Image d'une dérivation généralisée et opérateurs D-symétriques

Faculté des Sciences, 4 Avenue Ibn Battouta B.P. 1014 RP, Rabat – Maroc Tel +212 (0) 37 77 18 34/35/38, Fax : +212 (0) 37 77 42 61, http://www.fsr.ac.ma

N° d’ordre :2473

TH_ÉSE DE DOCTORAT

Présentée par

MOHAMED ECH-CHAD

Discipline : Mathématiques. Spécialité : Analyse Fonctionnelle.

Image d’une Dérivation Généralisée et

Opérateurs D-symétriques.

Soutenue le Vendredi 22 Janvier 2010

Devant le Jury :

Président :

EL KHADIRI ABDELHAFED, P.E.S., ( FS, Kénitra ).

Examinateurs :

M. BOUALI SAID, P.E.S., ( FS, Kénitra ).

M. ZEROUALI EL HASSAN, P.E.S., ( FS, Rabat ). M. BOUSSEJRA ABDELHAMID, P.E.S., ( FS, Kénitra ). M. KHAOULANI BOUCHTA, P.E.S., ( ENIM, Rabat ). M. INTISSAR AHMED, P.E.S., ( FS, Rabat ).

M. BENLARBI DELAI M’HAMMED, P.E.S., ( FS, Rabat ). M. EL FELLAH OMAR., P.E.S., ( FS, Rabat ).

Ce travail a été réalisé au sien de l’unité de formation doctorale "Théorie des opérateurs et Théorie des fonctions" siégeant au Département de Mathématiques et informatique de la faculté des sciences de Rabat.

Je voudrais exprimer mes sincères remerciements à Monsieur Said Bouali, professeur à la faculté des sciences de Kénitra, pour m’avoir assuré la direction de ce travail, et pour m’avoir apporté la rigueur scientifique nécessaire à son bon déroulement. Je tiens également à le remercier pour sa gentillesse, sa patience, sa grande disponibilité et ses encouragements tout au long de ce travail, et je le prie de croire en ma profonde reconnaissance.

Que Monsieur El Hassan Zerouali, professeur à la faculté des sciences de Rabat, trouve ici le témoignage de ma profonde reconnaissance et je le remercie vivement d’avoir examiné ce travail et me faire partager son grand intérêt pour la recherche.

Je tiens à remercier chaleureusement Monsieur le professeur Abdelhamid Boussejra de la Faculté des sciences de Kénitra d’avoir accepté de rapporter les résultas de ma thèse, sa présence dans ce jury me fait un grand honneur. Je souhaite aussi le remercier pour son soutien et pour sa relecture très attentive du manuscrit.

J’ exprime mes sincères remerciements à Monsieur M’Hammed Benlarbi Delai professeur à la faculté des sciences de Rabat, pour le temps qu’il a consacré à examiner ce travail, et pour l’honneur qu’il me fait en acceptant de participer à ce jury.

Je tiens à remercier vivement Monsieur Omar El Fellah, professeur à la faculté des sciences de Rabat, pour sa participation au jury de thèse, pour sa lecture approfondie de la thèse et ses précieuses remarques.

remercier pour la qualité de ses suggestions et de son écoute. Qu’il trouve ici l’expression de ma gratitude.

Je remercie vivement Monsieur Khaoulani Bouchta, professeur à l’école nationale de l’industrie minérale, pour l’intérêt qu’il a manifesté à l’égard de ce travail. Il a ma reconnaissance d’avoir accepté de rapporter les résultats de ma thèse. Je lui suis redevable pour son aide, son soutien et ses remarques fructueuses sur le manuscrit.

Mes remerciements s’adressent aussi à Monsieur Abdelhafed El Khadiri professeur à la faculté des sciences de Kénitra, pour l’apport scientifique qu’il a accordé a ce travail, pour ses remarques, ses conseils et pour avoir participé au jury de ma thèse en tant que président.

Je remercie tous ceux qui m’ont accompagné au cours de ses années, et ceux qui m’ont aidé d’une manière ou d’une autre, à mener ce travail à terme.

J’adresse aussi mes remerciements à tous mes amis qui n’ont jamais cessé de me soutenir et de m’encourager.

Enfin, j’aurai une pensée particulière pour ma famille pour son soutien et les encouragements dont elle m’a fait bénéficier pendant cette période.

La synthèse de divers travaux sur l’image d’une dérivation, les opérateurs D-symétriques, et l’image numérique d’un opérateur fait l'objet du premier chapitre.

Au second chapitre, on a donné une extension du résultat principal de Weber pour une dérivation généralisée. On a obtenu une condition suffisante pour que

( ) ( )

(

f A f B

)

(

A B

)

R

δ

=

R

δ

. On déduit l'orthogonalité de l'image au noyau de la

dérivation _/

p AB C

δ

si

(

f A f B

( ) ( )

,

)

admet la propriété de Fuglede- Putnam dans

p

C

pour

p >

1

.

Au troisième chapitre on considère Les paires d'opérateurs

(

A B

,

)

telles que

(

A B

)

R

δ

est auto-adjoint, on a appelé ces paires D-symétriques. On a donné quelques propriétés de base concernant cette classe.

Au chapitre suivant on s'intéresse à l'étude de la classe des paires d’opérateurs D*-symétriques,

(

A B

,

)

est D*-symétrique si

R

(

δ

_{A B}

)

=

R

(

δ

_{A B}* *

)

. On a prouvé que :

si

A

et

B

sont deux opérateurs D-symétriques de spectres disjoints, alors

(

A B

,

)

est D*-symétrique. On a tenu à démontrer des caractérisations de cette classe. On déduit qu’elle contient les paires d’opérateurs normaux et les paires d’isométries.

Au dernier chapitre on a initie l'étude sur l’image numérique généralisée

( )

{

,

,

1

}

g

Sommaire

Introduction générale……….3

Références………...6

0. Terminologies………8

1. Preliminaries………10

1.1. The range of a derivation and Orthogonality……….10

1.2. D-symmetric operators………...11

1.3. Numerical range……….13

References……….13

2.

Analytic Functions, Derivations and Orthogonality………15

2.1. Introduction………15

2.2. Analytic Functions and Derivation Ranges………...16

2.3. Range-kernel Orthogonality………...19

References……….20

3.

Generalized D-symmetric Operators I……….22

3.1. Introduction………22

3.2. Properties of D-symmetric Pairs………….………...23

3.3. Properties and Descriptions of

C A B

(

,

)

_and

I A B

(

_,

)

_{……….……...25}

4. Generalized D-symmetric Operators II………28

4.1. Introduction………28

4.2. D*-symmetric Pairs………...29

References……….33

5. Generalized Numerical Range……….35

5.1. Introduction………35

5.2. Properties of Generalized Numerical Range………..36

5.3. Generalized Numerical Range of Compact Operators………...38

5.4. Generalized Numerical Range of Derivation……….39

References……….………...40

Introduction générale

Une dérivation sur une algèbre A est un endomorphisme δ de A vérifiant ; δ(XY ) = δ(X)Y + Xδ(Y )

pour tout (X, Y ) ∈ (L(H))2. Dans le cas A = L(H) ; où H est un espace de Hilbert complexe séparable de dimension infinie, et L(H) l’algèbre des opérateurs linéaires bor-nés sur H, on sait que toute dérivation est une dérivation intérieure (voir [19; 20; 22] ), c’est à dire, de la forme δAavec

δA(X) = AX − XA A, X ∈ L(H).

Pour A et B ∈ L(H), nous définissons la dérivation généralisée δAB sur L(H) comme

suit :

δAB(X) = AX − XB

pour tout X ∈ L(H). Les propriétés de ces opérateurs, leur spectre (voir [9; 11; 12 et 13]), norme [25], et image (voir [2; 10; 14; 15; 18; 21; 24 et 29]) ont été examinés minutieusement ces dernières années, et plusieurs problèmes restent encore sans réponses (voir [28 et 29]).

Le contenu de cette thèse est composé de cinq chapitres. Dans le chapitre un, on rappelle des résultats sur l’image d’une dérivation, les opérateurs D-symétriques, et l’image numé-rique d’un opérateur. Il est prouvé par Weber [26] que ; si f est une fonction analytique au voisinage du spectre de A, alors R(δf (A)) ⊂ R(δA). Au second chapitre, nous donnons

une extension de ce résultat pour une dérivation généralisée. Nous déduisons une condi-tion suffisante pour que R(δf (A)f (B)) = R(δAB).

Dans la deuxième partie, nous généralisons le résultat principal de [4], concernant l’or-thogonalité de R(δAB/Cp) au ker(δAB/Cp) au sens de la définition 1.2 [6]. Nous montrons

que si f est une fonction bianalytique sur un ouvert qui contient σ(A) ∪ σ(B) tel que (f (A), f (B)) admet la propriété de (F, P )Cp, 1 ≤ p < ∞, alors pour tout X ∈ L(H), et

pour tout T ∈ ker(δAB/Cp),

kδAB(X) + T kp ≥ kT kp.

Nous généralisons le résultat principal de Weber [27] en montrons que si B ∈ R(δA) w

∩ {A}0

( R(δA) w

un entier n ≥ 1 tel que Bnest compact, alors B est quasi-nilpotent.

Les opérateurs D-symétriques ( A est D-symétrique si R(δA) = R(δA∗)) ont été étudier

par J. H. Anderson, J. W. Bunce, J. A. Deddens and J. P. Williams [1], S. Bouali and J. Charles [7][8] and J. G. Stampfli [24].

