(XA, Y ). The graph norm on D(A) is clearly stronger than the X-norm onD(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). Thenthe 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 closedGraph 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 closeddoes 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 isthanks to the domain of the bounded operators which is always taken to be the wholeBanach space X. However, one has to be careful with those manipulations when dealingwith unbounded operators, this is essentially due to the domains. If A : D(A) ⊂ X → Yand 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) orD(CA) may become trivial, i.e it reduces to zero even if strong conditions are imposed onA,B and C.

In practice, most unbounded operators are closed and are densely defined. In the sequelwe 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 givenoperator A has a closed extension. If A is bounded, this always holds, since we can simplytake the operator A with graph G(A); here G(A) is a graph since xn −→ 0 ∈ X impliesAxn −→ 0 ∈ Y . But when A is unbounded, one cannot be certain that it has a closedextension. But if A has a closed extension A1, then G(A1) is a closed subspace of X × Ycontaining G(A), hence also containing G(A). In that case G(A) is a graph. It is in factthe 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 smallestclosed extension of A whose graph equals G(A) is denoted by A and called the closure of A.Every closable operator has a closure.

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Proposition 1.2 ([64]) Let A : X −→ Y be an operator. The following conditions areequivalent:

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 impliesy = 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 → yin Y as n → ∞. Since B is bounded, then xn = Bzn converges to Bz and so Axn → y asn → ∞. Since A is closed, we deduce that Bz ∈ D(A) and ABz = y. Now if A is closablethen 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 ′ with D(A) = C1([0, 1]) and ϕ ∈ C([0, 1]) such thatϕ = 0 on [0,1/2]. Define B ∈ L(X) by Bf = ϕf for all f ∈ X. Then the operator BA withD(BA) = D(A) is not closed. To see this, take functions fn ∈ D(A) such that fn = 1 on[1/2; 1] and fn → f in X with f /∈ C1([0, 1]). Then, BAfn = ϕf

′n = 0 converges to 0, but

f /∈ D(A).

1.2.2 Adjoint operator.

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

‖x‖X∗ = inf|x∗(x)| : x ∈ X, ‖x‖ = 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∗ ) suchthat

|y∗(Ax)| ≤ c‖x‖X , for all x ∈ D(A)

Since D(A) is dense in X, the mapping extends by continuity to X, so there is a uniquelydetermined 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)

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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, more-over, ‖A∗‖ = ‖A‖.

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 Xrespectively. 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 〈., .〉. We can define the adjointof A ∈ C(X, Y ) as follows

D(A∗) = y ∈ Y : ∃z ∈ Y ∀x ∈ D(A) : 〈Ax, y〉 = 〈x, z〉, A∗y = z.

Moreover, if A is a bounded, we have

for all x ∈ X, y ∈ Y : 〈Ax, y〉 = 〈x,A∗y〉 .

Theorem 1.6 Suppose that H is a Hilbert space and A is a densely defined operator fromH 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 toitself.

• 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)

〈Ax, y〉 = 〈x,Ay〉 .

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. ByTheorem 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, thenA = B . This is because A ⊂ B ⊂ B∗ ⊂ A∗ = A. We see that a symmetric operatorcan have different self-adjoint extension.

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1.3 Operators with closed rangeThe 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 numberassociated with A.

Definition 1.7 the reduced minimal modulus of A ∈ C(X) is defined to be

γ(A) = infx/∈N(A)

‖Ax‖dist(x,N(A))

where dist(x,N(A)) = infy∈N(A) ‖x− y‖. 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 nec-essarily closed). If Y +N(A) is closed then A(Y ) is closed.

Proof. Let us denote by x the equivalence class x + N(A) in the quotient space X/N(A)

and by A : X/N(A)→ X the canonical injection defined by A(x) = Ax, where x ∈ x. SinceA(X) is closed A has a bounded inverse A−1 : R(A)→ X/N(A). Let Y = y : y ∈ Y . ClearlyA(Y ) = A(Y ) is the inverse image of Y under the continuous map A−1, so A(Y ) is closed ifY is closed .It remains to show that Y is closed if Y + N(A) is closed. Suppose that the sequence (xn)

of Y converges to x ∈ X/N(A) . This implies that there exists a sequence (xn) with xn ∈ xnsuch 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 andsince by assumption Y + N(A) is closed, we have x ∈ Y + N(A). This implies x ∈ Y ; thusY is closed .

Theorem 1.8 ([45]) Let A ∈ C(X, Y ). If there is a closed subspace Y0 of Y for whichR(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 aone-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.

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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) istrue. 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 andhas closed range.

Theorem 1.9 ([64]) A ∈ C(X, Y ) is bounded below if and only if there exists c > 0 suchthat

‖Ax‖ ≥ c‖x‖ for all x ∈ D(A) (1.12)

The next result shows that the properties to be bounded below or to be surjective aredual 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 equalityR(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 hasclosed range. By the equality R(A∗)⊥ = N(A) = 0, we conclude that A∗ is injective andhence 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 ofγ(A) we obtain

γ(A)dist(x,N(A)) = γ(A)‖x‖ ≤ ‖Ax‖ for all x ∈ D(A)

From we obtain‖Ax‖ ≥ ‖Ax‖ − |λ| ‖x‖ ≥ (γ(A)− |λ|)‖x‖

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 operatorsDefinition 1.9 An operator A : X → Y is said to be compact if A(B) is relatively compactin Y for every bounded subset B ⊂ X. Equivalently, for every bounded sequence (xn)n inX 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 writeK(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 offinite-rank defined by

F(X, Y ) = A ∈ L(X, Y ) : dimR(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 operatorson 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 ifA∗ ∈ L(Y ∗, X∗) is compact.

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

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 spacesand let k(x, y) be a measurable function on X × Y with∫

X×Yk(x, y)dµ(x)dν(y) <∞

Then(Kf)(x) =

∫Y

k(x, y)f(y)dν(y)

defines a compact operator from L2(Y ; dν(y)) to L2(X; dµ(x)). Such an operator is calledHilbert-Schmidt.

Example 1.8 (Nuclear Operators) Let X and Y be Banach spaces and denote by X∗.If (x′n) is a bounded sequence in X∗,(yn) is a bounded sequence in Y and (cn) is a set ofcomplex numbers obeying

∑∞n=0 |cn| <∞, then

Kx =∞∑k=0

cnx′n(x)yn

23

is called a nuclear operator from X to Y . Since

∞∑k=0

|cn| |x′n(x)| |yn| ≤ (supn‖x′n‖X∗ sup

n‖yn‖Y

∞∑n=0

|cn|) ‖x‖X

the series defining Kx converges strongly and K is a bounded operator of norm at most

supn‖x′n‖X∗ sup

n‖yn‖Y

∞∑n=0

|cn| .

1.5 Perturbations of closed operatorsThe 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 operatorA+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 andAxn + Bxn → y in Y as n → ∞. Since B is bounded, then limn→∞Bxn = Bx and soAxn → 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 closedoperators

Example 1.9 Let X = Cb(R2) = f : R2 → 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

D(B) = D(A1) ∩D(A2) = C1b (R2) = f ∈ C1(R2) : f, ∂1f, ∂2f ∈ X.

We have A1 and A2 are closed, But B is not closed. Take ϕn ∈ C1b (R) converging uniformly

to some Cb(R) \ C1(R). Set fn(x, y) = ϕn(x − y) and f(x, y) = ϕ(x − y) for (x, y) ∈ R2

and n ∈ N. We then obtain f ∈ X, fn ∈ D(B), ‖fn − f‖∞ = ‖ϕn − ϕ‖∞ → 0 andBfn = ϕ

′n − ϕ

′n = 0, but f /∈ D(B).

The extension of the stability Theorem to a not necessarily bounded perturbation isbased 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 simplyA-bounded if D(A) ⊂ D(B) and there exist nonnegative constants a, b such that

‖Bx‖ ≤ a‖Ax‖+ b‖x‖ (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 andonly if B0 is bounded.

Theorem 1.14 Let A be closed and let B be bounded relatively to A with a < 1 . ThenA+B with the domain D(A) is closed.

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Proof. We know that (1.13) hold for some a < 1 and b. Hence

‖(A+B)x‖+ ‖x‖ ≤ (1 + a)‖Ax‖+ (1 + b)‖x‖

and

(1− a)‖Ax‖+ ‖x‖ ≤ ‖Ax‖ − ‖Bx‖+ (1 + b)‖x‖ ≤ ‖(A+B)x‖+ (1 + b)‖x‖

Hence the norms ‖Ax‖+ ‖x‖ and ‖(A+B)x‖+ ‖x‖ are equivalent on D(A).

Theorem 1.15 Let A be injective and D(A) ⊂ D(B). Then B is A-bounded with a =‖BA−1‖ in (1.13).If, moreover, ‖BA−1‖ < 1, then A+B with domain D(A) is closed, invertible and

(A+B)−1 =∞∑n=0

(−1)nA−1(BA−1)n

Proof. By the estimate ‖Bx‖ ≤ ‖BA−1‖‖Ax‖, x ∈ D(A) we have a = ‖BA−1‖. Assumenow ‖BA−1‖ < 1. Let

Cn =n∑i=0

(−1)iA−1(BA−1)i

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

(A+B)Cny = y + (−1)nA−1(BA−1)n+1y → y.

But A+B is closed, hence Cy ∈ D(A+B) and (A+B)Cy = y. If x ∈ D(A+B), then

Cn(A+B)x = x+ (−1)nA−1(BA−1)nAx→ 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 linearoperator having closed range.

Theorem 1.16 ([63]) Let A,B : X −→ Y and Let A be a closed operator with closedrange ( so that γ(A) > 0 ) and with least one of α(A) and β(A) finite. If B is boundedoperator such that D(A) ⊂ D(B) and ‖B‖ < 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

‖Axn‖+ ‖xn‖ ≤ c. (1.14)

the sequence (Bxn) contains a convergent subsequence.

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If B is A-compact, B is A-bounded. In fact, if B is not A-bounded, there is a sequence (xn)such that ‖Axn‖ + ‖xn‖ = 1 but ‖Bxn‖ ≥ n for all n ∈ N. It follows that (Bxn) hes noconvergent 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 calledstrictly singular if, for every infinite-dimensional subspace M , the restriction of A to M isnot 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 thenK(X) = S(X).

Definition 1.13 An operator A ∈ L(X, Y ) is said to be weakly compact if A(B) is relativelyweakly 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 idealof L(X) containing K(X) (cf. [43, 45]).

The class of weakly compact operators on L1-spaces (resp. C(K)-spaces with K a com-pact 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 byKato [63] as a generalization of the notion of compact operators. For a detailed study of theproperties of strictly singular operators we refer to [45, 63]. For our own use, let us recallthe 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 thenA+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 ifD(A) ⊂ D(B) and there is no infinite-dimensional subspace M such that

‖Bx‖‖Ax‖+ ‖x‖

≥ γ > 0 for all x ∈M. (1.15)

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

1.6 The spectrum of closed operatorsLet X be a complex Banach space and A : D(A) −→ X a closed operator with domainD(A) ⊂ X. The identity operator on X will be denoted by IX , or simply I if no confusioncan 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 propertiesof R(λ,A) will be basic for an understanding of the operator A itself. Naturally, manyproperties of (λI −A) (or simply (λ−A)) and R(λ,A) depend on λ, and spectral theory isconcerned with those properties. For instance, we shall be interested in the set of all λ inthe 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,

ρ(A) :=λ ∈ C : R(λ− A) = X and (λ− A)−1 : R(λ− A) −→ D(A) exists and bounded

(1.17)

its complementσ(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 someimportant facts about the spectrum, resolvent operator, and spectral radius in the followingtheorem.

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).

2. If λ ∈ ρ(A) and |λ− µ| ≤ ‖R(λ,A)‖−1 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

R(λ,A) =∞∑n=0

(−1)n(λ− λ0)n(R(λ0, A))n+1,

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

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

r(A) = limn→+∞

‖An‖1n = inf

n∈N‖An‖

1n

27

3. We haver(A) ≤ ‖A‖ ,

equality holds, for example, if X is a Hilbert space and A is normal, i.e. commuteswith its adjoint.

4. The Neumann series

R(λ,A) =1

λ(I − 1

λA)−1 =

∞∑n=0

1

λn+1An

converges in L(X) for each λ ∈ C with |λ| > r(A).

5. For every λ with |λ| > ‖A‖ we have λ ∈ ρ(A) and

‖R(λ,A)‖ ≤ 1

|λ| − ‖A‖(1.20)

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

Example 1.10 Let A = i ddx

on L2[0, 1] and

D0 = f : f ∈ AC2[0, 1] and f(0) = 0, D1 = f/f ∈ AC2[0, 1]

where AC2[0, 1] denotes the set of absolutely continuous functions on [0, 1] whose derivativesare 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 ε ∈ R, let Aε ∈ L(X) be defined byAε(x) = y, where x = (xn)n and y = (yn)n are related by

y =

xk−1 if k 6= 0εx−1 if k = 0

Then we haveσ(A0) = D

where D denotes the open complex unit disc. On the other hand,

σ(Aε) = S = ∂D = λ ∈ C : |λ| = 1 (ε 6= 0).

So, here the spectrum collapses when ε 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 xsuch 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 injectivebut its range is not closed.

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

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

i=1σpi(A).

N(λ− A) = 0 N(λ− A) 6= 0R(λ,A) exists R(λ,A) exists R(λ,A) does notand 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 butλ−A∗ is not: there exists then x∗ ∈ X∗ such that (λ−A∗)x∗ = 0 hence x∗(λ−A) = 0which 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 tablegives a useful characterizations of the different parts of the spectrum defined in Table 1.1 interms 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.

Example 1.12 Let X = `p, 1 ≤ p ≤ ∞, the space of of all sequences x = (x1, x2, x3, . . . )with finite norm

‖x‖p =

(∑∞

n=1 |xn|p)

1p if 1 ≤ p <∞

supn≥1 |xn| if p =∞we define the following operators on `p by

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

A1x = (x2, x3, x4, . . . )

A2x = (x1,1

2x2,

1

3x3, . . . )

A3x = (x2,1

2x3,

1

3x4, . . . )

A4x = (0, x1,1

2x2,

1

3x3, . . . )

30

A A0 A1 A2 A3 A4

`p 1 < p <∞ p = 1 or ∞ 1 ≤ p <∞ p =∞ 1 ≤ p ≤ ∞ 1 ≤ p ≤ ∞ 1 ≤ p ≤ ∞

σp(A) ∅ ∅ D D 1, 12, 1

3, . . . 0 ∅

σc(A) S ∅ S ∅ 0 ∅ ∅σr(A) D D ∅ ∅ ∅ ∅ 0σ(A) D D D D 1, 1

2, 1

3, . . . ∪ 0 0 0

Table 1.3:

Sometimes it is useful to relate the spectrum of a closed densely defined linear operator tothat of its adjoint. Building on classical existence and uniqueness results for linear operatorequations 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 adjointA∗ 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 dimX =∞ and K ∈ L(X) be compact. Then the following assertionshold.