Au troisième chapitre nous considérons Les paires d’opérateurs (A, B) telles que R(δAB)

est auto-adjoint, nous appellerons ces paires d’opérateurs D-symétriques. Nous donnerons quelques propriétés de base concernant cette classe.

Une extension du théorème 3 de J. P. Williams [29] nous a permis de caractériser les paires d’opérateurs D-symétriques. En conséquence, nous obtenons que : s’il existe λ ∈ C tel que (B − λ)(A − λ) = (A − λ)2 = 0, (A − λ) 6= 0 et (B − λ) 6= 0, alors la paire (A, B) n’est pas symétrique. On en déduit que l’ensemble des paires d’opérateurs D-symétriques n’est pas fermé en norme dans (L(H))2_.

Dans la section 2 nous introduisons les ensembles suivants :

C(A, B) = {C ∈ L(H), CL(H) + L(H)C ⊂ R(δAB)}

et

I(A, B) = {Z ∈ L(H), ZR(δAB) + R(δAB)Z ⊂ R(δAB)},

qui généralisent ceux introduits par J. P. Williams dans [28]. Nous donnons des proprié-tés et une description de ces ensembles. Nous obtenons C(A, B) = {0} et I(A, B) = {A}0 _{∩ {B}}0_{, si R(δ}

AB) ne contient aucun opérateurs positif non nul.

Au chapitre suivant, nous introduisons la notion de paires d’opérateurs D*-symétriques, (A, B) est D*-symétrique si R(δAB) = R(δA∗_B∗). En première partie, nous montrons

que : si A et B sont deux opérateurs D-symétriques de spectres disjoints, alors (A, B) est D*-symétrique. Nous donnons une caractérisation de la classe des paires d’opérateurs D*-symétriques. Nous montrons que les propriétés suivantes sont équivalentes :

(1). (A, B) est D*-symétrique ;

(2). δA∗(A)L(H) + L(H)δ_B∗(B) ⊆R(δ_AB) ∩ R(δ_A∗_B∗) ;

(3). A∗R(δAB) + R(δAB)B∗ ⊆ R(δAB) et AR(δA∗_B∗) + R(δ_A∗_B∗)B ⊆ R(δ_A∗_B∗).

En conséquence, nous obtenons que la classe des paires d’opérateurs D*-symétriques contient les paires d’opérateurs normaux et les paires d’isométries.

En second partie nous donnons une autre caractérisation des paires d’opérateurs D*-symétriques. Nous montrons l’équivalence entre les assertions suivantes :

(1). (A, B) est D*-symétrique ;

(2). a. ([A], [B]) est essentiellement D*-symétrique, et b. (A, B) et (B, A) admettent la propriété de (F, P )C1;

(3). c. ([A], [B]) est essentiellement D*-symétrique, et d. R(δAB) U = R(δA∗_B∗) U .

Comme conséquence nous déduisons que : si ils existent deux éléments f et g non nuls de H et λ ∈ IC, tels que B(f ) = λf , B∗(f ) 6= λf et A∗(g) = λg, alors (A, B) n’est pas D*-symétrique.

Une extension naturelle en dimension finie et infinie des formes quadratiques est celle de l’image numérique W (A) qui permet entre-autre de localiser le spectre d’un opérateur linéaire borné A défini sur un espace de Hilbert complexe H. Marshall Stone a considéré le nom (domaine numérique) pour W (A), où

W (A) = {< Ax, x >, kxk = 1}.

Toplitz et Hausdorff l’ont appelé (Wertvorrat) d’une forme bilinéaire et d’autres ont choisi (le domaine de Hausdorff) et (le corps des valeurs) de A. Des études sur l’image numé-rique sont développées par plusieurs auteurs, citons par exemple F. Bonsall et J. Ducan [5], K. Gustafson et D. Rao [16], P. Halmos [17] et J. G. Stampfli et J. P. Williams [23]. Au dernier chapitre, nous introduisons l’image numérique généralisée Wg(A) définie par :

Wg(A) = {< Ax, x >, kxk ≤ 1}.

Au paragraphe 1, nous donnons des propriétés de l’image numérique généralisée. Nous prouvons que Wg(A) est convexe. Nous obtenons une condition nécessaire et suffisante

pour que Wg(A) = W (A).

Il est prouvé par G. D. Barra, J. R. Giles et B. Sims [3], que si A est un opérateur compact normal, alors

W (A) = co(σp(A)),

l’enveloppe convexe du spectre ponctuel de A.

Au paragraphe 2, Nous montrons que Wg(A) est fermée pour tout opérateur compact A.

Ensuite, nous obtenons que si A est un opérateur compact normal, alors Wg(A) = co(σp(A) ∪ {0}).

Au pragraphe 3, Nous prouvons que si pour tout λ ∈ IC, kA − λk = ρ(A − λ) ( ρ(A) le rayon spectral de A ) et kB − λk = ρ(B − λ) , alors l’image numérique généralisée

R

ÉFÉRENCES

.

[1] J. H. ANDERSON, J. W. BUNCE, J. A. DEDDENS and J. P. WILLIAMS , C*-algebras

and derivation ranges, Acta Sci. Math. (Szeged),40(1978), 211-227.

[2] C. APOSTOLand L. FIALKOW, Structural properties of elementary operators, Cana-dian Journal of Mathematics,38 (1986), 1485-524.

[3] G. D. BARRA, J. R. GILES and B. SIMS, On the numerical range of compact opera-tors on Hilbert spaces.J. London Math. Soc. (2),5(1972), 704-706.

[4] M. BENLARBI, S. BOUALI and S. CHERKI, Une remarque sur l’orthogonalité de

l’image au noyau d’une dérivation généralisée, Proc. Amer. Math. Soc,126 (1998), 167-71.

[5] F. F. BONSALLand J. DUCAN, Numerical ranges of operators on normed spaces and elements of normed algebras. London Math. Soc. Lecture Note Series. Cambridge Univ. Press. Cambridge.1971.

[6] S. BOUALI, and S. CHERKI, Approximation by generalised commutators, Acta Sci.

Math. (Szeged)63 (1997), 273-8.

[7] S. BOUALI, et J. CHARLES, Extension de la notion d’opérateurs d-symétriques I, Acta Sci. Math. (Szeged),58(1993), 517-525.

[8] S. BOUALI, et J. CHARLES , Extension de la notion d’opérateurs d-symétriques II, Linear Algebra And Its Applications,225(1995), 175-185.

[9] L. A. FIALKOW ANDR. LOBEL, Elementary mapping into ideals operators, Illinois

J. Math.,28 (1984).

[10] L. A. FIALKOW, Essential spectra of elementary operators, Transaction of the Ame-rican Mathematical Society267 (1981), 157-74.

[11] L. A. FIALKOW, Spectral properties of elementary operators, Acta Scientiarum Ma-thematicarum (Szeged)46 (1983), 269-82.

[12] L. A. FIALKOW, Spectral properties of elementary operators II, Journal of the

Ame-rican Mathematical Society290 (1985), 415-29.

[13] L. A. FIALKOW, The range inclusion problem for elementary operators, Michigan Mathematical Journal34 (1987), 451-9.

[14] C. K. FONG and A.R. SOUROUR, On the operator identity ΣAkXBk, Canadian

Journal of Mathematics31 (1979), 845-57.

Journal of Fudan University23 (1989), 148-56.

[16] K. GUSTAFSONand D. RAO, Numerical range, Springer1996.on, N. J.(1967). [17] P. R. HALMOS, Hilbert space problem book, New York,1970.

[18] D. A. HERRERO, Approximation of Hilbert space operators. I, Pitman, Advanced publishing program, Boston - Melbourne1982.

[19] R. V. KADISON, Derivations on operators algebras, Ann. of Math.,83 (1966), 280-293.

[20] I. KAPLANSKY, Modules over operators algebras, Ann. of Math.,27 (1959), 839-859.

[21] M. MATHIEU, Spectral theory for multiplication operators on C*-algebras, Procee-dings of the Royal Irish Academy83A (1983), 231-49.

[22] S. SAKAI, Derivation on W* algebras, Ann. of Math.,83 (1966), 273-279.

[23] J. G. STAMPFLI and J. P. WILLIAMS, Growth conditions and the numerical range in

a Banach algebra, Tohoku Math. J.20(1968), 417-424.

[24] J. G. STAMPFLI, On self-adjoint derivation ranges, Pacific J. Math.,82 (1979), 257-77.

[25] J. G. STAMPFLI, The norm of a derivation, Pacific J. Math.,33 (1970), 737-47.

[26] R. E. WEBER, Analytic functions, ideals, and derivation ranges, Proc. Amer. Math

soc.40 (1973), 492-6.

[27] R. E. WEBER, Derivation and the trace-class operators, Proc. Amer. Math soc., 1 (1979), 79-82.

[28] J. P. WILLIAMS, Derivation ranges : open problems, Topics in modern operator theory, Birkhauser-Verlag,1981, 319-28.

0. Terminologies

1. On désigne par H un espace de Hilbert complexe séparable. Les opérateurs qu’on considère sur H sont toujours linéaires et bornés ; l’espace de ces opérateurs est noté L(H). si A ∈ L(H), R(A) ( resp. Ker(A) ) désigne l’image ( resp. le noyau ) de A. 2. Sur L(H), on utilise l’une des topologies suivantes :

(i). Topologie de la norme ( ou uniforme ). une suite (Tn)nd’opérateurs converge en norme

vers 0, ce qu’on symbolise par Tn −→ 0, si kTnk −→ 0 quand n −→ ∞, où kT k =

sup{kT xk; x ∈ H et kxk = 1}.