1. σ(K) = 0 ∪ λn : n ∈ J, where either J = ∅, or J = 1, . . . , n for some n ∈ N, orJ = 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 alwaysbelongs to the spectrum if the underlying space is infinite dimensional. But this three op-erators 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 x∗1, . . . , x∗n of λx∗ = Ax∗, and theequation (1.21) has a solution x ∈ X if and only if 〈y, x∗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 = yand 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 dimX = ∞, then a closed operator A with compact resolvent R(λ;A) cannot bebounded since otherwise R(λ;A) were bounded and thus I = (λI − A)R(λ;A) werecompact 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 dimX =∞ and A be a closed operator with compact resolvent.Then σ(A) is either empty or contains only at most countably many eigenvalues λn withα(λnI − A) <∞. If A has infinitely many eigenvalues λn, then |λn| → ∞ as n→∞.

LetH be a finite-dimensional Hilbert space, sayH = Cn. It is known from linear algebra thateigenvectors of a self-adjoint operator on H form an orthogonal basis of H. The followingtheorems generalize this result to infinite dimensional spaces.

Theorem 1.25 ([64]) Let X be a separable Hilbert space and A be densely defined, closedand 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 ofeigenvectors of A and

X = N(A)⊕∞n=0 N(λnI − A)

Furthermore , the sum

A =∞∑n=0

λnPλn

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, closedand 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 eigenvectorsof 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 ontoN(µnI − A). Finally, the sum

Ax =∑n∈N

µnQnx

converges in X for all x ∈ D(A).

1.7 Approximate point spectrumAnother important part of the spectrum is the approximate point spectrum σap(A) that isdefined 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 forthe term approximate point spectrum. The elements of it are sometimes referred to as theapproximate eigenvalues of A.

σap(A) = λ ∈ C : ∀ε > 0,∃x ∈ D(A) with ‖x‖ = 1 and ‖(λI − A)x‖ < ε= λ ∈ C : ∃(xn)n ⊂ D(A) with ‖xn‖ = 1 ∀n and ‖(λI − A)xn‖ −→ 0= λ ∈ C : ∃(λn)n ⊂ C with λn −→ 0 and ∃(xn)n ⊂ D(A)

with ‖xn‖ = 1 ∀n such that ‖(λnI − A)xn‖ −→ 0.

Clearly, σp(A) ⊂ σap(A) and

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

The quantityj(λI − A) := inf

‖x‖=1‖ (λI − A)x‖

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) = ‖R(λ,A)‖−1.

33

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

‖λx− Ax‖ ≤ |λ− µ|+ ‖µx− Ax‖

hencej(λI − A) ≤ |λ− µ|+ j(µI − A)

and thereforej(λI − A)− j(µI − A) ≤ |λ− µ|

the inequality follows by interchanging λ and µ.2. Setting S = R(λ,A) , if ‖x‖ = 1, then

‖x‖ = ‖S(λ− A)x‖ ≤ ‖S‖ ‖λ− A‖ ‖x‖

hence1

j(λI − A)≤ ‖S‖

Moreover, ∥∥∥∥(λ− A)Sx

‖Sx‖

∥∥∥∥ =‖x‖‖Sx‖

=1

‖x‖

From this it follows that ‖Sx‖ ≤ 1j(λI−A)

and consequently

‖S‖ ≤ 1

j(λI − A)

This shows the results.

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

Proof. Since the function j(λ − A) is continuous in λ and σap(T ) is the inverse image of0 by j, it follows that σap(A) is closed.Now, let λ ∈ ρ(A), then j(λI − A) = ‖R(λ,A)‖−1 > 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 thereexist subsequences of (λn)n, for which limn→+∞ ‖R(λn, T )‖ = +∞. Hence

limn→+∞

j(λn − A) = limn→+∞

‖R(λn, T )‖−1 = 0

Since the function j(.) is continuous in λ, we then conclude that j(λ − A) = 0. Henceλ ∈ σap(A).

Example 1.13 If A is closed densely defined and symmetric on a Hilbert space X, thenσap(A) ⊆ R.

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

1. σ(R(λ,A)) = 1λ−µ ;µ ∈ σ(A).

34

2. σi(R(λ,A)) = 1λ−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. Forinstance, the defect spectrum (or the surjectivity spectrum) σsu(A) and the compressionspectrum σco(A) are defined by

σsu(A) := λ ∈ C : λI − A is not surjective

andσco(A) := λ ∈ C : R(λI − A) is not dense in X.

By the closed range theorem we easily know that the approximate point spectrum andthe surjectivity spectrum are dual to each other, in the sense that σap(A) = σsu(A

∗) andσap(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 inTable 1.1 we note that

σco(A) = σp2(A) ∪ σp3(A) ∪ σr(A).

1.8 The Riesz projection and the singularities of the re-solvent

Let A a closed operator and its spectrum σ(A) decomposes into two non-empty disjointclosed subsets σ and τ . Suppose that σ is bounded, we can surrounding σ by a Jordancontour1 positively oriented Γ. Hence we can associate a Riesz integral with σ

Pσ(A) =1

2πi

∮Γ

(λ− A)−1 dλ, (1.23)

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 ofcomplex function theory shows that the definition of Pσ does not depend on the particularchoice of the contour Γ. A fundamental properties of the spectral projection Pσ are given inthe next results.

Proposition 1.11 Let A be a closed operator on X such that its spectrum σ(A) is thedisjoint union of two non-empty closed subsets σ and τ with σ is bounded. Let Pσ be asdefined 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

rectifiable Jordan curves positively oriented Γ1,Γ2, . . . ,Γn such that Γ = ∪ni=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

P 2σ = − 1

4π2

∫Γ1×Γ2

(λ− A)−1(µ− A)−1 dλ dµ.

By the first resolvent equation we have

P 2σ = − 1

4π2

∫Γ1×Γ2

1

µ− λ(R(λ,A)−R(µ,A)) dλ dµ.

Then

(Pσ)2 =1

2πi

∫Γ1

R(λ,A)

(1

2πi

∫Γ2

1

µ− λdµ

)dλ− 1

2πi

∫Γ2

R(µ,A)

(1

2πi

∫Γ1

1

µ− λdλ

)dµ.

The fact that∫

Γ1

1µ−λ dλ = 0, (µ ∈ Γ2) and

∫Γ2

1µ−λ dλ = 2πi, (λ ∈ Γ1), we obtain

P 2σ = Pσ. This prove the first statement.

The second statement follows from the fact that A and the resolvent of A commute onD(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 commuteswith A, which means that A is decomposed according to X = R(Pσ)⊕N(Pσ) and the partsA1 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

R(λ,A)Pσ =1

2πi

∫Γ

(A− λ)−1(A− µ)−1 dµ

=1

2πi[

∫Γ

(λ− µ)−1(A− λ)−1 dµ−∫

Γ

(λ− µ)−1(A− µ)−1 dµ]. (1.24)

The first integral in (1.24) vanishes since λ outside Γ, this gives

R(λ,A)Pσ = − 1

2πi

∫Γ

(λ− µ)−1(A− µ)−1 dµ (1.25)

The integral in (1.25) is analytic in λ outside of Γ, hence R(λ,A)Pσ = (λ−A1)−1 is holomor-phic outside Γ. By taking Γ close to the boundary of σ, and using openness of the resolventset we see that (λ − A1)−1 is analytic on C \ σ. Thus ρ(A1) contains the exterior of Γ andσ(A1) ⊂ σ. In a similar way, it follows from (1.24), if λ inside Γ, that

R(λ,A)Pσ = R(λ,A) +1

2πi

∫Γ

(λ− µ)−1(µ− A)−1 dµ.

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

(λ− A)−1 = (λ− A)−1Pσ + (λ− A)−1(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. Wedefine the Riesz projection of A associated to the point λ0 by

Pλ0(A) =1

2πi

∮Γλ0

(λ− A)−1 dλ, (1.27)

Proposition 1.12 Let A be a closed operator on X and λ0 an isolated point of the spectrumof 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 ontoN(λ0 − A).

Proof. (i) Let x ∈ N(λ0 − A), then for λ 6= λ0, we have (λ − A)−1x = (λ − λ0)−1x. Weshow that Pλ0x = x, so x ∈ R(Pλ0). By the definition of Pλ0

Pλ0x =1

2πi

∫Γλ0

(λ− A)−1x dλ =1

2πi

∫Γλ0

(λ− λ0)−1x dλ = x

(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 andλ = λ0 + reiθ. Then

Pλ0 =1

2π

∮ π

−π(λ0 + reiθ − A)−1 rdθ.

and

P ∗λ0 =1

2π

∮ π

−π((λ0 + reiθ − A)−1)∗ rdθ,

=1

2π

∮ π

−π(λ0 + reiθ − A∗)−1 rdθ,

=1

2π

∮ π

−π(λ0 + re−iθ − A)−1 rdθ. (1.28)

Reparametrizing (1.28) with θ1 = −θ, then

P ∗λ0 =1

2π

∮ π

−π(λ0 + eiθ1 − A)−1 rdθ1 = Pλ0 .

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

(λ0 − A)Pλ0 =1

2πi

∮Γλ0

(λ0 − A)(λ− A)−1 dλ,

=1

2πi

∮Γλ0

(λ− λ0)(A− λ)−1 dλ.

37

Let Vλ0 denote the interior of Γλ0 , the operator (λ − λ0)(A − λ)−1 is an analytic, operatorvalued function on Vλ0λ0 and satisfies

|λ− λ0|||(A− λ)−1|| ≤ |λ− λ0|d(λ, σ(A)

.

Now if we take the diameter of Γλ0 small such that

|λ− λ|||(A− λ)−1|| < 1,

then the function (λ− λ0)(A− λ)−1 is uniformly bounded on Vλ0λ0 and extends to ananalytic function on Vλ0 and hence, by the Cauchy theorem we obtain

1

2πi

∮Γλ0

(λ− λ0)(A− λ)−1 dλ = 0.

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) Pλ,µ = Pλ + Pµ.

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

(λI − A)−1 =+∞∑

n=−∞

(λ− λ0)nAn, (1.29)

whereAn =

1

2πi

∫Γλ0

1

(λ− λ0)n+1R(λ,A) dλ, (1.30)

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 satisfiesthe 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,

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

38

Proof. The commutativity of An with A follows from the commutativity of A and theresolvent of A. For simplicity, to prove (i), we may assume λ0 = 0, since 0 is an isolatedpoint 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

AnAm =1

(2πi)2

∫γr

∫γr1

λ−n−1µ−m−1R(λ,A)R(µ,A) dλdµ

=1

(2πi)2

∫γr

∫γr1

λ−n−1µ−m−1(µ− λ)−1[R(λ,A)−R(µ,A)] dλdµ

By computing the double integral on the right in any order and the fact that

1

2πi

∫γr

λ−n−1(µ− λ)−1dλ = ηnµ−n−1

1

2πi

∫γr1

µ−m−1(µ− λ)−1dµ = (1− ηm)λ−m−1

whereηn =

1 if n ≥ 0,0 if n < 0

We obtain

AnAm =1− ηn − ηm

2πi

∫γr

λ−n−m−2R(λ,A)dλ = (1− ηn − ηm)An+m+1 (1.31)

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

I = (λI − A)+∞∑

n=−∞

(λ− λ0)nAn

= [(λ− λ0)I + (λ0I − A)]+∞∑

n=−∞

(λ− λ0)nAn

=+∞∑

n=−∞

(λ− λ0)n+1An ++∞∑

n=−∞

(λ− λ0)n(λ0I − A)An

=+∞∑

n=−∞

(λ− λ0)n[An−1 + (λ0I − A)An].

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 straight-forward.

From the standard terminology of the complex theory, we call the operator A−1 in theLaurent 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 theRiesz projection Pλ0 associated to λ0. Furthermore, by setting D = A−2 and S = −A0, the

39

relation (1.31) gives A−k = Dk−1 for k ≥ 2 and An = −Sn+1 for n ≥ 0. From this notationsthe Laurent series (1.29) around λ0 is equivalent to

(λI − A)−1 =1

λ− λ0

Pλ0 +∞∑n=1

1

(λ− λ0)n+1Dn −

∞∑n=0

(λ− λ0)nSn+1; (1.32)

By Proposition 1.11 and compare (1.26) with (1.32) we haveR(λ,APλ0) = R(λ;A)Pλ0 = 1

λ−λ0Pλ0 +∑∞

n=11

(λ−λ0)n+1Dn,

R(λ,A(I − Pλ0)) = R(λ;A)(I − Pλ0) = −∑∞

n=0(λ− λ0)nSn+1,

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

hence

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

R(λ,A)(I − Pλ0)

and SPλ0 = Pλ0S = 0, DS = SD = 0, D = DPλ0 = Pλ0D. We conclude the following

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

(λI − A)−1 =1

λ− λ0

Pλ0 +∞∑n=1

1

(λ− λ0)n+1Dn −

∞∑n=0

(λ− λ0)nSn+1;

with the residue operator A−1 is a projection coincides with the Riesz projection associatedto λ0 and SPλ0 = Pλ0S = 0, DS = SD = 0, D = DPλ0 = Pλ0D.

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 thatA−1 = Pλ0 , then it follows by Proposition 1.11 that (λ0I − A)m−1Pλ0 = A−m 6= 0 and(λ0I − A)mPλ0 = A−m−1 = 0.

(2) ⇒ (3). First we prove that N((λ0I − A)m) = N((λ0I − A)m+1). Since we alreadyknow that N((λ0I −A)m) ⊆ N((λ0I −A)m+1), it suffices to prove the inverse inclusion. Weproceed by contradiction. Let x ∈ N((λ0I − A)m+1) and x /∈ N((λ0I − A)m), that is, thevector y = (λ0I − A)mx 6= 0. it follows that (λ0I − A)y = (λ0I − A)m+1x = 0. This impliesby Proposition 1.12 part (ii), Pλ0y = y. Consequently

0 = (λ0I − A)mPλ0x = Pλ0(λ0I − A)mx = Pλ0y = y

40

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

(λ0I − A)m−1x = (λ0I − A)m−1Pλ0x 6= 0

From (λ0I − A)mx = (λ0I − A)mPλ0x = 0, it follows that

N((λ0I − A)m) 6= N((λ0I − A)m−1) (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 =

λ0. The operator (λ0I − A)n is also invertible on N(Pλ0) for all n ∈ N. The identity(λ0I −A)mPλ0 = 0, implies that (λ0I −A)m = 0 on R(Pλ0). Consequently R((λ0I −A)m) =N(Pλ0) = R((λ0I − A)m+1). Thus λ0I − A has finite descent, and by Theorem 1.1 we haved(λ0I − A) = m.

(3) ⇒ (1). Assume that a(λ0I − A) = d(λ0I − A) = m < ∞, By Proposition 1.11 andTheorem 1.1 we have N(Pλ0) = R((λ0I − A)m) = R((λ0I − A)n) and R(Pλ0) = N((λ0I −A)m) = N((λ0I −A)n) for all n ≥ m. It follows that Dn = (A−λ0I)nPλ0 = 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 necessarilyk = m.

Corollary 1.3 Let A be a closed operator. If λ0 is a pole of order m of the resolvent aroundλ0. Then λ0 is an eigenvalue of A. Moreover

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

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

(λI − A)−1 =1

λ− λ0

Pλ0 +m−1∑n=1

1

(λ− λ0)n+1Dn −

∞∑n=0

(λ− λ0)nSn+1; (1.35)

Remark 1.7 Let λ0 a pole of order m of the resolvent around λ0. From Proposition 1.11it 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: Thealgebraic 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 thedimension of N(λ0 − A) and equal α(λ0 − A). In general, we have α(λ0 − A) ≤ m.

41

Chapter 2

Essential Fredholm, Weyl and Browderspectra

2.1 Essential Fredholm spectraWe now introduce some important classes of operators in Fredholm 1 theory. Let X and Ybe Banach spaces and let A be an operator from X into Y . We denote by D(A) ⊂ X itsdomain and R(A) ⊂ Y its range. For A ∈ C(X, Y ), the nullity, α(A), of A is defined as thedimension of N(A) and the deficiency, β(A), of A is defined as the codimension of R(A) inY .

Definition 2.1 Let X, Y be Banach spaces. The set of upper semi-Fredholm operators fromX 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 byK(X, Y ). If X = Y then Φ+(X, Y ), Φ−(X, Y ), Φ±(X, Y ), Φ(X, Y ) and Φb(X, Y ) are re-placed, respectively, by Φ+(X), Φ−(X), Φ±(X), Φ(X) and Φb(X).Observe that in the case X = Y the class Φ(X) is non-empty since the identity trivially isa Fredholm operator. But for certain different Banach spaces X, Y no bounded Fredholmoperators 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 thefollowing 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 ifBA ∈ Φ(X) then A ∈ Φ(X) and B ∈ Φ(X). because N(A) ⊂ N(BA) and R(BA) =R(AB) ⊂ R(A).

By Theorem 1.7A ∈ 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 arbitraryprescribed integer. Indeed if

A =

(Up 00 U∗q

): `2 ⊕ `2 → `2 ⊕ `2,

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-Fredholmoperators.

Theorem 2.2 Suppose that A ∈ Φ±(X, Y ). If B ∈ L(X, Y ) such that D(A) ⊂ D(B) and‖B‖ < 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 ) ifand 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 thecharacterization above the ideal F(X) may be replaced by the ideal K(X) of all compactoperators. In particular, A ∈ Φb(X) if and only if A is invertible in L(X) modulo the idealof finite-dimensional operators F(X).

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

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

The space C(X) with this additional operation is a Banach algebra with the identity I andfollowing the quotient algebra norm

‖A‖e = infK∈K(X) ‖A+K‖ . (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-Fredholmspectrum 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 Wolfessential spectrum of A [118, 101, 117]. σsf (.) is defined by Kato [64]. The two spectraσuf (.) 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) but0 /∈ σuf (A).

Let A ∈ C(X). If the perturbation B in Theorem 2.2 is caused by a multiple of theidentity we have the punctured neighborhood theorem: if A ∈ Φ+(X) then there existsε > 0 such that λI −A ∈ Φ+(X) and α(λI −A) is constant on the punctured neighborhood0 < |λ| < ε. Moreover,

α(λI − A) ≤ α(A) for all |λ| < ε (2.2)

andind(λI − A) = ind(A) for all |λ| < ε.

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

β(λI − A) ≤ β(A) for all |λ| < ε (2.3)

andind(λI − A) = ind(A) for all |λ| < ε.

It follows that σlf (A), σuf (A), σsf (A) and σef (A) are closed sets of C ( compact sets ifA ∈ L(X)). Moreover, the open set ρsf (A) = C\σsf (A), in general, is the union of countablenumber of connected open sets Ωn. Moreover, in each Ωn , with the possible exception ofisolated 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 ofA with finite algebraic multiplicities r(λnj). in the general case ( in which one or both ofαn, 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 intheir immediate neighborhood.

2.2 Fredholm perturbationsOne of the most important question is the invariance of the essential spectra under additiveperturbations. The first result, in this context, is due a H. Weyl when he proved the stabilityof the essential spectrum of the self-adjoint operator under compact perturbation in Hilbertspace. 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 ) andind(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 firststatement of th preceding lemma is not optimal and may be expressed in terms of strictlysingular 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 amore general context. Let us introduce some notations and definitions.

Let X be a Banach space. If N is a closed subspace of X, we denote by πXN the quotientmap X → X/N . The codimension of N , codim(N), is defined as the dimension of the vectorspace X/N .

Definition 2.2 Let X and Y be two Banach spaces and S ∈ L(X, Y ). S is said to bestrictly cosingular operator from X into Y , if there exists no closed subspace N of Y withcodim(N) =∞ such that πYNS : X → Y/N is surjective.

Let CS(X, Y ) denote the set of strictly cosingular operators from X into Y . This classof 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 shortproperty DP) if for each Banach space Y every weakly compact operator A : X → Y takesweakly 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 compactHausdorff 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, ifX is a Banach space whose dual has the DP property thenX has the DP property(see [53]). For more information we refer to the paper by Diestel [40] which contains a surveyand 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 weaklycompact subset is dense in X. For more details and results we refer to [40]. In particular, allseparable 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 ofX, there exists a closed and finite dimensional subspace N ⊂M and a continuous projectionfrom X onto N . Clearly any Hilbert space is subprojective. The spaces c0, lp, (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 Xis contained in a closed subspace W having infinite codimension in X and such that there isa bounded projection from X to W . The spaces lp, (1 < p < ∞), and Lp (1 < p ≤ 2), aresuperprojective (cf. [116]).

Definition 2.4 Let X and Y be two Banach spaces and and A be any class of operatorsfrom 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 introducedand investigated in [44]. In particular, it is shown that PΦb(X, Y ) is a closed subset ofL(X, Y ) and if X = Y , then PΦb(X) = PΦb(X,X) is a closed two-sided ideal of L(X). Thecomponent PA(X, Y ) of the perturbation class PA has sense only when the set A(X, Y ) isnon-empty. In the case A(X, Y ) = ∅. we could define PA(X, Y ) = L(X, Y ), the set of alloperators from X to Y . However, this is not useful.

Indeed, for 1 < p <∞, p 6= 2, both sets Φ+(Lp, `p) and Φ−(`p, Lp) are empty because Lpcontains subspaces isomorphic to `2.

Proposition 2.1 ([44, pp. 69-70]) Let X, Y , Z be Banach spaces. If at least one of thesets 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).

The sets of upper semi-Fredholm and lower semi-Fredholm perturbations are PΦb+(X, Y )

and PΦb−(X, Y ), respectively. In [44], it is shown that PΦb

+(X, Y ) and PΦb−(X, Y ) are closed

subsets of L(X, Y ), and if X = Y , then PΦb+(X) := PΦb

+(X,X) is a closed two-sided idealof 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).

• (ii) If A ∈ Φb+(X, Y ) and F ∈ PΦb

+(X, Y ), then A+F ∈ Φb+(X, Y ) and ind(A+F ) =

ind(A).

• (iii) If A ∈ Φb−(X, Y ) and F ∈ PΦb

−(X, Y ), then A+F ∈ Φb−(X, Y ) and ind(A+F ) =

ind(A).

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

1. K(X, Y ) ⊂ S(X, Y ) ⊂ PΦb+(X, Y ),

2. K(X, Y ) ⊂ CS(X, Y ) ⊂ PΦb−(X, Y ).

A counterexample was found in [46]: there exists a complex separable Banach space Xsuch that PΦb

+(X) 6= S(X) and PΦb−(X∗) 6= CS(X∗). However, the space X is very special:

it is a finite product of hereditarily indecomposable spaces. The existence of hereditarilyindecomposable Banach spaces was only recently proved, see [1]. So the problem remainsopen for many spaces, specially classical Banach spaces.

It is known that PΦb+(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 weaklysequentially complete, and when 2 ≤ p ≤ ∞, Y containing a subspace isomorphic toLp [48]. X = Lp, Y = Lq, 1 ≤ q ≤ p < 2[48].

Moreover, PΦb−(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 theOrlicz 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

PΦb+(Lp) = S(Lp) = PΦb

−(Lp) = CS(Lp) = PΦb(Lp) (2.4)

for p ∈ [1,∞).

We observe that the perturbation classes PΦ+ and PΦ− studied in [115] correspond tonot necessarily bounded upper and lower semi-Fredholm operators. Weis proved that

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

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

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

• PΦ−(X) = CS(X) if every, infinite dimensional quotient space of the Banach space Xhas 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. Banachspace, 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 J the restriction of J to D(A). Moreover,if J ∈ L(XA, X) we say that J is A−bounded. One checks easily that if J is closed (orclosable). (cf.[64, Remark 1.5, p. 191]), then J is A−bounded.

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

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

α(A) = α(A), β(A) = β(A), R(A) = R(A)

α(A+ J) = α(A+ J)

β(A+ J) = β(A+ J), and R(A+ J) = R(A+ J)

(2.5)

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). IfA ∈ 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 thatA ∈ Φb(XA, X). Applying again the Theorem 2.4 we obtain that A0 ∈ Φb(X,XA). On theother 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) impliesthat J0 ∈ Φb(X), we conclude that (A + J)A0 ∈ Φb(X), it follows that A + J ∈ Φb(XA, X)because A0 ∈ Φb(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) isa 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, thefact that PΦ(X, Y ) is equal or not to PΦb(X, Y ) seems to be unknown. Moreover,whether or not PΦ+(X) (resp.PΦ−(X)) is equal to PΦb

+(X) (resp. PΦb−(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

PΦ+(Lp) = S(Lp) = PΦ−(Lp) = CS(Lp) = PΦb(Lp) (2.8)

for p ∈ [1,∞).

Lemma 2.4 ([74]) Let X be a Banach space. Then PΦ+(X) and PΦ−(X) are closed sub-sets 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 relationA+J = A−(Jn+J)−Jn, together with the fact Jn ∈ PΦ+(X) (resp. PΦ−(X)) we concludethat 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 consequentlyα(A + J) <∞. Conversely, assume that J /∈ PΦ+(X). Then there exists A ∈ Φ+(X) suchthat A + J /∈ Φ+(X). Therefore, A + J /∈ Φb

+(XA, X). Next, applying [79, Lemma 4.3] weinfer that there exists an operator K such that K ∈ K(XA, X) (i.e., K is A-compact) andα(A + J + K) = ∞. By using (2.5) we have α(A + J + K) = ∞. On the other hand, byLemma 2.1 we have A + K ∈ Φ+(X), and therefore α(A + J + K) < ∞. This contradictsthe 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) < ∞.Conversely, suppose that α(A+ J) <∞ for each A ∈ Φ(X). By 2.5, 1

µ(A+K) ∈ Φ(X) for

each K ∈ K(X) and µ an arbitrary nonzero complex number. Hence α(A+µJ +K) is finitefor all scalar µ. Thus by lemma 2.1 (i), we see that A+µJ ∈ Φ+(X). Now arguing as in theproof of [45, Theorem 2.1, p.117] and using the compactness of the interval [0, 1] we obtainβ(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 thereexists A0 ∈ L(X) and F ∈ F(X) such that

AA0 = I + F on X. (2.10)

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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(2.11) implies that AA0 and (A+ J)A0 are in Φb(X) and

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

On the other hand, proceeding as in the proof of Lemma 2.3 we see that A0 ∈ Φb(X,XA)and A+ J ∈ Φb(XA, X). Next, applying Atkinson theorem to both AA0 and (A+ J)A0 andusing (2.12 we obtain ind(A) = −ind(A0) and ind(A + J) = −ind(A0) which implies thatind(A) = ind(A+ J). 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 thatα(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 theresults of Proposition 2.2 are valid:

1. If I(X) satisfies the hypothesis (2.9), then F(X) ⊆ I(X). Hence the ideal of finiterank 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 onlythe 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) arew.c.g., consequently we have PΦ+(Lp) = S(Lp) and PΦ−(Lp) = CS(Lp). In Lp(Ω, dµ)and in C(E) (the Banach space of continuous scalar-valued function on E with thesupremum norm) provided that E is a compact Hausdorff space we have a strongerresult, 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 properideal 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 idealof L(X) and K(X) = PΦ+(X) = PΦ−(X) = PΦ(X)

(a) X is a separable Hilbert space. See Calkin [32].

(b) X = `p, 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

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

As consequence of the Proposition 2.2, the following important stability theorem of theessential 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 inExample 1.12, as follows

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

A1x = (x2, x3, x4, . . . )

A2x = (x1,1

2x2,

1

3x3, . . . )

A3x = (x2,1

2x3,

1

3x4, . . . )

A4x = (0, x1,1

2x2,

1

3x3, . . . )

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

(x1, 0, . . .).

and hence, α(A1) = 1. Moreovere, R(A1) = `p = N(A1)⊕X0, where X0 is the closed suspaceconsisting 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 andR(A0) = X0.Since the operators A2, A3 and A4 are compact, it follows therefore, that A0 +Ai and A1 +Aiare Fredholm operators with ind(A0 +Ai) = ind(A0) = −1 and ind(A1 +Ai) = ind(A1) = 1,i = 2, 3, 4.

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The following definition gives the concept of inessential operators to operators actingbetween 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 areBanach spaces, then R1TR2 ∈ I(W,Z).

In general, for every pair X, Y of Banach spaces for which PΦb−(X, Y ) and PΦb

−(X, Y ) aredefined, we have

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

2. K(X, Y ) ⊂ CS(X, Y ) ⊂ PΦb−(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 issaid to be completely continuous if A transforms relatively weakly compact sets intorelatively compact sets. Note that if X or Y is reflexive then every A ∈ L(X, Y ) isweakly compact, see Goldberg [45]. A Banach spaceX has the Dunford-Pettis propertyif any weakly compact operator A from X into another Banach spaces Y is completelycontinuous.

(b) X has the reciprocal Dunford-Pettis property and Y has the Schur property. Recallthat X is said to have the reciprocal Dunford-Pettis property if every completelycontinuous operator from X into any Banach spaces is weakly compact, whilst Y hasthe 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 comple-mented copies of `1 and Y = L1.