La fermeture en norme d’un sous-ensemble E de L(H) est E.

(ii). Topologie faible des opérateurs. une suite (Tn)n d’opérateurs converge faiblement

vers 0, ce qu’on note Tn w

−→ 0, si < Tnx, y >−→ 0 quand n −→ ∞, pour tout x, y ∈ H.

La fermeture faible d’un sous-ensemble E de L(H) est Ew.

(iii). Topologie ultrfaible des opérateurs. une suite (Tn)n d’opérateurs converge

ultrafai-blement vers 0, lorsque f (Tn) −→ 0 quand n −→ ∞, pour toute forme linéaire f sur

L(H).

Si E est un sous-ensemble de L(H), EU est la fermeture ultrafaible de E.

3. Le spectre de l’opérateurs A, noté σ(A), est l’ensemble des scalaires λ ∈ IC tels que A − λ est non inversible dans L(H). σp(A) est le spectre ponctuel de A.

4. Différentes classes d’opérateurs dans L(H). Un opérateurs A est dit 4.1. de rang fini si R(A) est de dimension finie ;

4.2. compact si < Axn, xn>−→ 0, pour toute suite orthonormée (xn)ndans H ;

4.3. de classe trace si P | < Axn, xn > | < ∞, pour toute base orthonormée (xn)n de

H ;

4.4. nilpotent si An= 0 pour un certain entier naturel n ; 4.5. quasi-nilpotent si son spectre est réduit à {0} ;

4.6. positif si < Ax, x >≥ 0, pour tout x ∈ H, ce qui est noté A ≥ 0; 4.7. auto-adjoint si A∗ = A, où A∗ est l’adjoint hilbertien de A; 4.8. normal si A∗A = AA∗;

4.9. sous-normal s’il admet une extension normale ; 4.10. hyponormal si A∗A − AA∗ est positif ;

5. K(H) désigne l’espace des opérateurs compact.

6. Cpest la classe de Von Neumann-Schatten, et k.kpsa norme.

7. [A] est la classe de l’opérateurs A dans l’algèbre de Calkin C(H) = L(H)/K(H). 8. La dérivation intérieure induite par A ∈ L(H) est l’opérateur δA sur L(H) défini par

δA(X) = AX − XA, X ∈ L(H). Le noyau de δAest appelé le commutant de A, et il est

noté {A}0.

9. Le bicommutant de A ∈ L(H), noté {A}00, est l’ensemble des opérateurs qui com-mutent avec tout opérateur dans {A}0.

10. La dérivation généralisée induite par A, B ∈ L(H) est l’opérateur δAB sur L(H)

défini comme suit δAB(X) = AX − XB pour tout X ∈ L(H).

11. Pour A ∈ L(H), LAet RAsont les opérateurs de multiplications à gauche et à droite

définies par LA(X) = AX et RA(X) = XA pour tout X ∈ L(H).

12. Pour A, B ∈ L(H), l’opérateur élémentaire ∆AB est donné par ∆AB(X) = AXB −

Chapter 1

Preliminaries

In this preliminary chapter, we present some backgrounds on essential results on the orthogonality of the range and the kernel of a derivation on L(H), D-symetric operators and numerical range.

1

The range of a derivation and Orthogonality

It follows from the elementary properties of derivations that the set of all B such that R(δB) ⊂ R(δA) is a subalgebra of L(H). ( see [13].) Therefore if B is a polynomial in

A, then R(δB) ⊂ R(δA). Weber generalized this to analytic functions.

Théorème 1.1. [12] Let A ∈ L(H) and f be an analytic function on an open set contai-ningσ(A). Then R(δf (A)) ⊂ R(δA).

Corollaire 1.1. [12] Let A _{= 0 be an element of L(H) with 0 /}∈ σ(A). Then R(δ

A12) =

R(δA).

Given subspaces E and F of a Banach space B with norm k.k, E is said to be ortho-gonal to F if

kX + Y k ≥ kY k for all X ∈ E and Y ∈ F .

Anderson [2] has shown that ;

Théorème 1.2. [2] If A and B are normal operators. Then for every operator T such that AT = T B, we have

kδAB(X) + T k ≥ kT k

The above inequality says that the range R(δAB) of the generalized derivation δAB is

orthogonal to the kernel ker(δAB) of δAB.

Définition 1.1. [4] Let A, B be in L(H) and J be a two sided ideal of L(H). The pair (A, B) is said to possess the Fuglede-Putnam property (F, P )J if,AT = T B and T ∈ J

impliesA∗T = T B∗.

Let J be the norm ideal associated with the unitarily invariant norm k.kJ. In the same

direction, it should be noted that F. Kittaneh established that ;

Théorème 1.3. [8] Let A, B in L(H). If the pair (A, B) has the property (F.P )L(H), then

kδAB(X) + T kJ ≥ kT kJ

for allX ∈ J and T ∈ ker(δAB/J).

Théorème 1.4. [4] Let A, B in L(H). If the pair (A, B) has the property (F.P )J, then

kδAB(X) + T kJ ≥ kT kJ

for allX ∈ J and T ∈ ker(δAB/J).

It has been shown in theorem 4 [7] that ;

Théorème 1.5. [7] If A is a cyclic subnormal operator and T ∈ C2(H) ∩ {A}0, then for

allX ∈ L(H)

kδA(X) + T k22 = kδA(X)k22+ kT k 2 2.

Théorème 1.6. [4] For A, B in L(H) the following are equivalent : (1). (A, B) has the property (F.P )C2(H);

(2). for all X ∈ L(H) and T ∈ ker(δAB/C2(H)) ;

kδAB(X) + T k22 = kδAB(X)k22+ kT k 2 2.

2

D-symmetric operators

A is said to be D-symmetric if R(δA) is closed under taking adjoint. This concept of

D-symmetry of an operator was introduced by J. H. Anderson, J. W. Bunce, J. A. Deddens and J. P. Williams [1].

Définition 2.1. [1] Let A ∈ L(H). If R(δA) = R(δA∗) (i.e. R(δ_A) is self-adjoint), we say

thatA is D-symmetric.

Théorème 2.1. [1] For A in L(H) the following are equivalent : (1). A is D-symmetric ;

(2). δA∗(A)L(H) + L(H)δ_A∗(A) ⊆R(δ_A) ;

(3). A∗R(δA) + R(δA)A∗ ⊆ R(δA).

Corollaire 2.1. [1] Every normal operator is D-symmetric. Corollaire 2.2. [1] Every isometry U is D-symmetric.

Théorème 2.2. [1] An operator A on H is D-symmetric if and only if (a) A is essentially D-symmetric, and

(b) AT = T A for an operator T in the trace class implies AT∗ = T∗A. Corollaire 2.3. [1]

(a) An essentially normal operator A is D-symmetric if and only if AT = T A for an operatorT in the trace class implies AT∗ = T∗A.

(b) An operator in the trace class is D-symmetric if and only if it is normal.

Théorème 2.3. [6] Let A ∈ L(H) be a subnormal operator with a cyclic vector. Then A is D-symmetric.

Théorème 2.4. [11] Let A ∈ L(H) be a hyponormal weighted shift ( unilaterial or bila-terial ) with no point spectrum. ThenA is D-symmetric.

Théorème 2.5. [11] Let A, B ∈ L(H). If A and B are D-symmetric operators with disjoint spectra, thenA ⊕ B is D-symmetric.

Théorème 2.6. [9] An essentially normal weighted shift S ( Sen = wnen+1 ) is

D-symmetric if and only if it satisfies the total products condition, that is,P

kwk.wk+1...wk+n−1 =

∞.

Théorème 2.7. [9] A hyponormal ( in particular subnormal ) weighted shift Sen =

3

Numerical range

Let A be a complex Banach algebra with identity e, and let P = {f ∈ A∗, f (e) = 1 = kf k} be the set of states on A. The numerical range [10] of an element A in A is by definition the set ;

Wo(A) = {f (A), f ∈ P }.

If A = L(H) is the algebra of bounded operators on a Hilbert space H, then Wo(A) =

W (A) is precisely the closure of the ordinary numerical range, W (A) = {< Ax, x >, kxk = 1}.

In the following section we present several properties of The numerical range.

Théorème 3.1. [10] If A ∈ A, then Wo(A) is convex, compact and contains the spectrum

ofA.

Théorème 3.2. [3] Let A ∈ K(H). If 0 ∈ W (A), then W (A) is closed.

Théorème 3.3. [3] For a compact normal operator A on H, W (A) = co(σp(A)), the

convex hull of the point spectrum ofA.

Théorème 3.4. [14] If A ∈ L(H) such that kA − λk = ρ(A − λ), for all λ in IC, then Wo(A) = co(σ(A)).

Théorème 3.5. [5] If A, B ∈ L(H) such that kA − λk = ρ(A − λ) and kB − λk = ρ(B − λ) for all λ in IC, then Wo(δAB) = co(σ(δAB)).

R

EFERENCES

.

[1] J. H. ANDERSON, J. W. BUNCE, J. A. DEDDENSand J. P. WILLIAMS, C*-algebras

and derivation ranges, Acta Sci. Math. (Szeged),40(1978), 211-227.

[2] J. H. ANDERSON, On normal derivations, Proc. Amer. Math. Soc.,38(1973), 135-140. [3] G. D. BARRA, J. R. GILES and B. SIMS, On the numerical range of compact opera-tors on Hilbert spaces.J. London Math. Soc. (2),5(1972), 704-706.

[4] M. BENLARBI, S. BOUALI and S. CHERKI, Une remarque sur l’orthogonalité de

l’image au noyau d’une dérivation généralisée, Proc. Amer. Math. Soc,126 (1998), 167-71.

[5] S. BOUALI, and J. CHARLES, Generalized derivation and numerical range, Acta Sci. Math. (Szeged),63(1997), 563-570.

[6] S. BOUALI, et J. CHARLES, Extension de la notion d’opérateurs d-symétriques I, Acta Sci. Math. (Szeged),58(1993), 517-525.

[7] F. KITTANEH, Normal derivations in Hilbert-Schmidt type, Glasgow Math. J.,29(1987), 245-248.

[8] F. KITTANEH, Normal derivations in norm ideals, Proc. Amer. Math. Soc., 123 : n6(1995), 1779-1785.

[9] C. ROSENTRATER, Compact operators and derivations induced by weighted shifts, Pacific J. Math.,104(1983), 465-470.

[10] J. G. STAMPFLI and J. P. WILLIAMS, Growth conditions and the numerical range in

a Banach algebra, Tohoku Math. J.20(1968), 417-424.

[11] J. G. STAMPFLI, On self-adjoint derivation ranges, Pacific J. Math., 82(1979), 257-277.

[12] R. E. WEBER, Analytic functions, ideals, and derivation ranges, Proc. Amer. Math. soc.40 (1973), 492-6.

[13] R. E. WEBER, Derivation ranges, Thesis, Indiana University, Bloomington, Ind., 1972.

Chapter 2

Analytic Functions, Derivations and Orthogonality*

1

Abstract. Let H be a separable infinite-dimensional complex Hilbert space. For A ∈ L(H), δA

denote the inner derivation induced by the operator A, defined by δA(X) = AX − XA in [22].

Weber [19] has shown that, if B is an analytic function of A, then R(δB) ⊂ R(δA), R(δA) is the

range of δA. In the first part, we extend Weber’s result to generalized derivation. This allows us

to generalize the principal result in [4]. We also prove that ; if B ∈ {A}0∩ R(δ_A)w ( R(δA) w

the weak closure of R(δA) ) and there exists a natural number n such that Bnis compact, then B is

quasinilpotent.

4

Introduction

Let H be a separable infinite-dimensional complex Hilbert space and let A, B ∈ L(H), where L(H) is the algebra of all bounded operators on H into itself. Let δAB,

∆AB and τAB be the operators on L(H) defined by δAB(X) = AX − XB, ∆AB(X) =

AXB − X and τAB(X) = AXB for all X ∈ L(H).

The properties of these operators, their spectrum (see [5; 7; 8 and 9]), norm (see [18]) and ranges (see [2; 6; 10; 12; 13; 15; 17 and 22]) have been much studied, and many of their problems remain also open (see [21 and 22]).

It has been shown by Weber in theorem 1 [19] that if B is an analytic function of A, then the range of δB is contained in the range of δA. In the first section we extend

Weber’s result to generalized derivation. We obtain a sufficient condition under which R(δf (A)f (B)) = R(δAB). We also turn our attention to the range of ∆AB and τAB.

1. Mathematics Subject Classification (2000) : 47B47, 47B10 ; secondary 47A30.

Key words and phrases : generalized derivation, elementary operator, Analytic function, range-kernel or-thogonality.

In the second section we generalize the principal result of [4] on the range-kernel orthogo-nality of the operator δAB/Cp, where Cpdenote the Von Neumann-Schatten class for p > 1.

We show that if f is a bi-analytic function on an open set containing σ(A) ∪ σ(B) such that (f (A), f (B)) has the property (F, P )Cp, 1 ≤ p < ∞, then

kδAB(X) + T kp ≥ kT kp

for all X ∈ L(H) and T ∈ ker(δAB/Cp).

Finally we generalize the principal result of Weber [20]. We prove that ; if B ∈ R(δA) w

∩ {A}0

( R(δA) w

the weak closure of R(δA) ) and there exists a natural number n such that

Bnis compact, then B is quasinilpotent. We conclude this section with some notations.

Notations.Let K(H) be the ideal of all compact operators. Let C1(H) be the ideal of trace

class operators. For 1 < p < ∞ we denote Cp(H) the Von Neumann-Schatten class and

k.kpits associated norm. For A ∈ L(H), let LAand RAbe the operators on L(H) defined

by LA(X) = AX and RA(X) = XA. Let A be a commutative Banach algebra with

maximal ideal space MAand let a and b belong to A, the joint spectrum of a and b is the

set σ(a, b) = { (ϕ(a), ϕ(b)); ϕ ∈ MA } (see Gamelin [11, p. 76]).

In addition to the notation already introduced, we shall use the following notation. Given X ∈ L(H), we shall denote the kernel, the range and the spectrum of X by ker(X), R(X) and σ(X) respectively. For A ∈ L(H), the weak closure of R(δA) will be denoted

by R(δA) w

.

5

Analytic Functions and Derivation Ranges

Théorème 5.1. Let A, B ∈ L(H) and f be an analytic function on an open set contai-ningσ(A) ∪ σ(B). Then R(δf (A)f (B)) ⊂ R(δAB).

Proof. Let X ∈ L(H). Consider the operators on H ⊕ H

T = A 0 0 B ! and Y = 0 X X 0 ! .

Since σ(T ) ⊂ σ(A) ∪ σ(B), then f is an analytic function on an open set containing σ(T ). Thus R(δf (T )) ⊂ R(δT) by [19, theorem 1]. It follows that there exists

Z = Z1 Z2

Z3 Z4

!

∈ L(H ⊕ H)

such that δf (T )(Y ) = δT(Z). A simple calculation shows that δf (A)f (B)(X) = δAB(Z2).

Corollaire 5.1. Let A, B ∈ L(H) and f be a bi-analytic function on an open set contai-ningσ(A) ∪ σ(B). Then R(δf (A)f (B)) = R(δAB).

Corollaire 5.2. Let A, B ∈ L(H). Then there exists n0 ∈ IN such that

R(δAB) = R(δ_eA

n_eBn) for all n ≥ n0.

Proof. Let D1 = { z ∈ IC / z /∈ IR−} and D2 = { z ∈ IC / z = x + ıy, −π <

y < π, x ∈ IR }. Let L : D1 −→ D2 the branch of logarithm defined on D1 by :

L(z) = ln(|z|) + ıθ(z), with θ(z) ∈] − π, π[. Then e : D2 −→ D1

z 7−→ ez

is a bi-analytic function. Let K = σ(A) ∪ σ(B), K is compact. Then there exists M > 0 such that |z| < M for all z ∈ K. Take n0 a natural number such that M_n0 < π. Let n ≥ n0

and z ∈ σ(A_n) ∪ σ(B_n) = 1_nK. Hence nz ∈ K, so that z ∈ D2. Thus σ(A_n) ∪ σ(B_n) ⊂ D2

for each n ≥ n0. It follows from corollary 5.1 that

R(δ

eAn_eBn) = R(δA_nB_n) = R(δAB).

for all n ≥ n0. ♦

Lemme 5.1. [11, p. 566] Let A be a commutative Banach algebra. There exists a unique rule assigning to every ordered pair(a, b) of elements in A and to every complex valued function of two complex variablesf (z, w) analytic in a neighborhood of σ(a, b), an ele-mentf (a, b) ∈ A satisfying the following conditions :

(1) If f (z, w) = Σcidjziwj is a polynomial, thenf (a, b) = Σcidjaibj.

(2) If f (z, w) and g(z, w) are analytic in a neighborhood of σ(a, b), then f + g(a, b) = f (a, b) + g(a, b) and f g(a, b) = f (a, b)g(a, b).

(3) If f (z) is analytic in a neighborhood U of σ(a) and if f1(z, w) is the extension of f (z)

to U × IC defined by f1(z, w) = f (z), then f1(a, b) = f (a), where f (a) is an analytic

function of the elementa in the sense of the Riesz-Dunford functional calculus.

Théorème 5.2. Let A, B be in L(H). Let f, g be two analytic functions on the open sets U and V containing σ(A) and σ(B) respectively. Then

under one of the following hypotheses : (1). f (0) = g(0) = 0.

(2). f (0) = 0 and B is invertible. (3). A is invertible and g(0) = 0. (4). A and B are invertible.

Proof. Let A be the maximal abelian subalgebra of L(L(H)) containing LA, RBand

the identity. We have

σ(LA, RB) = { (ϕ(LA), ϕ(RB)); ϕ ∈ MA}.

Then

σ(LA, RB) ⊂ σA(LA) × σA(RB) = σ(LA) × σ(RB).

Thus

σ(LA, RB) ⊂ σ(A) × σ(B) ⊂ U × V.

Assume (2). Then there exists f1an analytic function on U such that f (z) = zf1(z) for all

z ∈ U . Since 0 /∈ σ(B), there exists an open set V0 _{∈ IC such that 0 /∈ V}0 _{and σ(B) ⊂ V}0_.

Hence there exists g1 an analytic function on V ∩ V0 such that g(z) = zg1(z) for all

z ∈ V ∩ V0. Thus

f (z)g(w) = zwf1(z)g1(w)

for all (z, w) ∈ U × (V ∩ V0). It follows from lemma 5.1 that f (LA)g(RB) = LARBf1(LA)g1(RB).

Using [14, p. 33], we obtain

Lf (A)Rg(B)= LARBf1(LA)g1(RB).

Hence

f (A)Xg(B) = A(f1(LA)g1(RB)X)B

for all X ∈ L(H). Thus R(τf (A)g(B)) ⊂ R(τAB).