(f) X or Y are `1 with 1 ≤ p ≤ ∞ or c0, and X, Y are different.

Another important class in connection with the Fredholm perturbation and not necessaryan 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 tobe 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 characterizationof 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).

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Riesz operators are a generalization of compact and strictly singular operators and exhibitmany 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 idealcontained in R(X). Moreover, In [34], Cardus proved that every inessential operator liesin R(X). For further information on the family of Riesz operators we refer to [34, 62] andthe references therein. Now, we state the result of the Riesz perturbation of the Fredholmoperators.

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 spectraAn 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 ornegative 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

A =

(U 00 U∗

): `2 ⊕ `2 → `2 ⊕ `2,

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 inσ(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 operatorbounded below.

The various classes of operators defined above motivate the definition of several essentialspectra:

• 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],and σeb(.) the Browder4 essential spectrum [31, 55, 62, 101]. σuw(.) and σlw(.) are theessential 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 anotheressential version of the approximate spectrum and the defect spectrum respectively. Notethat 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 isbounded. 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 betweenthese 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). Supposethat ρef (A) ∩ σew(A) 6= ∅, then there exists λ0 ∈ ρef (A) ∩ σew(A), since ρ(A) 6= ∅, thereexists 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). Thiscontradict 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)), ) thenα(A − λ) < ∞ and β(A − λ) = +∞, (resp. α(A − λ) = +∞ and β(A − λ) < ∞, ), henceind(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 isa Weyl operator. we know that the operator K given by

Kx = (x1,1

2x2,

1

3x3, . . . )

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 spectrumsuch that the corresponding Riesz projectors Pλ(A) are finite dimensional. By Corollary 1.3the points of σdis(A) being poles of finite rank, i.e., around each of these points there is apunctured disk in which the resolvent has a Laurent expansion (1.35) whose principal parthas only finitely many nonzero terms, the coefficients in these being of finite rank. It followsthat 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 bethe 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 anequivalent 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 theRiesz projection associated to the point λ0.

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

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

(λI − A)−1 =1

λ− λ0

Pλ0 +m−1∑n=1

1

(λ− λ0)n+1Dn −

∞∑n=0

(λ− λ0)nSn+1; (2.19)

with D = (λ0I −A)Pλ0, S = − 12πi

∫Γλ0

(λ− λ0)−1(λI −A)−1 dλ and Γλ0 is a positivelyoriented 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 whichthe resolvent is finitely meromorphic.

For λ ∈ ρB(A), let Pλ be the corresponding finite rank Riesz projector. From the factthat 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 wedenote by RB(λ,A) and called the Browder resolvent, i.e.,

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

Clearly RB(λ,A) = (λ − A)−1, for λ ∈ ρ(A) and RB(A, λ) may be viewed as an exten-sion of the usual resolvent from ρ(A) to ρB(A) and retains many of its important prop-erties. For example, because PλAλ = Pλ on D(A) and AλPλ = Pλ on X it follows thatPλRB(λ,A) = Pλ = RB(λ,A)Pλ, and we also have the following version of the resolventidentity 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 op-erator with dimR(MA(λ, µ)) = dim(R(Pλ)) + dim(R(Pµ)). Furthermore, in case λ 6= µthe 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 essentialspectrum

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 theunbounded connected component of ρeb(A). Since C\C is compact, we find λ0 ∈ C\C suchthat

|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, butalso satisfy a Gel’fand-type formula with the norm ‖A‖ 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

re(A) = r(π(A)) = limn→+∞

‖An‖1ne = inf

n∈N‖An‖

1ne .

58

2.4 Characterizations of the essential spectraIn the following we give some characterizations of the essential spectra based in the Fredholmperturbations (not necessary an ideals) and considered as a useful equivalent definitions ofthis essential spectra.

Theorem 2.16 ([17]) Let A ∈ C(X). Then

σew(A) =⋂

K∈K(X)

σ(A+K)

=⋂

F∈F(X)

σ(A+ F )

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 spaceN(A). We have, N(A)∩N(P ) = 0 and we can represent the finite rank operator P in theform

Px =i=1∑n

fi(x)xi,

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 Pxi = xi for every i = 1, . . . , n, from which weobtain that fi(xj) = δi,j, where δi,j denote the delta of Kronecker. Denote M the topologicalcomplement of R(A). Then dimM = n, so we can choose a basis y1, . . . , yn of M . Set

Fx =i=1∑n

fi(x)yi.

Then F is an operator of finite rank, by Theorem 2.7, A + F ∈ W(X). Now, letx ∈ N(A+F ), then Ax = Fx = 0, and this easily implies that fi(x) = 0 for all i = 1, . . . , n.From this it follows that Px = 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.this show

⋂F∈F(X) σ(A+ F ) ⊆ σew(A) and the fact that F(X) ⊆ K(X) we have⋂

K∈K(X)

σ(A+K) ⊆⋂

F∈F(X)

σ(A+ F ) ⊆ σew(A)

Now, suppose that A + K is invertible with K is compact, obviously A + K ∈ W(X), andhence by Theorem 2.7 we conclude that A ∈ W(X).

Theorem 2.17 ([74]) Let A ∈ C(X). Then

σew(A) =⋂

K∈M(X)

σ(A+K) (2.22)

whereM(X) be any subset of L(X) satisfying K(X) ⊆M(X) ⊆ PΦ(X).

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) andind(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 therelation λI −A = (λI −A−K)(I + (λI −A−K)−1K together with Atkinsons theorem weget λI − A is a Weyl operator. This shows that λ /∈ σew(A). The opposite inclusion followsfrom K(X) ⊆M(X).

59

Theorem 2.18 ([59]) Let A ∈ C(X). Then

σuw(A) =⋂

K∈K(X)

σap(A+K). (2.23)

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) anoperator bounded below. Hence by (1.12) there exists c > 0 such that ‖Ux‖ ≥ c‖x‖ for allx ∈ D(A). Thus λ /∈ σap(A+K) and therefore λ /∈

⋂K∈K(X) σap(A+K).

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 Proposition2.2 that λI − A ∈ Φ+(X) with ind(A) ≤ 0. This completes the proof.

Theorem 2.19 ([59]) Let A ∈ C(X). Then

σlw(A) =⋂

K∈K(X)

σsu(A+K). (2.24)

Theorem 2.20 ([59]) Let A ∈ C(X) with nonempty resolvent set. Then

σuw(A) =⋂

K∈∆A(X)

σap(A+K). (2.25)

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

Proof. Since K(X) ⊂ ∆A(X), we refer that⋂K∈∆A(X) σap(A+K) ⊂ σuw(A). Conversely,

suppose that there exists K ∈ ∆A(X) such that λI − A − K is bounded below, henceλI −A−K ∈ Φ+(X). Since Y = R(λI −A−K) is closed subspace of X, then Y itself is aBanach space with the same norm. Therefore (λ− A− K)−1 ∈ L(Y,XA). let µ ∈ ρ(A) suchthat K(µ− A)−1 ∈ PΦb

+(X), then we have

K(λ− A− K)−1 = [J + (µ− λ+ K)(λ− A− K)−1] (2.26)

where J denotes the embedding operator which maps every x ∈ Y on to the same elementin X. Since (λ−A− K) ∈ L(XA, X) and K(λ− A)−1 ∈ PΦb

+(X), then it follows from [44,p. 70] and Eq. (2.26) that

K(λ− A− K)−1 ∈ PΦb+(Y,X) (2.27)

clearly, J is injective and R(J) = Y . So, J ∈ Φb+(Y,X) and ind(J) ≤ 0. Therefore, we can

deduce from (2.27) and Lemma 2.2 that

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

The fact that λ− A = (J + K(λ− A− K)−1)(λ− A− K) and by using (2.28) together withTheorem 2.1 we get λ− A ∈ Φb

+(XA, X) and ind(λ− A) ≤ 0. Now using (2.5) we infer thatλ /∈ σuw(A).

Theorem 2.21 ([59]) Let A ∈ C(X). Then

σlw(A) =⋂

K∈PΦ−(X)

σsu(A+K). (2.29)

60

Proof. Suppose that there exists K ∈ PΦ−(X) such that λI −A−K is surjective, henceλI − 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.Thusλ /∈ 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 LetM(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) =⋂

K∈K(X),KA=AK

σ(A+K) =⋂

F∈F(X),FA=AF

σ(A+ F ) =⋂

R∈R(X),RA=AR

σ(A+R),

σub(A) =⋂

K∈K(X),KA=AK

σap(A+K) =⋂

F∈F(X),FA=AF

σap(A+F ) =⋂

R∈R(X),RA=AR

σap(A+R),

and

σlb(A) =⋂

K∈K(X),KA=AK

σsu(A+K) =⋂

F∈F(X),FA=AF

σsu(A+ F ) =⋂

R∈R(X),RA=AR

σsu(A+R).

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 isisolated, 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 existsa compact operator K such that λ ∈ ρ(A + K). Denoting by E the connected componentof 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 thereforealso of R(µ;A). This contradicts our assumption, and so λ ∈ ∂σew(A). To prove the secondinclusion in (2), suppose we can find λ ∈ ∂σew(A) \ ∂σef (A). Then λI −A is Fredholm, andso there exists δ > 0 such that also µI − A is Fredholm for |λ− µ| < δ, by the stability ofthe Fredholm property. Moreover, ind(λI − A) = ind(µI − A) for such µ. But we can takeλ ∈ ρew(A), this implies that ind(µI−A) = 0, and so ind(λI−A) = 0, a contradiction. Nowsuppose λ ∈ ∂σef (A) \ ∂σsf (A). Then λI − A is semi-Fredholm, and so there exists δ > 0such that also µI−A is semi-Fredholm for |λ− µ| < δ, by the stability of the semi-Fredholmproperty. Moreover, ind(λI−A) = ind(µI−A) for such µ. We can take λ ∈ ρef (A), so thatind(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, ingeneral, a closed (or closable) A-defined linear operator. The operatorK does not necessarilysatisfy the hypotheses of the previous results. For some physical conditions on K, we haveinformation about the operator (λI−A)−1− (λI−B)−1 (λ ∈ ρ(A)∩ρ(B)). So the followinguseful stability result.

Theorem 2.23 ([59, 74]) Let A,B ∈ C(X) such that ρ(A) ∩ ρ(B) 6= ∅. Let I(X) be anynonzero 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).

This shows that µ − A ∈ Φ+(X)(resp.µ − A ∈ Φ−(X) ) if and only if µ−1 − A−1 ∈ Φ+(X)(resp. µ−1 − A−1 ∈ Φ−(X) ), in this case we have ind(µ− A) = ind(µ−1 − A−1). 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). Thisproves (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) foreach µ /∈ σuf (A) (resp. µ /∈ σlf (A)). This concludes the proof of (ii) (resp.(iii). Finally, bycombining (ii) and (iii), then by Proposition 2.2 (iv) we have (iv).

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2.5 Left-right Fredholm and Left-right Browder spectraWe use Gl(X) and Gr(X), respectively, to denote the set of all left and right invertibleoperators on X. It is well-known that A ∈ Gl(X) if and only if A is injective and R(A) isa closed and complemented subspace of X. Also, A ∈ Gr(X) if and only if A is onto andN(A) is a complemented subspace of X. The set of all invertible operators on X is denotedby 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 ifABA = 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 aprojection 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)) suchthat ABA = A.

(c) A ∈ W+(X) if and only if there exists B ∈ Φ(X), β(B) = 0 (hence B ∈ Gr(X)) suchthat ABA = A.

(c) A ∈ B(X) if and only if there are m ≥ 1 and B ∈ G(X) such that AmBAm = Am andAmB = BAm.

Generalized invers are useful in solving linear equations. Suppose that B is a generalizedinverse of A. If Ax = y is solvable for given y ∈ X; then By is a solution (not necessaryonly 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) <∞,

andΦl(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) Weyloperators.

The operator A ∈ L(X) is left Browder if it is left Feredholm of finite ascent, and A isright Browder if it is right Fredholm of finite ascent. Let Bl(X) (Br(X)) denote the set ofall left (right) Browder operators. The following theorem gives a characterization of left andright Browder operators

63

Theorem 2.25 ([123]) Let A ∈ L(X). Then A is left (right) Browder operator if and onlyif there exist closed subspaces X1 and X2 invariant with respect to A such that X = X1⊕X2,dimX1 < ∞, 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 followinginclusions

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 Nbe 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) andσlb(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) andσlf (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 Rieszperturbations:

Theorem 2.26 ([122]) If A ∈ L(X) and R ∈ L(X) is a Riesz operators which commuteswith 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, Theorem8.8] 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 givesσlef (A+R) = σlef (A). For the right Fredholm spectrum, we replace in (2.31) the left spec-trum by the right.

Note that, if A ∈ L(X) and R ∈ L(X) is a Riesz operators which commutes with A, bya 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

X = `2 × `2, A =

(T 00 S

), B =

(S 00 T

), N =

(0 I0 0

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

(B −N)(A+N) = I −(

0 00 I − ST

), (A+N)(B −N) = I −

(I − ST 0

0 0

)both products are Fredholm of index zero, that is Weyl, and we can check that A+N is oneone 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 asthe 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) =⋂

R∈R(X),RA=AR

σl(A+R),

andσrb(A) =

⋂R∈R(X),RA=AR

σri(A+R).

Theorem 2.29 ([123]) Let A ∈ L(X) and J(X) any non zero ideal of Riesz operators.Then

σlew(A) =⋂

R∈R(X),RA−AR∈J(X)

σl(A+R),

andσrw(A) =

⋂R∈R(X),RA−AR∈J(X)

σri(A+R).

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 alsoσif (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 alsoσib(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 spectraNow we want to study the influence of perturbations on the spectrum. Our hope is thatat least some parts of the spectrum remain invariant under additive perturbations, such asoperators 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

‖Qn‖1n −→ 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 notnilpotent is the operator A4 defined in Example 1.12.An example for quasi-nilpotent but neither nilpotent nor compact:

Q = Q1 ⊕Q2 : `2 ⊕ `2 −→ `2 ⊕ `2,

where

Q1x = (0, x1, 0, x3, 0, x5, . . . )

Q2x = (0, x1,1

2x2,

1

3x3, . . . ).

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 invertibleoperators.

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 R is quasi-nilpotentin the Calkin algebra C(X).

Now, we will consider, in the following and in the next chapters, for every A ∈ L(X), Xinfinite 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 ‖U‖ < ε for some ε > 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 analyticfunction 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 ofessential 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 theboxes 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, ewsee [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]. Theinterested reader may find further results on the spectral mapping Theorem also inSchmoeger [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 pertur-bation 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 closeddensely defined linear operators on X (see the results of this Chapter, and for moredetails see also [105], [64] and [59]).