We obtain (1), (3) and (4) by a similar argument. ♦

Remarque 5.1. If f (z) = z, g(z) = 1 for all z ∈ IC, A is an invertible operator and B is not left invertible. Suppose that R(τf (A)g(B)) ⊂ R(τAB). It follows that there exists

Y ∈ L(H) such that A = AY B. Then Y B = I, this is a contradiction. Thus the condition [ f (0) = 0 or A is invertible ] and [ g(0) = 0 or B is invertible ] is essential.

Théorème 5.3. Let A, B be in L(H). Let f, g be two analytic functions on the open sets U and V containing σ(A) and σ(B) respectively. If

σ(A) × σ(B) ⊂ { (z, w) ∈ IC2 / zw 6= 1 } = W,

thenR(∆f (A)g(B)) ⊂ R(∆AB).

Proof. Let A be the maximal abelian subalgebra of L(L(H)) containing LA, RBand

the identity. We have

σ(LA, RB) ⊂ σ(A) × σ(B) ⊂ (U × V ) ∩ W.

Let

h : (U × V ) ∩ W −→ IC

(z, w) 7−→ f (z)g(w) − 1 zw − 1 h is an analytic function on (U × V ) ∩ W ⊃ σ(LA, RB), and

(zw − 1)h(z, w) = f (z)g(w) − 1

for all (z, w) ∈ (U × V ) ∩ W . It follows from lemma 5.1 that there exists h(LA, RB) ∈ A

such that (LARB− 1)h(LA, RB) = f (LA)g(RB) − 1. Using [14, p. 33], we obtain (LARB− 1)h(LA, RB) = Lf (A)Rg(B)− 1. Hence f (A)Xg(B) − X = A(h(LA, RB)(X))B − h(LA, RB)(X)

for all X ∈ L(H). Which completes the proof. ♦

6

Range-kernel Orthogonality

Définition 6.1. [3] Let A, B be in L(H) and J be a two sided ideal of L(H). The pair (A, B) is said to possess the Fuglede-Putnam property (F, P )J if,AT = T B and T ∈ J

Théorème 6.1. Let A, B in L(H) and f be a bi-analytic function on an open set U containingσ(A) ∪ σ(B). If the pair (f (A), f (B)) has the property (F.P )Cp,1 ≤ p < ∞

then :

(1). kδAB(X) + T kp ≥ kT kpfor allX ∈ L(H) and T ∈ ker(δAB/Cp).

(2). ker(δn

AB/Cp) = ker(δAB/Cp) for all n ≥ 1.

Proof. (1). let T be in ker(δAB_/Cp), then AT = T B. Hence

(λ − A)−1T = T (λ − B)−1

for each λ /∈ σ(A). A simple calculation shows that f (A)T = T f (B), that is, T ∈ ker(δf (A)f (B)/Cp). Let X ∈ L(H). Corollary 5.1 asserts that there exists Y ∈ L(H) such

that

δAB(X) = δf (A)f (B)(Y ).

Since (f (A), f (B)) has the property (F.P )Cp, it follows from theorem 2.2 in [4] that

kδf (A)f (B)(Y ) + T kp ≥ kT kp.

Thus

kδAB(X) + T kp ≥ kT kp

for all X ∈ L(H) and T ∈ ker(δAB/Cp).

(2). If the pair (f (A), f (B)) has the property (F.P )Cp, then (1) implies that

R(δAB/Cp)

k.kp

∩ ker(δAB/Cp) = { 0 }.

Using lemma 2.3 in [4], we obtain

ker(δ_AB/Cn _p) = ker(δAB/Cp) for all n ≥ 1. ♦

Théorème 6.2. Let A be in L(H) and B ∈ R(δA) w

∩ {A}0_{. If there existsn ∈ IN}∗ _such

thatBn_{is compact, thenB is quasinilpotent.}

Proof. If B ∈ R(δA) w

∩ {A}0

, then there exists a sequence {Xα}α ⊂ L(H) such that

AXα− XαA w −→ B. Then Bn−1AXα− Bn−1XαA w −→ Bn_{∈ K(H).} Hence Bn∈ R(δA) w ∩ {A}0 ∩ K(H).

Remarque 6.1. The result ceases being true if, we replace the assumption Bn_{is compact}

by P (B) is compact for some polynomial P . Indeed ; it is known [16, theorem 3] that there exists B ∈ L(H) and K ∈ K(H) such that K ∈ R(δB) ∩ {B}0. It is known

[1, corollary 5] that there exists A ∈ L(H) such that I ∈ R(δA). Thus there exists two

sequences{Xn}n,{Yn}ninL(H) such that

AXn− XnA k.k −→ I and BYn− YnB k.k −→ K. We put A1 = A 0 0 B ! , Zn = Xn 0 0 Yn ! and K1 = I 0 0 K ! .

A simple calculation shows thatδA1(Zn)

k.k

−→ K1, that is, K1 ∈ R(δA1). It follows that

K1 ∈ R(δA1)

w

∩ {A1}0 and K12− K1 ∈ K(H ⊕ H).

But in this caseI − K1is not invertible, so thatσ(K1) 6= { 0 }.

R

EFERENCES

.

[1] J. H. ANDERSON, Derivation ranges and the identity, Bull. Amer. Math. Soc., 79 (1973), 705-9.

[2] C. APOSTOLand L. FIALKOW, Structural properties of elementary operators, Canadian

Jour-nal of Mathematics, 38 (1986), 1485-524.

[3] M. BENLARBI, S. BOUALIand S. CHERKI, Une remarque sur l’orthogonalité de l’image au noyau d’une dérivation généralisée, Proc. Amer. Math. Soc, 126 (1998), 167-71.

[4] S. BOUALI, and S. CHERKI, Approximation by generalised commutators, Acta Sci. Math. (Szeged) 63 (1997), 273-8.

[5] L. A. FIALKOW ANDR. LOBEL, Elementary mapping into ideals operators, Illinois J. Math.,

28 (1984).

[6] L. A. FIALKOW, Essential spectra of elementary operators, Transaction of the American Ma-thematical Society 267 (1981), 157-74.

[7] L. A. FIALKOW, Spectral properties of elementary operators, Acta Scientiarum Mathematica-rum (Szeged) 46 (1983), 269-82.

[8] L. A. FIALKOW, Spectral properties of elementary operators II, Journal of the American

[9] L. A. FIALKOW, The range inclusion problem for elementary operators, Michigan Mathema-tical Journal 34 (1987), 451-9.

[10] C. K. FONG and A.R. SOUROUR, On the operator identity ΣAkXBk,Canadian Journal of

Mathematics 31 (1979), 845-57.

[11] F. W. GAMELIN, Uniform algebras, Prenlice-Hall, Englewood Cliffs, N. J., 1969.

[12] Z. GENKAI, On the operator δA,B : X 7−→ AX − XB and τA,B : X 7−→ AXB − X,

Journal of Fudan University 23 (1989), 148-56.

[13] D. A. HERRERO, Approximation of Hilbert space operators. I, Pitman, Advanced publishing program, Boston - Melbourne 1982.

[14] G. LUMER and M. ROSENBLUM, Linear operator equations, Proc. Amer. Math. Soc. 10 (1959), 32-41.

[15] M. MATHIEU, Spectral theory for multiplication operators on C*-algebras, Proceedings of

the Royal Irish Academy 83A (1983), 231-49.

[16] J. G. STAMPFLI, Derivations on B(H) : the range ., Illinois J. Math., 17 (1962), 518-24. [17] J. G. STAMPFLI, On self-adjoint derivation ranges, Pacific J. Math., 82 (1979), 257-77. [18] J. G. STAMPFLI, The norm of a derivation, Pacific J. Math., 33 (1970), 737-47.

[19] R. E. WEBER, Analytic functions, ideals, and derivation ranges, Proc. Amer. Math soc. 40

(1973), 492-6.

[20] R. E. WEBER, Derivation and the trace-class operators, Proc. Amer. Math soc., 1 (1979), 79-82.

[21] J. P. WILLIAMS, Derivation ranges : open problems, Topics in modern operator theory, Birkhauser-Verlag, 1981, 319-28.

Chapter 3

Generalized D-symmetric Operators I*

2

Abstract. Let H be an infinite-dimensional complex Hilbert space and let A, B ∈ L(H), where L(H) is the algebra of operators on H into itself. Let δAB: L(H) → L(H) denote the generalized

derivation δAB(X) = AX − XB. This note will initiate a study on the class of pairs (A, B) such

that R(δAB) = R(δB∗_A∗) ; i.e. R(δ_AB) is self-adjoint.

7

Introduction

Let L(H) the algebra of all bounded operators on an infinite dimensional complex Hilbert space H. The generalized derivation operator δABassociated with (A, B), defined

on L(H) by δAB(X) = AX − XB was systematically studied for the first time in [6].

The properties of such operators have been studied extensively ( see for example [2], [5], [8], [9] and [10] ).

The D-symmetric operators ( A is D-symmetric if R(δA) is self-adjoint, where R(δA) is

the closure of the range R(δA) of δAin the norm topology ) were studied by J. H.

Ander-son, J. W. Bunce, J. A. Deddens and J. P. Williams [1], S. Bouali and J. Charles [3][4] and J. G. Stampfli [8].

We consider the class of pairs (A, B) such that R(δAB) is self-adjoint, we call such pairs

symmetric. In this work we extend the results of the symmetric operators to D-symmetric pairs.

In the first part we give some properties and characterizations which concern the

D-2. Classification (2000) : 47B47, 47B10 ; secondary 47A30.