7. Consider the identity operator in a Hilbert space and let P be a 1−dimensional or-thogonal 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 Fx = 〈x, e0〉 e1. Then d(A − F ) = ∞ so that σub(A) andσlb(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 unilateralshift, consider

A =

(U + I 0

0 U∗ − I

): `2 ⊕ `2 → `2 ⊕ `2,

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,

p(A) = (A+ I)(A− I) =

(U + 2I 0

0 U∗

)(U 00 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 versionThe semi-regular spectrum was first introduced by Apostol [8] for operators on Hilbertspaces and successively studied by several authors Muller[89]and Rakocevic [98], Mbekhtaand Ouahab [86]and Mbekhta [82] in the more general context of operators acting on Banachspaces.

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 theequivalent conditions of lemma 1.2 or Corollary 1.1. Trivial examples of semi-regular oper-ators are surjective operators as well as injective operators with closed range.

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

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

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

Observe that jump(A) ≥ 0 and the continuity of the index ensures that both definitionsof 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 gapmetric. let us defined this concept.

Let M,N be two closed linear subspaces of the Banach space X and set

δ(M,N) = supdist(x,N) : x ∈M, ‖x‖ = 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

δ(M,N) = maxδ(M,N), δ(M,N)

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

70

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

Note that if X is Hilbert space, then the gap metric is defined in terms of the orthogonalprojection as follows:

δ(M,N) = ‖(I − PN)PM‖and

δ(M,N) = ‖PM − PN‖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 anoperator 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. δ(N(λnI − A), N(λ0I − A)) ≤ 1α|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. minγ(µI − A), γ(λI − A)δ(N(µI − A), N(λI − A)) ≤ |µ− λ|.

Proof. (1). The statement is trivial for λ = µ. Suppose that λ 6= µ and consider anelement 0 6= x ∈ N(µI − A). Then x /∈ N(λI − A) and hence

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

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

γ(λI − A) supx∈B

dist(x,N(λI − A)) ≤ |µ− λ|

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 < ε < 1 there existsx0 ∈ X such that (x− x0) ∈M and

dist(x0, N) ≥(

(1− ε)1− δ(M,N)

1 + δ(M,N)

)‖x0‖ . (3.1)

71

Proof. If x ∈M it suffices to take x0 = 0.Assume therefore that x /∈M . Let X = X/M denote the quotient space and put x = x+Mthe equivalence class of x. Evidently, ‖x‖ = infz∈x ‖z‖ > 0. We claim that there exists anelement x0 ∈ X such that

‖x‖ = dist(x0,M) ≥ (1− ε) ‖x0‖

Indeed, when it is not so, then

‖x‖ = ‖z‖ < (1− ε) ‖z‖ for every z ∈ x

and therefore‖x‖ ≤ inf

z∈x‖z‖ = (1− ε) ‖x‖

This is impossible since ‖x‖ > 0.Let µ = dist(x0, N) = infu∈N ‖x0 − u‖. We know that there exists y ∈ N such that‖x0 − y‖ ≤ µ+ ε ‖x0‖. From that we obtain ‖y‖ ≤ (1 + ε) ‖x0‖+ µ.On the other hand, we have dist(y,M) ≤ δ(N,M) ‖y‖ and hence

(1− ε) ‖x0‖ ≤ dist(x0,M)

≤ ‖x0 − y‖+ dist(y,M)

≤ µ+ ε ‖x0‖+ δ(N,M) ‖y‖≤ µ+ ε ‖x0‖+ δ(N,M)[(1 + ε) ‖x0‖+ µ]

From this we obtain that

µ ≥[

1− ε− δ(N,M)

1 + δ(N,M)− ε]‖x0‖ .

Since ε > 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

δ(N(λnI − A), N(λmI − A)) ≤ 1

minγ(λnI − A), γ(λmI − A)|λn − λm|

Since by assumption γ(λnI−A) ≥ α, for all n ∈ N we have minγ(λnI−A), γ(λmI−A) ≥ αand

δ(N(λnI − A), N(λmI − A)) ≤ 1

α|λn − λm| (3.2)

The sequence (N(λnI −A))n) is a Cauchy sequence in the complet metric space F(X), thusit converges. Let F = limn→∞N(λnI − A).Let x ∈ N(λ0I − A), by proposition 3.1 part (1) we find

δ(N(λ0I − A), N(λnI − A)) ≤ 1

α|λn − λ0|

From this estimate we deduce that x ∈ F and N(λ0I − A) ⊂ F .conversely, let x ∈ F , by proposition 3.2 for 0 < ε < 1, N = F and M = N(λnI − A) thereexists xn ∈ X such that (x− xn) ∈ N(λnI − A) and

dist(xn, F ) ≥(

(1− ε)1− δ(N(λnI − A), F )

1 + δ(N(λnI − A), F )

)‖xn‖ .

72

From this we obtain

δ(F,N(λnI − A)) ≥ dist(xn, F ) ≥(

(1− ε)1− δ(N(λnI − A), F )

1 + δ(N(λnI − A), F )

)‖xn‖ .

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 endthe proof of (1) we take n→∞ in (3.2).2. Suppose that λ0 /∈ R(α), then there exists x ∈ X and 0 < ε < 1 ; ‖x‖ = 1, x /∈ N(λ0I−A)and

‖(λ0I − A)x‖ < (1− ε)α ‖x‖We can find xn /∈ N(λnI − A) such that (x− xn) ∈ N(λnI − A) and take n ∈ N such that

|λn − λ0| <ε

2α

Then‖(λnI − A)x‖ ≤ ‖(λ0I − A)x‖+ |λn − λ0| ‖x‖

and therefore

‖(λnI − A)x‖ ≤ (1− ε

2)α ‖x‖ (3.3)

On other hand, We have (x− xn) ∈ N(λnI − A) and hence

‖x− xn‖ ≤ supdist(y,N(λnI − A)); y ∈ N(λ0I − A), ‖y‖ = 1≤ δ(N(λ0I − A), N(λnI − A)) ‖x‖≤ δ(N(λ0I − A), N(λnI − A)) ‖x‖

≤ 1

α|λn − λ0| ‖x‖

≤ ε

2‖x‖

From the last inequality it fllows that

‖x‖ ≤ 1

(1− ε2)‖xn‖ (3.4)

From (3.3) and (3.4) we obtain

‖(λnI − A)xn‖ < α ‖xn‖

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

dist(x,R((λ0I − A)k)) ≤ dist(x,N(λnI − A)) ‖x‖≤ δ(N(λ0I − A), N(λnI − A)) ‖x‖≤ δ(N(λ0I − A), N(λnI − A)) ‖x‖

73

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

A semi-regular operator A has a closed range. It is evident that the reduced minimummodulus of A is useful to find conditions which ensure that R(A) is closed. The followingtheorem 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 gaptopology.

4. R(λ0I − A) is closed in a neighborhood of λ0 and the mapping λ → R(λI − A) iscontinuous at λ0 in the gap topology.

For an essential version of semi-regular operators we use the following notation. Forsubspaces M,L ⊂ X write M ⊂e L if there exists a finite-dimensional subspace F of X forwhich 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 closedand N(An) ⊂e R(A), for all n ≥ 0.

Now, setV0(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-groupsnor 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 ifand only if there exists a closed subspace V of X such that TV = V and the operatorA : 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 neighbor-hood 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),supn∈N(dimN(λI − A) +N((λI − A)n))/N(λI − A) <∞ and R(λI − A) is closed.

Theorem 3.4 ([98]) Let A ∈ L(X). Then

σes(A) =⋂

K∈K(X),KA=AK

σse(A+K) =⋂

F∈F(X),FA=AF

σse(A+ F )

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 supn∈N

σse(An) ⊂ σse(A) and lim supn∈N

σes(An) ⊂ σes(A)

3.2 Closed-range spectrumMost of the classes of operators considered before require that the operators have closedranges. Thus it is natural to consider the closed-range spectrum or the Goldberg spectrumof 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 butR(λI −A) is not closed for all λ ∈ D(0, 1) \ 0. For example, the right shift operatorA 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

A(x1, x2, x3, . . . ) = (0, x1, 0,1

3x2, 0,

1

5x3, 0, . . . )

The operator A is compact and R(A) is not closed, A2 = 0 and R(A2) is closed.

75

4. Conversely, it is also possible that R(A) is closed but R(A2) is not. let A =

(V I0 0

)be an operator defined on `2 ⊕ `2, where V has the following proprieties that V 2 = 0and 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 thenilpotent 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 mostcountable. Thus the essentially semi-regular spectrum can be considered as a nice completionof the closed-range spectrum. However, the spectrum σec(A) can be used (see Remark 3.1and Corollary 3.1 below) to obtain information on the location in the complex plane of thevarious types of essential spectra, Fredholm, Weyl and Browder spectra etc... , for largeclasses 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 wehave

σc(A) ⊂⋂i∈Λ

σi(A).

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 spectrumQuasi-Fredholm operators have been introduced by J.P Labrousse [70] as a generalization ofthe semi-Fredholm operators. Define

∆(A) = n ∈ N : ∀m ∈ N : m ≥ n⇒ R(An) ∩N(A) ⊆ R(Am) ∩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-Fredholmoperators are semi-regular operators (quasi-Fredholm of degree 0), essentially semi-regularoperators, Fredholm operators and semi-Fredholm operators. Some other examples of quasi-Fredholm operators operators may be found in Mbekhta [86], Labrousse [70] and in Chapter4. We give now a fundamental characterizations of quasi-Fredholm operators.

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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 Hsuch 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 re-strictions,

Theorem 3.7 A ∈ L(X) is quasi-Fredholm operator if and only if there exists n ∈ N suchthat R(An) is closed and An is semi-regular operator. In this case Tn is semi-regular for allm ≥ n.

Proof. Suppose that there exists n ∈ N such that R(An) is closed and An is semi-regular operator. Then Amn is semi-regular for all m ≥ 1, it follows that R(Ap) is closed forall p ≥ n. Since An is semi-regular we have N(An) = N(A) ∩ R(An) = N(A) ∩ R(Am) forall 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 isquasi-Fredholm operator.

Conversely, suppose that A is a quasi-Fredholm operator and let d = dis(A). Thus R(Ad)is closed. Consider the operator Ad : R(Ad)→ R(Ad), then R(Ad) = R(Ad+1) is closed andN(Ad) = N(A) ∩ R(Ad) = N(A) ∩ R(Am) ⊂ R(Amd ) for all m ≥ d. So Ad is a semi-regularoperator. Moreover, for all m ≥ d

N(Am) ⊆ N(Ad) ⊆ R(And) = R(Ad+1+n)) for all n ∈ N,

In particular

N(Am) ⊆ R(Ad+(m−d)+1+n)) = R(Am+1+n) = R(Anm) for all n ∈ N.

Moreover, since And is semi-regular for all n ∈ N, it then follows R(Am) is closed for allm ≥ 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 isthe 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)

andσec(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 mostcountable.

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.

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3.4 Generalized Kato spectrumNow, we introduce an important class of bounded operators which involves the concept ofsemi-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 forwhich (A/N)d = 0. In this case T is said to be of Kato type of order d. An operator isof Kato type if it is of Kato type of some degree d. Operators which admit a generalizeddecomposition 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] andthe work of Q. Jiang-H. Zhong [60]. The operators which satisfy this property form a classwhich 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 Theorem3.14. In particular, if 0 is a pole of the resolvent of A, see Theorem 1.27 or Theorem3.15.

6. Riesz operators. If A is a Riesz operator, then A = A1 ⊕ A2 with A1 is compact andA2 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 =

(A1 00 A2

): R(P )⊕N(P ) −→ R(P )⊕N(P )

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 invertibleand 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

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10. Left Browder operators. If A is a Browder operator, then A = A1 ⊕A2 with A1 is leftinvertible and A2 is nilpotent operator, see Theorem 2.25.

11. Semi-Fredholm operators. Kato proved that a closed semi-Fredholm operator is ofKato type [63, Theorem 4].

12. Quasi-Fredholm operators. In Hilbert spaces, this class coincide with Kato type oper-ators, Theorem 3.6. The same decomposition exists also for quasi-Fredholm operatorson 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 op-erators coincides with the set of Kato type operators. But in the case of Banach spacesthe Kato type operator is also quasi-Fredholm (see Theorem 3.9), according to the remarkfollowing 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 generalizedKato 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 decompositionσgk(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 isa 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 opendisc D(0, ε) for which λI − A is semi-regular for all λ ∈ D(0, ε) \ 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 opendisc D(0, ε) for which λI − A is semi-regular for all λ ∈ D(0, ε) \ 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 3.10and Theorem 3.11 , we easily obtain that these spectra differ from each other on at mostcountably many isolated points.

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Proposition 3.4 ([1, 60]) The sets σse(A) \ σgk(A), σes(A) \ σk(A), σes(A) \ σgk(A) andσk(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),

andσec(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 considerboundary 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 operatorthere exists ε1 > 0 for which λI − A ∈ Φ±(X) for all 0 ≤ |λ| < ε1. On other hand, A is ofKato type, then by Theorem 3.11, there exists ε2 > 0 such that λI −A is semi-regular for all0 < |λ| < ε2. Consequently, jump(λI − A) = 0 for all 0 < |λ| < ε with ε = minε1, ε2. Weconclude that if A ∈ Φ±(X) such that jump(A) > 0, then there exists ε > 0, such that λI−Ais semi-regular operator for all 0 < |λ| < ε, Hence there exists a sequence of semi-regularoperators (An)n≥1 such that An −→ A.

For A ∈ L(X), there are two linear subspaces of X defined in [83], the quasi-nilpotentpart H0(A) of A:

H0(A) =x ∈ X : lim

n→∞‖Anx‖

1n = 0

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

Ax1 = x,Axn+1 = xn and ‖xn‖ ≤ δn‖x‖ for all n ∈ N

It easily follows, from the definitions, that H0(A) and K(A) are generally not closed andA(K(A)) = K(A). Observe that if Y is a closed subspace of X such that A(Y ) = Y , thenY ⊂ 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).

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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 assertionsare 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 assertionsare equivalent:

1. 0 is a pole of the resolvent of A;

2. There exists p ∈ N such that K(A) = R(Ap) and H0(A) = N(Ap).

Motivated by the relation of the essential quasi-Fredholm spectrum and the closed rangespectrum given in Theorem 3.8, we study this relation in the case of the generalized Katospectrum. 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, ε) such that µI −A is semi-regular inD(λ, ε)\λ, so that R(µI−A) is closed if µ ∈ D(λ, ε)\λ. This contradicts our assumptionthat 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) =⋃∞m=1R( 1

m) and

σgk(A) ∩ (C \ σec(A)) =∞⋃m=1

(σgk(A) ∩R(1

m)).