Key words and phrases : generalized derivation, self-adjoint derivation ranges, D-symmetric operators. * Serdica Mathematical Journal 34 (2008), 557-562.

symmetric pairs. The second part contains a description of the sets : C(A, B) = {C ∈ L(H), CL(H) + L(H)C ⊂ R(δAB)}

and

I(A, B) = {Z ∈ L(H), ZR(δAB) + R(δAB)Z ⊂ R(δAB)}

which generalize those introduced by J. P. Williams in [10]. Notations.

1. Let K(H) be the ideal of all compact operators. For A ∈ L(H), let [A] denote the coset of A in the Calkin algebra C(H) = L(H)/K(H).

2. C1(H) is the ideal of trace class operators.

3. For A, B ∈ L(H), R(δAB) U

denotes the ultraweak closure of R(δAB), and L(H)0U

denotes the bounded linear forms in ultraweak topology.

4. Let M be a subspace of L(H). We denote the orthogonal of M in the duality L(H), L(H)0 by Mo.

5. For g and ω two vectors in H, we define g ⊗ ω ∈ L(H) as follows : g ⊗ ω(x) =< x, ω > g for all x ∈ H.

8

Properties of D-symmetric Pairs

Définition 8.1. Let A, B ∈ L(H).

(1) IfR(δAB) is self-adjoint i.e. R(δAB) = R(δB∗_A∗), we say that (A, B) is D-symmetric

pair of operators. We denote the set of such pairs byGD(H).

(2) Let δ[A][B] the generalized derivation operator defined on C(H) by δ[A][B]([X]) =

[δAB(X)]. If R(δ[A][B]) is self-adjoint i.e. R(δ[A][B]) = R(δ[B∗_][A∗_]), we say that ([A], [B])

is D-symmetric inC(H).

Lemme 8.1. If A, B ∈ L(H), then

R(δAB)0 = R(δAB)0∩ K(H)0⊕ ker (δBA) ∩ C1(H).

The proof of lemma 8.1 is the same as the proof of theorem 3 in [11]. Théorème 8.1. For A, B ∈ L(H) the following are equivalent :

(1). (A, B) is D-symmetric ;

b.BT = T A implies BT∗ = T∗A for all T ∈ C1(H) ;

(3). c. ([A], [B]) is D-symmetric in C(H), and d.R(δAB)

U

= R(δB∗_A∗)

U

. Proof. Note that R(δAB)

U

is self-adjoint if and only if R(δAB)0 ∩ L(H)0U is

self-adjoint. Using lemma 8.1 we have

R(δAB)0∩ L(H)0U ' ker (δBA) ∩ C1(H).

Consequently we obtain : R(δAB) U

is self-adjoint if and only if ker (δBA) ∩ C1(H) is

self-adjoint. Thus (2) ⇔ (3).

The equivalence of (1) and (2) is a consequence of lemma 8.1. ♦

Théorème 8.2. Let A, B ∈ L(H). If there exists λ ∈ IC such that (B − λ)(A − λ) = (A − λ)2 _{= 0, A − λ 6= 0 and B − λ 6= 0, then (A, B) is not D-symmetric.}

Proof. Since for all λ ∈ IC, R(δAB) = R(δ(A−λ)(B−λ)), we may assume without loss

of generality that λ = 0. The condition A∗A 6= 0 (A 6= 0) implies that there exists an vector f = Ah 6= 0, such that A∗f 6= 0. Then Bf = 0. Since A∗B∗ = 0, we choose g 6= 0 such that A∗g = 0. We put A∗f = ω ;

< ω, f >=< A∗f, f >=< f, Af >=< f, A2h >= 0

i.e. ω and f are orthogonal. If X = kωk−2(g ⊗ ω) and Y ∈ L(H), then it follows that : < (B∗X − XA∗)f, g > = < B∗Xf, g > − < XA∗f, g > = < 0, g > − < Xω, g > = − < g, g > = −kgk2 and < (AY − Y B)f, g >=< Y f, A∗g > − < 0, g >= 0. Suppose that B∗X − XA∗ ∈ R(δAB) U

. Then there exists a net (Yα)α ⊂ L(H) such that,

for all x and y in H, we have :

< (AYα− YαB)x, y >−→< (B∗X − XA∗)x, y > .

So that,

0 =< (AYα− YαB)f, g >−→< (B∗X − XA∗)f, g >= −kgk2.

It follows that g = 0 ; this proves that B∗X − XA∗ ∈ R(δ/ AB) U

. Consequently we obtain that (A, B) is not D-symmetric by theorem 8.1. ♦

Théorème 8.3. If H is separable, then GD(H) is not norm-closed in (L(H))2_.

Proof. Let {en}n≥1 be an orthonormal basis for H. Define a sequence of operators

(Sn)n≥1as follows :

Sn(ek) =

( ₁

ne2, if k = 1;

ek+1, if k ≥ 2.

Corollary 3 in [7] asserts that for every n ≥ 1 K(H) ⊂ R(δSn). It follows from [11,

corollary 1, p. 277] that {Sn}0∩ C1(H) = {0}, then theorem 8.1 implies that (Sn, Sn) ∈

GD(H) for all n ≥ 1. Let

S(ek) =

(

0, if k = 1;

ek+1, if k ≥ 2.

It is clear that k(Sn, Sn) − (S, S)k −→ 0. Let f = e1 + e2, ω = e3 and g = e1. Since

S∗f = 0, Sf = ω and Sg = 0, It follows from the proof of theorem 8.2 that (S∗, S∗) is not D-symmetric. Thus (S, S) /∈ GD(H), which completes the proof. ♦

9

Properties and Descriptions of C(A, B) and I(A, B).

Consider the natural closed subalgebras of L(H) associated with (A, B) : C(A, B) = {C ∈ L(H), CL(H) + L(H)C ⊂ R(δAB)}

and

I(A, B) = {Z ∈ L(H), ZR(δAB) + R(δAB)Z ⊂ R(δAB)}

It is clear that ; if R(δAB) is norm-dense in L(H), I(A, B) = C(A, B) = L(H) ( for

example A = 2B = 2I ). Thus C(A, B) 6= {0} and I(A, B) contains non-scalar operators in general.

Théorème 9.1. If (A, B) is D-symmetric, then :

ı. C(A, B) and I(A, B) are norm closed C∗−algebras in L(H) ; ıı. C(A, B) is a two-sided ideal of I(A, B).

Proof. ı. It is clear that C(A, B) and I(A, B) are norm closed algebras in L(H). Since R(δAB) is self-adjoint, C(A, B) and I(A, B) are C∗−algebras.

ıı. If Z ∈ I(A, B) and C ∈ C(A, B), then for all X ∈ L(H) we have : X(CZ) = (XC)Z ∈ R(δAB)Z ⊂ R(δAB),

and (CZ)X = C(ZX) ∈ R(δAB). Thus C(A, B) is a right ideal of I(A, B). Since

Lemme 9.1. Let A, B ∈ L(H), then ;

I(A, B) = {Z ∈ L(H), δZ(A)L(H) + L(H)δZ(B) ⊂ R(δAB)}.

Proof. If Z ∈ I(A, B) and X ∈ L(H), then

δZ(A)X = ZδAB(X) − δAB(ZX), and XδZ(B) = δAB(X)Z − δAB(XZ).

This implies that δZ(A)X ∈R(δAB) and XδZ(B) ∈ R(δAB). Thus

δZ(A)L(H) + L(H)δZ(B) ⊂ R(δAB).

The reverse inclusion follows from the identities :

ZδAB(X) = δZ(A)X + δAB(ZX), and δAB(X)Z = XδZ(B) + δAB(XZ). ♦

Théorème 9.2. Let A, B ∈ L(H). If R(δAB) does not contain any nonzero positive

operator, thenC(A, B) = {0} and I(A, B) = {A}0_{∩ {B}}0_.

Proof. If C ∈ C(A, B) then CC∗ ∈ R(δAB) ; consequently we have C = 0. Thus

C(A, B) = {0}.

Let Z ∈ I(A, B), δZ(A)L(H) ⊂ R(δAB) and L(H)δZ(B) ⊂ R(δAB) by lemma 9.1.

Consequently we obtain δZ(A)(δZ(A))∗ = (δZ(B))∗δZ(B) = 0. Thus Z ∈ {A}0∩ {B}0.

Conversely ; if Z ∈ {A}0 ∩ {B}0_{, then δ}

Z(A) = δZ(B) = 0. It follows from lemma 9.1

that Z ∈ I(A, B). ♦

R

EFERENCES

.

[1] J. H. ANDERSON, J. W. BUNCE, J. A. DEDDENS and J. P. WILLIAMS , C*-algebras

and derivation ranges, Acta Sci. Math. (Szeged),40(1978), 211-227.

[2] J. H. ANDERSON and C. FOIAS, Properties which normal operators share with nor-mal derivation and related operators, Pacific J. Math.,61(1976), 313-325.

[3] S. BOUALI, et J. CHARLES, Extension de la notion d’opérateurs d-symétriques I, Acta Sci. Math. (Szeged),58(1993), 517-525.

[4] S. BOUALI, et J. CHARLES , Extension de la notion d’opérateurs d-symétriques II,

Linear Algebra And Its Applications,225(1995), 175-185.

[5] D. A. HERRERO, Approximation of Hilbert space operators. I, Pitman, Advanced pu-blishing program, Boston - Melbourne1982.

[6] M. A. ROSENBLUM, On the operator equation BX − XA = Q, Duke Math. J.,

23(1956), 263-269.

[7] C. ROSENTRATER, Compact operators and derivations induced by weighted shifts, Pacific J. Math.,104(1983), 465-470.