To finish the proof we prove that the set σgk(A) ∩ R( 1m

) is at most countable. Let λ0 bea non-isolated point of σgk(A) ∩ R( 1

m). Then there exists (λn)n ⊂ σgk(A) ∩ R( 1

m) such

that λn → λ0, by Theorem 3.1 λ0I − A is semi-regular operator, hence λ0 /∈ σgk(A). Thiscontradicts the closedness of σgk(A).

Proposition 3.7 σse(A) \ (σgk(A) ∩ σec(A)) is at most countable.

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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 impliesthat 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) isnon-isolated point, hence λ ∈ σgk(A).

Example 3.1 Let X = l2 the space of complex square-summable sequences and the linearoperator A defined by

Ax = (0, x1, 0,1

3x2, 0,

1

5x3, 0, . . . ), x = (xn) ∈ `2

The operator A is compact and R(A) is not closed, then 0 ∈ σec(A). It easy to see thatA2 = 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 apolynomial growth condition. An operator A satisfies this condition if there exists K > 0,and δ > 0 for which

‖exp(iλA)‖ ≤ K(1 + |λ|δ) for all λ ∈ R,

Examples of operators which satisfy a polynomial growth condition are Hermitian operatorson 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 realspectra. 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 suchthat ∥∥(λI − A)−1

∥∥ ≤ K(1 + |Imλ|−δ) 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).

2. If λ ∈ σec(A) and R((λI − A)p) is closed for some p ∈ N, then λ is a pole of theresolvent of A.

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Proof. 1. If we assume that A ∈ Pg(X) and R((λI−A)) is closed for some λ ∈ C then alsop = 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, Theorem2.5] 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). Thisimplies 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 thatR((λI−A)p) is closed, by Proposition 3.10 part 2, λ is a pole of the resolvent and λ /∈ σgk(A)),thus σec(A)\σgk(A) is at most countable . Now if R((λI−A)p) is not closed for every p ∈ N,then R(((λI − A)/M)p) is not closed for every A-invariant closed subset M and p ∈ N, soλI − 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 Katospectrum σk(A),see [15].

3.5 Saphar operators, essentially Saphar operators andcorresponding spectra

In this section we study a class of operators studied by P. Saphar [100], as the "comple-mented" version of the semi-regular operators.

Definition 3.6 An operator A ∈ L(X) is called Saphar operator if A has a generalizedinverse and N(An) ⊆ R∞(A).A is called essentially Saphar operator if A is has a generalized inverse and N(An) ⊆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, inHilbert 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

andSe(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 0and 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

F (λ) =∞∑n=0

λnBn+1

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where B ∈ L(X) is a generalized inverse of A, and U = λ ∈ C : |λ| < ‖B‖−1. 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 inverseof (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 manyauthors, 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 aneighborhood 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:

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(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 (forinfinite 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 provedin [68], for σgk is proved in [16].

4. The stability of essentially semi-regular spectrum under commuting compact pertur-bation was shown in [98], and under not necessary commuting finite rank perturbationin [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 hencefor σk). For Banach spaces the theorem hold for every function f non constant on eachcomponent of its domain of definition (see[69]).

8. The boxes marked by "?" represent open problems.

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Chapter 4

Essential spectra defined by means ofrestrictions

4.1 Descent spectrum and essential descent spectrumLet A ∈ L(X), and consider the decreasing sequence cn(A) = dimR(An)/R(An+1), n ∈ N,see [62]. the esential descent of A is

de(A) = infn ∈ N : cn(A) <∞

(the infimum of an empty set is defined to be ∞). We say that A has finite essentialdescent if de(A) < ∞. Clearly, every lower semi-Fredholm operators has finite essentialdescent and we have de(A) = 0. This class of operators contain also every operator of finitedescent. 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 respectivelyby

σd(A) = λ ∈ C : d(λI − A) =∞;

σed(A) = λ ∈ C : de(λI − A) =∞;

We begin by the following result which shows that an operator with finite essential descentis 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 existsan open disc D(0, ε) and a positive integer d such that for all λ ∈ D(0, ε) \ 0, we have thefollowing 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 con-sequently X/R(A) is isomorph to R(An−1)/R(An). On other hand, it is well-known that

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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 = infn ∈ N : cn(A) =cp(A) for all p ≥ n. Clearly, de(A) ≤ d, and if d(A) < ∞ then we have d(A) = d. Wedenote by Ad the restriction of A ∈ L(X) on the subspace R(Ad). We define a new norm onR(Ad) by

|y| = ‖y‖+ inf‖x‖ : x ∈ X and y = Ax, for all y ∈ R(Ad)

It is easy to verify that R(Ad)) equipped with this norm is a Banach space, and that Ad isa bounded operator on (R(Ad), |.|). Hence it follows that Ad is both lower semi-Fredholmand semi-regular. In fact, Ad is lower semi-Fredholm because R(Ad) = R(Ad+1) is of finitecodimension in R(Ad). Moreover, since de(A) <∞ , [50, Theorem 3.1] ensures that

N(Ad) = N(A) ∩R(Ad) = N(A) ∩R(An+d) ⊆ R(An+d) = R(And) for all n ∈ N.

Let ε > 0 be such that for every 0 < |λ| < ε, λI −Ad both semi-Fredholm and semi-regular,it follows then that, since N(λI −A) = N(λI −Ad) ⊆ R((λI −Ad)k) ⊆ R((λI −A)k) for allk ∈ N, λI −A is also semi-regular. On the other hand, by Corollary 1.2 and Lemma 4.1 weobtain

β((λI − A)n) = β((λI − Ad)n) = nβ(Ad) = n dim(R(Ad)/R(Ad+1)

In prticular, λI − A is lower semi-Fredholm. For the last statement, we have

α((λI − A)n) = α(λI − Ad)n)

= ind((λI − Ad)n) + β((λI − Ad)n)

= 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 obtainthe desired result.

As a direct consequence of the preceding theorem and Lemma 1.4, the following resultshows that an operator with finite descent is either surjective or 0 is an isolated point of itssurjective spectrum.

Corollary 4.1 Let A ∈ L(X) be an operator with finite descent d = d(A), then there existsan open disc D(0, ε) for which λI − A is onto and α(λI − A) = dim(N(A) ∩ R(Ad)) for allλ ∈ D(0, ε) \ 0

Also as immediate consequence of Theorem 4.1 and Corollary 4.1, we have

Corollary 4.2 If A ∈ L(X), then σed(A) and σd(A) are a compact subset of C. Moreoverσd(A) \ σed(A) is an open set.

Proof. The first assertion follows directly from Theorem 4.1 and Corollary 4.1. For thesecond, let λ ∈ σd(A) \ σed(A) and d be as in the proof of Theorem 4.1. Then there exist apunctured open neighborhood V of λ such that V ⊆ ρed(A) and for all µ ∈ V and n ∈ N,

β((µI − A)n) = n dim(R(µI − A)d/R(µI − A)d+1)

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Since µI − A has infinite descent, dim(R(µI − A)d/R(µI − A)d+1) is non-zero, and conse-quently β((µI−A)n)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 ofσ(A), not identically constant on each connected component of its domain, then

σed(f(A)) = f(σed(A)) and σd(f(A)) = f(σd(A)).

It is clear that the descent spectrum, and therefore the essential descent spectrum, of anoperator 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

ρed(A) ∩ ∂σ(A) = ρd(A) ∩ ∂σ(A) = σdis(A).

Moreover, the following assertions are equivalent:

1. σd(A) = ∅,

2. σed(A) = ∅,

3. ∂σ(A) ⊆ ρd(A),

4. ∂σ(A) ⊆ ρed(A),

5. A is algebraic.

By [11, Theorem 2.9] and [11, Corollary 2.10], we have

Theorem 4.4 Let A ∈ L(X) and Ω be a connected component of ρed(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,

3. σed(A) is at most countable,

In this case, σed(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 ⊆σdis(A)) if and only if σd(A) ⊆ 0, if and only if σed(A) ⊆ 0.

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4.2 Ascent spectrum and essential ascent spectrumAssociated to an operator A on X we consider the non-increasing sequence

c′

n(A) = dimN(An+1)/N(An).

It follows from [62] that, for every n ∈ N,

c′

n(A) = dimN(A) ∩R(An).

The esential ascent of A is

ae(A) = infn ∈ N : c′

n(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 operatorscontain also every operator of finite ascent. Operators with finite essential ascent was studiedby many authors, for instance, we cite [12, 49, 50, 51, 87]. In [51], it was established that ifA ∈ L(X) has finite essential ascent then

R(An) is closed for some n > ae(A) if and only if R(An) 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) aredefined respectively by

ρa(A) = λ ∈ C : a(λI − A) <∞ and R(Aa(A)+1) is closed ;

ρea(A) = λ ∈ C : ae(λI − A) <∞ and R(Aae(A)+1) is closed;

The complementary sets σa(A) = C \ ρa(A) and σea(A) = C \ ρea(A) are the ascent spectrumand essential ascent spectrum of A, respectively. It is clear that σea(A) ⊆ σa(A) ⊆ σ(A).

The next theorem shows that an operator with finite essential ascent is either semi-regularor 0 is an isolated point of its semi-regular spectrum. The proof of this theorem requires thefollowing 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), andconsequently α(An) = α(An−1) +α(A). Thus, a successive repetition of this argument leadsto α(An) = nα(A).

Theorem 4.5 Let A ∈ L(X) be an operator with 0 ∈ ρea(A), then there exists an open discD(0, ε) and a positive integer d such that for all λ ∈ D(0, ε) \ 0, we have the followingassertions:

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)).

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Proof. If ae(A) < ∞ , we put d = infn ∈ N : c′n(A) = c

′p(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 ofA ∈ L(X) on the subspace R(Ad). Hence it follows that Ad is both upper semi-Fredholmand semi-regular. In fact, Ad is upper semi-Fredholm because N(Ad) has finite dimension,and R(Ad) = (R(A)+N(Ad))/N(Ad) = A−d(N(Ad+1))/N(Ad) is closed. On the other hand,we have

N(Ad) = N(A) ∩R(Ad) = N(A) ∩R(An+d) ⊆ R(An+d) = R(And) for all n ∈ N,

which proves that Ad is semi-regular. Hence there exists ε > 0 such that for every 0 < |λ| < ε,λI −Ad is both semi-regular and upper semi-Fredholm with α(λI −Ad) = α(Ad). For λ ∈ Cwith 0 < |λ| < ε, we have

N((λI − Ad)n) = N((λI − A)n)/N(Ad) = (N((λI − A)n)⊕N((Ad)))/N((Ad)) (4.2)

R(λI − Ad) = (R(λI − A) +N((Ad))/N((Ad) = R((λI − A))/N((Ad) (4.3)

Consequently, R(λI−A) is closed and contains the finite dimensional subspace N((λI−Ad)n)for all n ∈ N. This implies the first statement. For the second, by (4.2) and the previousLemma,

α((λI − A)n) = α(λI − Ad)n)

= 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)

= β((λI − Ad)n)

= α(λI − Ad)n)− ind((λI − Ad)n)

= n[α(Ad)− ind(Ad)] = nβ(Ad)

= n dimX/(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 anoperator with finite ascent is either bounded below or 0 is an isolated point of its approximatespectrum.

Corollary 4.3 Let A ∈ L(X) be an operator with 0 ∈ ρea(A), then there exists an open discD(0, ε) 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, ε) \ 0

Also as immediate consequence of Theorem 4.5 and Corollary 4.3, we have

Corollary 4.4 If A ∈ L(X), then σea(A) and σa(A) are a compact subset of C. Moreoverσa(A) \ σea(A) is an open set.

The spectral mapping theorem holds for the ascent and the essential ascent spectrum(see [87]):

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Theorem 4.6 Let A ∈ L(X) and f be an analytic function an an open neighborhood ofσ(A), not identically constant on each connected component pf its domain, then

σea(f(A)) = f(σea(A)) and σa(f(A)) = f(σa(A)).

The ascent spectrum and the essential ascent spectrum of an operator is not necessarilynon-empty, this occurs for algebraic operators.

Theorem 4.7 Let A ∈ L(X). Then

ρea(A) ∩ ∂σ(A) = ρa(A) ∩ ∂σ(A) = σdis(A).

Moreover, the following assertions are equivalent:

1. σa(A) = ∅,

2. σea(A) = ∅,

3. ∂σ(A) ⊆ ρa(A),

4. ∂σ(A) ⊆ ρea(A),

5. A is algebraic.

By [12, Theorem 2.9] and [12, Corollary 2.10], we have

Theorem 4.8 Let A ∈ L(X) and Ω be a connected component of ρea(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,

3. σea(A) is at most countable,

In this case, σea(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 ⊆σdis(A)) if and only if σa(A) ⊆ 0, if and only if σea(A) ⊆ 0.

4.3 Essential spectrum and Drazin invertible operatorsAn 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 isunique. The smallest k in the previous definition is called as the Drazin index of A anddenoted by i(A). For A ∈ L(X), if a(A) < ∞ and R(Aa(A)+1) is closed, then A is said tobe left Drazin invertible. If d(A) < ∞ and R(Ad(A)) is closed, then A is said to be rightDrazin invertible. If a(A) = d(A) < ∞, then A is said to be Drazin invertible. Clearly,

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A ∈ L(X) is both left and right Drazin invertible if and only if A is Drazin invertible. Ifae(A) <∞ and R(Aae(A)+1) 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 isleft 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, theDrazin spectrum, the left essentially Drazin spectrum, right essentially Drazin spectrum,essentially Drazin spectrum and semi-essentially Drazin spectrum the of A as follows, re-spectively:

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;

σeLD(A) = λ ∈ C : λI − A is not a left essentially Drazin invertible operator;σeRD(A) = λ ∈ C : λI − A is not a right essentially Drazin invertible operator,

σeD(A) = λ ∈ C : λI − A is not a essentially Drazin invertible operator,σeSD(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

σD(A) = σLD(A) ∪ σRD(A), σLD(A) = σa(A) ⊂ σap(A), σeLD(A) = σea(A) ⊂ σap(A).

andσeSD(A) = σeRD(A) ∩ σeLD(A), σeD(A) = σeRD(A) ∪ σeLD(A).

It is well know that A is Drazin invertible if and only if A is finite ascent and descent, whichis 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)

Let Ad the restriction of A ∈ L(X) on the subspace R(Ad). An immediate consequence ofTheorem 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, ε) and a positive integer d such that for all λ ∈ D(0, ε)\0,we have the following assertions:

1. λI − A is both semi-regular and semi-Fredholm operator,

2. β((λI − A)n) = nβ(Ad),

3. α((λI − A)n) = nα(Ad).