[8] J. G. STAMPFLI, On self-adjoint derivation ranges, Pacific J. Math.,82(1979), 257-277. [9] J. G. STAMPFLI, The norm of a derivation, Pacific J. Math.,33(1970), 737-747.

[10] J. P. WILLIAMS , Derivation ranges : Open problems, Topics in Modern Operator Theory, Birkhauser-Verlag,1981, pp. 319-328.

Chapter 4

Generalized D-symmetric Operators II*

3

Abstract. Let H be a separable infinite-dimensional complex Hilbert space and let A, B ∈ L(H), where L(H) is the algebra of all bounded linear operators on H. Let δAB : L(H) → L(H)

denote the generalized derivation δAB(X) = AX − XB. This note will initiate a study on the

class of pairs (A, B) such that R(δAB) = R(δA∗_B∗).

10

Introduction.

Let L(H) be the algebra of all bounded linear operators on an infinite dimensional complex Hilbert space H. For an operator A in L(H), the inner derivation on A, δA, is

defined on L(H) by δA(X) = AX − XA for each X in L(H).The generalized derivation

operator δABassociated with (A, B), defined on L(H) by δAB(X) = AX − XB has been

much studied, and many of its spectral and metric properties are known ( see [2], [6], [7] and [9] ).

J. G. Stampfli [8], J. H. Anderson, J. W. Bunce, J. A. Deddens and J. P. Williams [1], and S. Bouali and J. Charles [4][5] gave some properties and characterizations of D-symmetric operators, the class of operators that induce derivations for which the norm closures of their ranges are self-adjoint. In order to generalize these results, we initiate the study of a more general class of D-symmetric operators, in other words, the class of pairs of opera-tors A, B ∈ L(H) that have R(δAB) = R(δA∗_B∗), where R(δ_AB) is the norm closure of

the range of δAB. We call such pairs D*-symmetric.

Notations.

1. For A ∈ L(H), σ(A) is the spectrum of A.

3. Classification (2000) : 47B47, 47B10 ; secondary 47A30.

Key words and phrases : generalized derivation, adjoint, D-symmetric operator, normal operator. * à apparaitre dans Canadian Mathematical Bulletin (Canadian Mathematical Society, Ottawa).

2. Let K(H) be the ideal of all compact operators. For A ∈ L(H), let [A] denote the coset of A in the Calkin algebra C(H) = L(H)/K(H).

3. C1(H) is the ideal of trace class operators.

4. For A, B ∈ L(H), R(δAB) U

denotes the ultraweak closure of R(δAB), and L(H)0U

denotes the continuous linear forms in the ultraweak topology.

5. Let M be a subspace of L(H). We denote the orthogonal of M in the dual space of L(H), L(H)0

, by Mo.

6. For g and ω two vectors in H, we define g ⊗ ω ∈ L(H) as follows : g ⊗ ω(x) =< x, ω > g for all x ∈ H.

11

D*-symmetric Pairs

Définition 11.1. Let A, B ∈ L(H). If R(δAB) = R(δA∗_B∗), we say that (A, B) is

D*-symmetric.

Théorème 11.1. Let A, B ∈ L(H). If A and B are D-symmetric operators with disjoint spectra, then(A, B) is D*-symmetric.

Proof. Let X ∈ R(δAB). There exists a sequence (Xn)n ⊂ L(H), such that kδAB(Xn)−

Xk −→ 0. Consider the operators on H ⊕ H

M = 0 X 0 0 ! , Mn= 0 Xn 0 0 ! , and T = A 0 0 B ! . It follows that δT(Mn) = 0 δAB(Xn) 0 0 ! k.k −→ 0 X 0 0 ! = M.

Thus M ∈ R(δT). Since A and B are D-symmetric operators with disjoint spectra, then

T is D-symmetric by J. G. Stampfli [8; P : 260]. Hence there exists a sequence (Nn)n⊂

L(H ⊕ H), such that δT∗(N_n)

k.k

−→ M . A simple calculation proves that there exists a sequence (Yn)n ⊂ L(H), such that δA∗_B∗(Y_n)

k.k

−→ X. Thus R(δAB) ⊂ R(δA∗_B∗). We

have the reverse inclusion by the same way. ♦

Remarque 11.1. Let A and B two cyclic subnormal operators with disjoint spectra. A andB are D-symmetric operators by [4; Th 2.5]. Since σ(A) ∩ σ(B) = ∅, Theorem 11.1 implies that(A, B) is D*-symmetric.

Théorème 11.2. For A, B in L(H) the following are equivalent : (1). (A, B) is D*-symmetric ;

(2). δA∗(A)L(H) + L(H)δ_B∗(B) ⊆R(δ_AB) ∩ R(δ_A∗_B∗) ;

(3). A∗R(δAB) + R(δAB)B∗ ⊆ R(δAB) and AR(δA∗_B∗) + R(δ_A∗_B∗)B ⊆ R(δ_A∗_B∗).

Proof. (1) ⇒ (2). For all X ∈ L(H) we have :

δA∗(A)X = δ_A∗_B∗(AX) − Aδ_A∗_B∗(X) and Xδ_B∗(B) = δ_AB(X)B∗− δ_AB(XB∗).

Since AR(δA∗_B∗) ⊆ AR(δ_AB) ⊆ R(δ_AB) and R(δ_AB)B∗ ⊆ R(δ_A∗_B∗)B∗ ⊆ R(δ_AB), it

follows that

δA∗(A)L(H) + L(H)δ_B∗(B) ⊆R(δ_AB).

The implication (2) ⇒ (3) is a consequence of the following identities : for all X and Y in L(H),

A∗δAB(X) + δAB(Y )B∗ = δAB(A∗X + Y B∗) + δA∗(A)X + Y δ_B∗(B)

and

AδA∗_B∗(X) + δ_A∗_B∗(Y )B = δ_A∗_B∗(AX + Y B) − δ_A∗(A)X − Y δ_B∗(B).

(3) ⇒ (1). Suppose that (3) holds. Then A∗nR(δAB) ⊆ R(δAB) for each n in IN . We

always have the inclusion AmR(δAB) ⊆R(δAB) for each m in IN .

We shall prove that R(δAB)o = R(δA∗_B∗)o. Let f ∈ R(δ_AB)o and X ∈ L(H). Observe

that

A∗nAX − AA∗nX = A∗nδAB(X) − δAB(A∗nX)

for each n in IN . Hence A∗nAX −AA∗nX ∈ R(δAB) for each n in IN . A similar argument

using mathematical induction on m shows that A∗nAm_{X − A}m_A∗n_{X ∈ R(δ}

AB) for each

n and m in IN . Thus f (A∗nAmX) = f (AmA∗nX) for each n and m in IN . It follows that f (eαAeβA∗X) = f (eβA∗eαAX)

for all complex numbers α and β. An induction argument shows that

f ((αA + βA∗)nX) = n X k=0 n k ! f ((αA)k(βA∗)n−kX)

for each n in IN and for all complex numbers α and β. Hence

for each X in L(H) and for all complex numbers α and β. A similar argument using R(δAB)B∗ ⊆ R(δAB) shows that

f (XeαB+βB∗) = f (XeαBeβB∗) = f (XeβB∗eαB) for each X in L(H) and for all complex numbers α and β.

Since f (AX) = f (XB), it follows by induction that f (An_{X) = f (XB}n_{) for all n ∈ IN ,}

and hence f (eαA_{X) = f (Xe}αB_{) or f (e}αA_Xe−αB_{) = f (X) for all α ∈ IC and X ∈ L(H).}

These relations yield, for all λ ∈ IC, the equations

f (eıλA∗Xe−ıλB∗) = f (eıλAeıλA∗Xe−ıλB∗e−ıλB) = f (eı(λA+λA∗)Xe−ı(λB∗+λB)). Define the function g on IC as follows :

g(λ) = f (eıλA∗Xe−ıλB∗).

Since λA + λA∗and λB∗+ λB are self-adjoint operators, then eı(λA+λA∗)and e−ı(λB∗+λB) are unitary operators . Thus for all λ ∈ IC,

|g(λ)| ≤ kf k kXk.

By Liouville’s theorem, the entire function g side must be constant. In particular, the derivative vanishes at λ = 0. This gives f (A∗X − XB∗) = 0 for all X ∈ L(H). Thus R(δAB)o ⊆ R(δA∗_B∗)o. We have the reverse inclusion by the same way. ♦

Corollaire 11.1. If A and B are normal operators, then (A, B) is D*-symmetric. Corollaire 11.2. Let U and V two isometries, then (U, V ) is D*-symmetric.

Proof. Let P = I − U U∗. Then for all X ∈ L(H),

δU∗_V∗(X) = δ_{U V}(−U∗XV∗) − P XV∗.

Hence, to prove that R(δU∗_V∗) ⊆ R(δ_{U V}), it suffices to show that P X ∈ R(δ_{U V}) for all

X ∈ L(H). Let Tn= n−1 X k=0 (k n − 1)U k P XV∗k+1, n ∈ IN∗,

where IN∗ = IN \{0}. A simple calculation shows that

δU V(Tn) − P X = − 1 n n X k=1 UkP XV∗k.

Since < UjP x, Uk_{P y >= 0 for j 6= k and x, y in H, then} (1) k n X k=1 UkP XV∗kxk2 = n X k=1 kUkP XV∗kxk2 ≤ nkP Xk2kxk2. Thus kδU V(Tn) − P Xk ≤ n− 1 2kP Xk, that is, P X ∈ R(δ_{U V}).