Theorem 4.11 Let A ∈ L(X) be an operator essentially Drazin invertible operator, thenthere exists an open disc D(0, ε) and a positive integer d such that for all λ ∈ D(0, ε) \ 0,we have the following assertions:

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1. λI − A is both semi-regular and Fredholm operator,

2. β((λI − A)n) = nβ(Ad),

3. α((λI − A)n) = nα(Ad),

4. ind((λI − A)n) = 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, ε) forwhich λI − A is invertible for all λ ∈ D(0, ε) \ 0

Recently, Koliha introduced the concept of a generalized Drazin inverse [65]. The gen-eralized Drazin inverse of A ∈ L(X) exists if and only if 0 is not accumulation point of thespectrum of A and is described as follows. If 0 is not accumulation point of the spectrum ofA, then the spectral projection of A at 0 is the unique idempotent P ∈ L(X) such that (seesection 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 )

=+∞∑n=0

λ−n−1AnP −+∞∑n=0

λn((A+ P )−1)n+1(I − P )

=+∞∑n=0

λ−n−1AnP −+∞∑n=0

λn(AD)n+1.

whereAD = (A+ P )−1(I − P )

is called the generalized Drazin inverse (g-Drazin inverse) of A. In this case A is calledg-Drazin invertible. In this case, we can define the Drazin index i(A) of A as follows

i(A) =

0 if A is invertible,k if AP is nilpotent of degree k ≥ 1,∞ if AP is quasi-nilpotent but not nilpotent,

Observe thatAAD = ADA = (A+ P )(A+ P )−1(I − P ) = (I − P ),

that isP = I − AAD.

A g-Drazin invertible A ∈ L(X) has a unique decomposition A = S+Q, where S ∈ L(X) isg-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 resolventof A. The order of this pole is equal to i(A). Particularly, it follows that 0 is not the point

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of accumulation of the spectrum of A. Furthermore, the comparison between the Browderresolvent defined in the subsection 2.3.1 and the the Drazin inverse gives

RB(λ,A) = (λI − A)D + 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 Drazininvertible with index m.

4.4 B-Fredholm, B-Browder and B-Weyl spectraIn 1958 Kato proved that a closed semi-Fredholm operator is of Kato type. In 1987 J.PLabrousse [70] studied and characterized a new calss of operators named quasi-Fredholmoperators, in the case of Hilbert spaces, and he proved that this class coincides with theset of Kato type operators. In 1999 M. Berkani [19] studied a class of bounded linearquasi-Fredholm operators acting on a Banach space X called B-Fredholm operators andcharacterized 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-Fredholmoperators and proved, in Hilbert spaces, that every semi-B-Fredholm is a direct sum of anilpotent operator and a Fredholm operator. Recently (2008) in [29] Berkani extended thischaracterization of B-Fredholm bounded operators to the class of B-Fredholm closed linearoperators acting on a Hilbert space and study its properties.

For each integer n, define An to be the restriction of A to R(An) viewed as the mapfrom R(An) into R(An) (in particular A0 = A). If there exists n ∈ N such that R(An)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. uppersemi-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 followsfrom [26, Proposition 2.1] that if there exists n ∈ N such that R(An) is closed and An issemi-Fredholm, then R(Am) is closed, Am is semi-Fredholm and ind(Am) = ind(An) for allm ≥ n. This enables us to define the index of a semi-B-Fredholm operator A as the indexof the semi-Fredholm operator An, where n is an integer satisfying R(An) is closed and Anis semi-Fredholm. An operator A ∈ L(X) is called B-Weyl (resp. upper semi-B-Weyl, lowersemi-B-Weyl) if A is B-Fredholm and ind(An) = 0 (resp. A is upper semi-B-Fredholm andind(An) ≤ 0, A is lower semi-B-Fredholm and ind(An) ≥ 0). If A ∈ L(X) is upper or lowersemi-B-Weyl, then A is called semi-B-Weyl.

The following theorem shows that every semi B-Fredholm operator is quasi-Fredholmoperator

Theorem 4.12 ([26]) Let A ∈ L(X). Then A is an upper semi B-Fredholm (resp. a lowersemi B-Fredholm) operator if and only if there exists an integer d ∈ N such that A is aquasi-Fredholm and such that N(A) ∩ R(Ad) is of finite dimension (resp. N(Ad) + R(A) isof finite codimension.

We give now a fundamental characterizations of semi B-Fredholm operators which provesthat admits a Kato decomposition.

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Theorem 4.13 ([19]) An operator A ∈ L(X) is B-Frdholm operator if and only if thereexists 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-Frdholmoperator if and only if A = A1 ⊕ A2 such that A1 is a semi-Fredholm operator and A2 is anilpotent 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 operatorand 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 uppersemi-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

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• 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 haveσbf (A) = σubf (A) ∪ σlbf (A)

σbw(A)) = σubw(A) ∪ σlbw(A)

andσqf (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 elegantequalities:

σLD(A) = σubb(A), σRD(A) = σlbb(A);

σeLD(A) = σubf (A), σeRD(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) =⋂

F∈F (X)

σD(A+ F )

The following essential spectrum named the topological uniform descent spectrum, as weknow, 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 suchthat 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 includesall classes of operators introduced above. It also includes many other classes of operatorssuch as operators of Kato type, quasi-Fredholm operators, operators with finite descentand operators with finite essential descent, and so on. Especially, operators which havetopological uniform descent for n ≥ 0 are precisely the semi-regular operators. Discussionsof 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 operatorof 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 followinginclusion.

σud(A) ⊆ σsbf (A) ⊆ σsbw(A)σubf (A)=σeLD(A)⊆ σubw(A) ⊆ σbw(A)

σubb(A)=σLD(A)⊆ σbb(A) = σD(A),

σud(A) ⊆ σsbf (A) ⊆ σsbw(A)σlbf (A)=σeRD(A)⊆ σlbw(A) ⊆ σbw(A)

σlbb(A)=σRD(A)⊆ σbb(A) = σD(A),

σud(A) ⊆ σsbb(A) ⊆ σbb(A) = σD(A)

andσud(A) ⊆ σbf (A) ⊆ σbb(A) = σD(A).

The properties (P1)-(P7) for these sets of essential spectra defined above are summarizedin 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

σed(A) no yes no yes no no yes

σa(A) no yes no no no no yes

σea(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−Fwould be a B-Fredholm operator. Hence the class of B-Fredholm operators is not stableunder compact perturbation.

3. We have σa(0) = σed(0) = ∅. Since every operator commutes with 0, σa and σed cannothave properties (P1), (P3), (P5),(P6).

4. If A ∈ L(X) is the zero operator, then σ(A) = 0 = σ(A + Q), σk(A) = ∅ andσk(A) 6= 0 = σk(A+Q), k ∈ ubb, lbb, bb. Thus the invariance under perturbation bycommuting 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), ifN is a nilpotent operator with AN = NA and if either the Banach space is a Hilbertspace or the adjoint operator has the SVEP (single-valued extension property), thenthe 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 theoryLet A ∈ L(X). We say that A has the single-valued extension property at λ0 ∈ C, abbrevi-ated A has the SVEP at λ0, if for every neighborhood Uλ0 of λ0 the only analytic functionf : 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 aGKD(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 followingassertions 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 followingassertions 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)

andσeb(A) = σew(A) ∪ Ξ(A) = σew(A) ∪ Ξ(A∗) (4.5)

Note thatΞ(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) ). Allresults established above have a numerous of interesting applications. In the next theoremwe 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)

andσec(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∗ hasthe SVEP at λ ∈ ∂σ(A), then λI −A does not admit a generalized Kato decomposition andthus λ ∈ σgk(A). So

σsu(A) = ∂σ(A) ⊆ σgk(A) ⊆ σk(A) ⊆ σes(A) ⊆ σse(A) ⊆ σap(A)

andσgk(A) = σk(A) = σes(A) = σse(A) = σsu(A).

Thus we haveσec(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 andthe left B-Fredholm spectrum and by duality we find a similar result holds for the rightDrazin spectrum and the right B-Fredholm spectrum.

Theorem 4.22 ([6]) Let A ∈ L(X). Then

σLD(A) = σlbf (A) ∪ Ξ(A) (4.6)

andσRD(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)

andσlbb(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 thenσqf (A) = σubw(A) = σubb(A) (4.11)

andσbw(A) = σbb(A) = σlbb(A) = σlbw(A) (4.12)

3. If A∗ has the SVEP then

σqf (A) = σlbw(A) = σlbb(A) (4.13)

andσbw(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); anddually 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 everyλ ∈ ∂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 everyλ ∈ ∂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

(Cpf)(λ) =1

λ

∫ λ

0

f(λ)

1− λdµ, for all f ∈ Hp(D) and λ ∈ D.

The spectrum of the operator Cp is the closed disc Γp centred at p2with radius p

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 theoremH.Weyl examined the spectra of all compact perturbations of a hermitian operator on Hilbertspace and found in 1909 that their intersection consisted precisely of those points of thespectrum which were not isolated eigenvalues of finite multiplicity. This Weyl’s theorem hassince been extended to hyponormal and to Toeplitz operators [36], to seminormal and otheroperators [17] and recently to Banach spaces operators berkani6. Extended and variantsof this theorem have been discussed in [4, 25, 27, 41, 96]. In the following, we use theabbreviations 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 withrespect to Browder’s theorem or the properties. The following table summarizes the meaningof various theorems.

gaW σap(A)\σubw(A) = Ea(A) gaB σap(A)\σubw(A) = Πa(T )aW σap(A)\σuw(T ) = E0

a(T ) aB σa(A)\σuw(T ) = Π0a(T )

gW σ(A)\σbw(A) = E(A) gB σ(A)\σbw(A) = Π(A)W σ(A)\σew(T ) = E0(A) B σ(A)\σew(A) = Π0(A)(gw) σap(A)\σubw(A) = E(A) (gb) σap(A)\σubw(T ) = Π(A)(w) σap(A)\σuw(A) = E0(A) (b) σap(A)\σuw(A) = Π0(A)(gaw) σ(A)\σbw(A) = Ea(A) (gab) σ(A)\σbw(A) = Πa(A)(aw) σ(A)\σew(A) = E0

a(A) (ab) σ(A)\σew(A) = Π0a(A)

Where,

• E(A) = λ ∈ isoσ(A) : 0 < α(λI−A), the set of all eigenvalues of A isolated in σ(A).

• E0(A) = λ ∈ E(A) : α(λI − A) < ∞ = σdis(A), the set of all eigenvalues of A offinite multiplicity isolated in σ(A).

• Ea(A) = λ ∈ isoσap(A) : 0 < α(λI − A), the set of all eigenvalues of A isolated inσap(A).

• E0a(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 theresolvent of A.

• Π0(A) = λ ∈ Π(A) : α(λI − A) < ∞, the the set of the poles of the resolvent of Aof finite rank.

• Πa(A) = λ ∈ σap(A) : 0 < max(a(λI − A), d(λI − A)) <∞, the set of the left polesof the resolvent of A.

• Π0a(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 arereferences to the results to the bibliography therein.

103

gB (aw)[28]⇐= (gaw) gaB gW

⇑ [25] ⇑ [28] [27] ⇑ [5] ⇑

gaB[25]⇐= gaW

[25]=⇒ gW

[25]=⇒ gB

[28]⇐= (gab)[28]⇐= (gb)

[27]⇐= (gw)m [7] ⇓ [25] ⇓ [25] m [7] ⇓ [28] [27] ⇓ [5] ⇓

aB[25]⇐= aW

[96]=⇒ W

[10]=⇒ B

[28]⇐= (ab)[28]⇐= (b)

[27]⇐= (w)⇓ [41] ⇑ [28] [27] ⇓ [4] ⇓B (aw) aB W

Weyl-Browder type theorems and the various proprieties, in their and more recently intheir generalized form, have been studied by a large of authors and becomes a significantsector of the development of spectral theory. Two important studies in this important sectorare

• 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 thisproprieties defined in the table above, see [1, 17, 23, 25, 36, 96].

104

Chapter 5

Applications

5.1 One-dimensional transport equationIn this section, we shall apply the results of perturbations to the one-dimensional transport

equation on Lp-spaces, with p ∈ [1, ∞).Let

Xp = Lp((−a, a)× (−1, 1), dx dξ), a > 0 and p ∈ [1,∞).

We consider the boundary spaces :

Xop := Lp[−a × (−1, 0), |ξ|dξ]× Lp[a × (0, 1), |ξ|dξ] := Xo

1,p ×Xo2,p

and

X ip := Lp[−a × (0, 1), |ξ|dξ]× Lp[a × (−1, 0)], |ξ|dξ] := X i

1,p ×X i2,p

respectively equipped with the norms

‖ψo‖Xop

=(‖ψo1‖

pXo

1,p+ ‖ψo2‖

pXo

2,p

) 1p

=

[∫ 0

−1

|ψ(−a, ξ)|p|ξ| dξ +

∫ 1

0

|ψ(a, ξ)|p|ξ| dξ] 1p

and

‖ψi‖Xip

=(‖ψi1‖

p

Xi1,p

+ ‖ψi2‖p

Xi2,p

) 1p

=

[∫ 1

0

|ψ(−a, ξ)|p|ξ| dξ +

∫ 0

−1

|ψ(a, ξ)|p|ξ| dξ] 1p

.

Let Wp the space defined by:

Wp =

ψ ∈ Xp : ξ

∂ψ

∂x∈ Xp

.

It is well-known that any function ψ in Wp has traces on −a and a in Xop and X i

p. Theyare denoted, respectively by ψo and ψi, and represent the outgoing and the incoming fluxes,with

ψo =

(ψo1ψo2

)and ψi =

(ψi1ψi2

),

given by

ψo1(ξ) = ψ(−a, ξ), ξ ∈ (−1, 0),

ψo2(ξ) = ψ(a, ξ), ξ ∈ (0, 1),

ψi1(ξ) = ψ(−a, ξ), ξ ∈ (0, 1),

ψi2(ξ) = ψ(a, ξ), ξ ∈ (−1, 0).

105

We define the operator TH by:

TH : D(TH) ⊆ Xp −→ Xp

ψ −→ THψ(x, ξ) = −ξ ∂ψ∂x

(x, ξ)− σ(ξ)ψ(x, ξ)

D(TH) =ψ ∈ Wp such thatψi = Hψo

Where σ(.) ∈ L∞(−1, 1) and H is bounded from Xo

p to X ip.

The function ψ(x, ξ) represents the number density of gas particles having the position xand the direction cosine of propagation ξ. The variable ξ may be thought of as the cosine ofthe angle between the velocity of particles and the x-direction. The function σ(.), is calledthe 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|ξ|→0

σ(ξ) ,(for more detail see

[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). ForReλ+ λ∗ > 0, the solution of (5.3) is formally given by:

ψ(x, ξ) =

ψ(−a, ξ) e−

(λ+σ(ξ))|a+x||ξ| + 1

|ξ|

∫ x

−ae−(λ+σ(ξ))|x−x

′|

|ξ| ϕ(x′, ξ) dx′ if 0 < ξ < 1,

ψ(a, ξ) e−(λ+σ(ξ))|a−x|

|ξ| +1

|ξ|

∫ a

x

e−(λ+σ(ξ))|x−x

′|

|ξ| ϕ(x′, ξ) dx′ if − 1 < ξ < 0.

where

ψ(a, ξ) = ψ(−a, ξ) e−2a(λ+σ(ξ))

|ξ| + 1|ξ|

∫ a

−ae−−(λ+σ(ξ))|a−x|

|ξ| ϕ(x, ξ) dx

ψ(−a, ξ) = ψ(a, ξ) e−2a(λ+σ(ξ))|ξ| + 1

|ξ|

∫ a

−ae−(λ+σ(ξ))|a+x|

|ξ| ϕ(x, ξ) dx

In the sequel we shall consider the following operators:

106

Mλ : X ip −→ Xo

p ,Mλu := (M+λ u,M

−λ u) where

M+λ u(−a, ξ) = u(−a, ξ) e

−2a|ξ| (λ+σ(ξ)) if 0 < ξ < 1

M−λ u(a, ξ) = u(a, ξ) e

−2a|ξ| (λ+σ(ξ)) if 0 < ξ < 1

Bλ : X ip −→ Xp;Bλu = χ(−1,0)(ξ)B

−λ u+ χ(0,1)(ξ)B

+λ u where

(B+λ u)(−a, ξ) = u(−a, ξ) e

−(λ+σ(ξ))|ξ| |a+x|, if 0 < ξ < 1

(B−λ u)(−a, ξ) = u(−a, ξ) e−(λ+σ(ξ))|ξ| |a−x|, if − 1 < ξ < 0

Gλ : Xp −→ Xop , Gλϕ := (G+

λϕ,G−λϕ) where

G+λϕ =

1

|ξ|

∫ a

−ae−(λ+σ(ξ))|ξ| |a−x|ϕ(x, ξ) dx, if 0 < ξ < 1

G−λϕ =1

|ξ|

∫ a

−ae−(λ+σ(ξ))|ξ| |a+x|ϕ(x, ξ) dx, if − 1 < ξ < 0

Cλ : Xp −→ Xp;Cλϕ = χ(−1,0)(ξ)C+λ ϕ+ χ(0,1)(ξ)C

−λ ϕ where

C−λ ϕ =1

|ξ|

∫ x

−ae−(λ+σ(ξ))|ξ| |x′−x|ϕ(x′, ξ) dx′ if 0 < ξ < 1

C+λ ϕ =

1

|ξ|

∫ a

x

e−(λ+σ(ξ))|ξ| |x′−x|ϕ(x′, ξ) dx′ if − 1 < ξ < 0

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 respectivedomains respectively, by e−2a(Reλ+λ∗), [p(Reλ + λ∗)]

−1p , [(Reλ + λ∗)]

−1q and [(Reλ + λ∗)]−1

where q denotes the conjugate of p. We define the real λ0 by

λ0 =

−λ∗, if ||H|| ≤ 1

12a

log ||H|| − λ∗ if ||H|| > 1

It follows from the norm estimate of Mλ that, for Reλ > λ0, ||MλH|| < 1 and conse-quently

ψ0 =+∞∑n=0

(MλH)nGλϕ (5.4)

On the other hand, we have

ψ = BλHψ0 + Cλϕ

= (BλH+∞∑n=0

(MλH)nGλ + Cλ)ϕ

107

Hence, λ ∈ C such that Reλ > λ0 ⊂ ρ(TH) and for Reλ > λ0

(λ− TH)−1 = BλH(I −MλH)−1Gλ + Cλ (5.5)

On the other hand, observe that the operator Cλ is nothing else but (λ− T0)−1 . Obviously,if the operator I − MλH is boundedly invertible , (for example, if Reλ > λ0), then λ ∈ρ(TH) ∩ ρ(T0) and

(λ− TH)−1 − (λ− T0)−1 = Dλ, (5.6)

where

Dλ = BλH+∞∑n=0

(MλH)nGλ

5.1.1 Application of the Fredholm perturbations to transport equa-tions

Next we consider the transport operator

AH = TH +K

where K is the collision operator given by

K : Xp −→ Xp

ψ −→∫ 1

−1

κ(x, ξ, ξ′)ψ(x, ξ′) dξ′

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 intro-duced 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∫ 1

−1

|(Kψ)(x, ξ)|p dξ ≤ ‖K(.)‖pL∞(]−a,a[,Z)

∫ 1

−1

|ψ(x, ξ)|p dξ

This leas to the estimate

‖K‖Xp ≤ ‖K(.)‖L∞(]−a,a[,Z). (5.11)

108

4. The assumptions (5.8) and (5.9) mean that K(x, ξ, ξ′) is the kernel compact operator

on L∞(]−1, 1[ ,Z) (for x fixed in [−a, a]) and that this family of operators, indexedby x ∈ [−a, a] , lies in compact subset of Z.

Proposition 5.1 ([75, Proposition 3.2]) Assume that K is a regular operator, then forany λ ∈ C such that Reλ ≤ −λ∗, the operator (λ−TH)−1K 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

κn(x, ξ, ξ′) =n∑i=1

αi(x)fi(ξ)gi(ξ′)

where αi(.) ∈ L∞(]−a, a[), fi(.) ∈ Lp(]−1, 1[) and gi(.) ∈ Lq(]−1, 1[) (q denote the conjugateof 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)

Proof. Let λ be such that r[(λ− TH)−1K] < 1 then λ ∈ ρ(AH) and

(λ− AH)−1 − (λ− TH)−1 =+∞∑n=1

[(λ− TH)−1K]n(λ− TH)−1 (5.14)

By Proposition 5.1 (λ−AH)−1−(λ−TH)−1 is compact (resp. weakly compact) onXp, 1 < p <∞ (resp. X1). Threfore, it follows from the inclusionK(Xp) ⊆ S(Xp) ⊆ PΦ+(Xp) ⊆ PΦ(Xp)for 1 < p <∞ and the fact that the set of weakly compact operators consides with the set ofstrictly singular on L1 spaces that (λ−AH)−1−(λ−TH)−1 ∈ 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

(λ− AH)−1 − (λ− T0)−1 = Dλ ++∞∑n=1

[(λ− TH)−1K]n(λ− TH)−1

Therefore the strict singularity of H implies that of (λ − AH)−1 − (λ − T0)−1. Now, theequality (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 transportequations

Next we consider the transport operator

AH = TH +K

where K is the bounded operator given byK : Xp −→ Xp

ψ −→∫ ξ

−1

κ(x, ξ, ξ′)ψ(x, ξ′) dξ′

and κ satisfies the following assumptions:

(H)

κ(., ., .) is a measurable function form [−a, a]× [−1, 1]× [−1, 1] to R and|κ(x, ξ, ξ′)| ≤ c <∞, a.e.

Lemma 5.2 If κ satisfies (H) then, for any integer n ≥ 1

‖Kn‖ ≤ 21q

+n+1

n!cn

where 1p

+ 1q

= 1.

Proof. Let ψ ∈ Xp. Holder’s inequalities implies that

|Kψ(x, ξ)| =

∣∣∣∣∫ ξ

−1

κ(x, ξ, ξ′)ψ(x, ξ′)dξ′∣∣∣∣

≤(∫ ξ

−1

|κ(x, ξ, ξ′)|q dξ′) 1

q(∫ 1

−1

|ψ(x, ξ′)|p dξ′) 1

p

≤ c(ξ + 1)1q

(∫ 1

−1

|ψ(x, ξ′)|p dξ′) 1

p

and ∣∣K2ψ(x, ξ)∣∣ =

∣∣∣∣∫ ξ

−1

κ(x, ξ, ξ′)Kψ(x, ξ′)dξ′∣∣∣∣

≤ c2

∫ ξ

−1

(ξ′ + 1)1q dξ′

(∫ 1

−1

|ψ(x, ξ′)|p dξ′) 1

p

≤ c2 11q

+ 1(ξ + 1)

1q

+1

(∫ 1

−1

|ψ(x, ξ′)|p dξ′) 1

p

we proceed by induction to obtain

|Knψ(x, ξ)| ≤ cn1

(1q

+ 1)(1q

+ 2) . . . (1q

+ n)(ξ + 1)

1q

+n

(∫ 1

−1

|ψ(x, ξ′)|p dξ′) 1

p

then, by Fubini’s theorem we deduce∫ a

−a

∫ 1

−1

∣∣∣∣∫ ξ

−1

κ(x, ξ, ξ′)ψ(x, ξ′)dξ′∣∣∣∣p dξ′dx ≤ 2cn

1

n!(ξ + 1)

1q

+n ‖ψ‖pXp

this shows the result.

110

Theorem 5.2 Let p ∈ [1,∞[ and suppose that the collision operator satisfies (H) on Xp

and KTH − THK ∈ 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 followsas in the proof of Theorem 5.1.

5.2 Singular transport operatorsIn this section we are concerned with the essential spectra of singular transport operators

Aψ(x, ξ) = −ξ ·∇xψ(x, ξ)−σ(ξ)ψ(x, ξ) +

∫Rnκ(x, ξ′)ψ(x, ξ′) dξ′ (x, ξ) ∈ Ω×V, (5.17a)

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 measuredµ on Rn and ψ ∈ Lp(Ω × V, dxdµ(ξ)) (1 ≤ p < ∞). In (5.17b) Γ− denotes the incomingpart 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 thetransport of particle (neutrons, photons, molecules of gas, etc.) in the domain Ω. Thefunction ψ represents the number (or probability) density of particles having the positionx and the velocity ξ. The functions σ(.) and κ(., .) are called, respectively, the collisionfrequency and the scattering kernel. Let us first introduce the functional setting we shalluse in the sequel. Let

Xp = Lp(Ω× V, dxdµ(ξ)),

Xσp = Lp(Ω× V, σ(ξ)dxdµ(ξ)) 1 ≤ p <∞,

Lpσ(Rn) = Lp(Rn, 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−,

Wp = ψ ∈ Wp ; ψ|Γ− = 0.

The streaming operator T associated with the boundary condition (5.17b) isT : D(T ) ⊂ Xp → Xp

ψ 7→ Tψ(x, ξ) := −ξ · ∇xψ(x, ξ)− σ(ξ)ψ(x, ξ),

111

with domainD(T ) := Wp ∩Xσ

p .

The transport operator (5.17) can be formulated as follows A = T + K, where K denotesthe following collision operator K : Xp → Xp

ψ 7→∫Rnκ(x, ξ′)ψ(x, ξ′) dξ′

We will assume that the scattering kernel κ(., .) is non-negative and there exists a closedsubset E ⊂ Rn with zero dµ measure and a constant σ0 > 0 such that

σ(.) ∈ L∞loc(Rn \ E), σ(ξ) > σ0; (5.18a)[∫Rn

(κ(., ξ′)

σ(ξ′)1p

)dµ(ξ′)q

] 1q

∈ Lp(Rn). (5.18b)

Using boundedness of Ω and the assumption (5.18b) we can fined that K ∈ L(Xσp , Xp) with

‖K‖L(Xσp ,Xp) ≤

∥∥∥∥∥∥[∫

Rn(κ(., ξ′)

σ(ξ′)1p

)qdµ(ξ′)

] 1q

∥∥∥∥∥∥Lp(Rn

(5.19)

Note that a simple calculation using the assumption (5.18a) shows that Xσp is a subset of

Xp and the the embedding Xσp → 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

ψ(x, ξ) =

∫ t(x,ξ)

0

e−(λ+σ(ξ))sϕ(x− sξ, ξ) ds (5.21)

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 ∫

Ω

∣∣(λI − T )−1ψ(x, ξ)∣∣p dx ≤ 1

(Reλ+ σ(ξ))p

∫Ω

|ψ(x, ξ)|p dx

Therefore, ∥∥(λI − T )−1ψ∥∥Xσp≤ sup

ξ∈Rn

σ(ξ)

(Reλ+ σ(ξ))p‖ψ‖Xp (5.22)

Hence, (λI −T )−1 ∈ L(Xp, Xσp ). 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.,for each, e ∈ Sn−1, dµξ ∈ Rn, ξ.e = 0 = 0,

(5.25)

where Sn−1 denotes the unit ball of Rn and the collision operator K : Lpσ(Rn) −→ Lp(Rn) iscompact. Then for any λ ∈ C such that Reλ > −σ0, the operator K(λI − T )−1 is compacton 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,

∥∥(λI − T )−1∥∥L(Xp,Xσ

p )≤ sup

ξ∈Rn

σ(ξ)

(Reλ+ σ(ξ))p.

So, since Xσp is continuously embedded in Xp, we infer that limλ→+∞ ‖K(λI − T )−1‖L(Xp) =

0. Therefore, there exists λ ∈ ρ(T ) such that r(K(λI − T )−1) < 1. For such λ we have

(λI − A)−1 − (λI − T )−1 =+∞∑n=1

(λ− T )−1[K(λI − T )−1]n (5.27)

By using Proposition 5.2, (λI−A)−1−(λI−T )−1 is compact onXp, 1 < p <∞. Therefore,it follows from the inclusion K(Xp) ⊆ S(Xp) = PΦ+(Xp) = PΦ(Xp) for 1 < p <∞. Henceby 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; itsaim is to give a survey of various characteristic stability properties of different notions ofessential spectrum under perturbations belonging to a large of class operators ( Fredholmperturbations, compact, quasi-nilpotent, . . . ), as well as some equivalent descriptions of thesespectra, and many cases when these essential spectra coincide or differ from each other onat most countably many isolated points. The results obtained are used for describing theessential spectra of some transport operators.

By the preceding chapters, some questions do, however, remain open to further investi-gation. This is not intended to be a complete survey of such problems but a sample of thosethat 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 underadditive 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) wasproved in [24] .

2. In view of chapter 5, an interest research is the use of essential spectra and theirperturbations in the spectral analysis of the Cauchy abstract problem and associatedperturbed Cauchy abstract problem, to derive sufficient or necessary (or both ) con-ditions for well-posedness, exponential stability and norm continuity of the solutions.As an applications of these topics: transport equations, infinite linear systems, controltheory of infinite dimensional linear systems.

114

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The main thrust of this thesis is in the spirit of the spectraltheory and operator theory; its aim is to give a survey of variouscharacteristic stability properties of different notions of essentialspectrum under perturbations belonging to a large of class operators( Fredholm perturbations, compact, quasi-nilpotent, . . .), as well assome equivalent descriptions of these spectra, and many cases whenthese essential spectra coincide or differ from each other on at mostcountably many isolated points. The results obtained are used fordescribing the essential spectra of some transport operators.

Conclusion