For the reverse inclusion, first prove that if Q = I − V V∗, then P X ∈ R(δU∗_V∗) and

XQ ∈ R(δU∗_V∗) for all X ∈ L(H). Let

Sn= n−1 X k=0 (k n − 1)U k+1_{P XV}∗k , n ∈ IN∗.

A simple calculation shows that

δU∗_V∗(S_n) + P X = 1 n n X k=1 UkP XV∗k.

It follows from (1) that kδU∗_V∗(S_n) + P Xk ≤ n− 1 2kP Xk. Thus P X ∈ R(δ_U∗_V∗). Consi-der Rn = n−1 X k=0 (k n − 1)U k+1_XQV∗k , n ∈ IN∗, Then δU∗_V∗(R_n) + XQ = 1 n n X k=1 UkXQV∗k. Hence (δU∗_V∗(R_n) + XQ)∗ = 1 n n X k=1 VkQX∗U∗k. Thus kδU∗_V∗(R_n) + XQk ≤ n− 1 2kQX∗k, and so XQ ∈ R(δ_U∗_V∗). Since U δU∗_V∗(X) = δ_U∗_V∗(U X) − P X and δ_U∗_V∗(X)V = δ_U∗_V∗(XV ) − XQ, then U R(δU∗_V∗) + R(δ_U∗_V∗)V ⊆ R(δ_U∗_V∗).

It follows from the proof of Theorem 11.2 that R(δU V) ⊆ R(δU∗_V∗). Thus (U, V ) is

D*-symmetric. ♦

Définition 11.2. [3] Let A, B be in L(H) and J be a two sided ideal of L(H). The pair (A, B) is said to possess the Fuglede-Putnam property (F, P )J if,AT = T B and T ∈ J

Théorème 11.3. For A, B ∈ L(H) the following are equivalent : (1). (A, B) is D*-symmetric ;

(2). a. ([A], [B]) is D*-symmetric in C(H), and

b.(A, B) and (B, A) have the property (F, P )C1;

(3). c. ([A], [B]) is D*-symmetric in C(H), and d.R(δAB)

U

= R(δA∗_B∗)

U

. Proof. Note that R(δAB)

U

= R(δA∗_B∗)

U

if and only if

R(δAB)o∩ (L(H))0U = R(δA∗_B∗)o∩ (L(H))0U.

On the other hand

(I) R(δAB)o ' R(δAB)o∩ K(H)o⊕ ker (δBA) ∩ C1(H), [10, Th 3.]

In particular,

R(δAB)0∩ L(H)0U ' ker (δBA) ∩ C1(H).

This proves that R(δAB) U = R(δA∗_B∗) U if and only if ker (δBA) ∩ C1(H) = ker (δB∗_A∗) ∩ C₁(H). Thus (2) ⇔ (3).

Clearly the above shows that (1) ⇒ (3). Suppose that (3) holds. Let f ∈ R(δAB)0. Then

by (I), we have f = f0+ fT such that f0 ∈ R(δAB)o∩ K(H)oand T ∈ ker (δBA) ∩ C1(H)

( where fT(X) = tr(XT ) for each X in L(H) ). Since R(δAB) U

= R(δA∗_B∗)

U

, it follows that T ∈ ker (δB∗_A∗) ∩ C₁(H). Let Z ∈ R(δ_A∗_B∗). Then [Z] ∈ R(δ_[A∗_][B∗_]). Since

([A], [B]) is D*-symmetric in C(H), then [Z] ∈ R(δ[A][B]). There exists a sequence of

operators (Xn)nin L(H) and a sequence (Kn)nof compact operators in K(H) such that

AXn− XnB + Kn−→ Z.

But

f0(AXn− XnB + Kn) = f0(AXn− XnB) + f0(Kn) = 0,

and thus f0(Z) = 0. It follows that f0 ∈ R(δA∗_B∗)o∩ K(H)o, and hence f ∈ R(δ_A∗_B∗)o.

Therefore, R(δAB)o ⊆ R(δA∗_B∗)o. We obtain the reverse inclusion by a similar argument.

♦

Proof. (U, V ) is D*-symmetric by Corollary 11.2. It follows from Theorem 11.3 that (U, V ) has the property (F, P )C1. ♦

Théorème 11.4. Let A, B ∈ L(H). If there exists two nonzero elements f and g in H, and λ ∈ IC, such that B(f ) = λf , B∗(f ) 6= λf and A∗(g) = λg, then (A, B) is not D*-symmetric.

Proof. Since for all λ ∈ IC, R(δAB) = R(δ(A−λ)(B−λ)), we may assume without loss

of generality that λ = 0. Note that B∗f = ω 6= 0 where ω ⊥ f . If X = kωk−2(g ⊗ ω) and Y ∈ L(H), then < (A∗X − XB∗)f, g > = < A∗X(f ), g > − < XB∗f, g > = < 0, g > − < X(ω), g > = − < g, g > = −kgk2 and < (AY − Y B)f, g >=< Y f, A∗g > − < 0, g >= 0. Suppose that A∗X − XB∗ ∈ R(δAB) U

. Then there exists a net (Yα)α in L(H) such that

for all x and y in H, we have :

< (AYα− YαB)x, y >−→< (A∗X − XB∗)x, y > .

So that

0 =< (AYα− YαB)f, g >−→< (A∗X − XB∗)f, g >= −kgk2.

It follows that g = 0. This proves that A∗X − XB∗ ∈ R(δ/ AB) U , that is, R(δAB) U 6= R(δA∗_B∗) U

. Consequently we obtain that (A, B) is not D*-symmetric by Theorem 11.3.♦

R

EFERENCES

.

[1] J. H. ANDERSON, J. W. BUNCE, J. A. DEDDENS and J. P. WILLIAMS, C*-algebras

[2] J. H. ANDERSON and C. FOIAS, Properties which normal operators share with nor-mal derivation and related operators, Pacific J. Math.,61(1976), 313-325.

[3] M. BENLARBI, S. BOUALI and S. CHERKI, Une remarque sur l’orthogonalité de l’image au noyau d’une dérivation généralisée, Proc. Amer. Math. Soc, 126(1998), 167-171.

[4] S. BOUALI, et J. CHARLES, Extension de la notion d’opérateurs d-symétriques I, Acta

Sci. Math. (Szeged),58(1993), 517-525.

[5] S. BOUALI, et J. CHARLES, Extension de la notion d’opérateurs d-symétriques II, Li-near Algebra And Its Applications,225(1995), 175-185.

[6] D. A. HERRERO, Approximation of Hilbert space operators. I, Pitman, Advanced

pu-blishing program, Boston - Melbourne1982.

[7] M. A. ROSENBLUM, On the operator equation BX − XA = Q, Duke Math. J., 23(1956), 263-269.

[8] J. G. STAMPFLI, On self-adjoint derivation ranges, Pacific J. Math.,82(1979), 257-277. [9] J. P. WILLIAMS, Derivation ranges : Open problems, Topics in Modern Operator Theory, Birkhauser-Verlag,1981, pp. 319-328.

Chapter 5

Generalized Numerical Range*

4

Abstract. In this paper we will introduce the generalized numerical range Wg(A) of an operator

A on a separable Hilbert space. We will give some properties of Wg(A), and study the situation in

which Wg(A) = W (A) ( the ordinary numerical range of A ). We also shed light on the

generali-zed numerical range of derivation.

12

Introduction

Let A be a complex Banach algebra with identity e, and let P = {f ∈ A∗, f (e) = 1 = kf k} be the set of states on A. The numerical range [7] of an element A in A is by definition the set ;

Wo(A) = {f (A), f ∈ P }.

Wo(A) is convex, compact and contains the spectrum of A [7].

If A = L(H) is the algebra of bounded operators on a Hilbert space H, then Wo(A) =

W (A) is precisely the closure of the ordinary numerical range, W (A) = {< Ax, x >, kxk = 1}.

The numerical range was systematically studied by several authors, for example F. Bonsall and J. Ducan [2], K. Gustafson and D. Rao [4], and P. Halmos [5].

For A ∈ A, we define the generalized numerical range of A by Wog(A) = {f (A), f ∈ Pg},

4. Classification (2000) : 47B47, 47B10 ; secondary 47A30.

Key words and phrases : generalized numerical range, generalized derivation, compact operator, hyponor-mal operator.

where Pg = {f ∈ A∗; f (e) = kf k ≤ 1}. If A ∈ L(H), let

Wg(A) = {< Ax, x >; kxk ≤ 1}.

The motivation of the numerical range which has just been introduced is based on one of Halmos’s results [5].

In the first part, we give some properties of the generalized numerical range. We prove that Wg(A) is convex, and obtain some equivalent ( sufficient ) conditions for Wg(A) =

W (A).

It has been shown in theorem 2 [1] that if A ∈ L(H) is a compact normal operator, then W (A) = co(σp(A)), the convex hull of the point spectrum of A. In the second part, we

show that for a compact operator A ; Wg(A) is closed, and obtain that for a compact

normal operator A ; Wg(A) = co(σp(A) ∪ {0}).

In the third part, we prove that, if for all λ in IC, kA − λk = ρ(A − λ) ( ρ stands for the spectral radius ) and kB − λk = ρ(B − λ), then the generalized numerical range Wog(δAB) = co(σ(δAB) ∪ {0}). δAB is the generalized derivation operator associated

with (A, B) ∈ (L(H))2, defined on L(H) by δAB(X) = AX − XB for all X ∈ L(H).

In addition to the notation already introduced, we shall use the following notation. We shall denote the ideal of all compact operators by K(H). Given X ∈ L(H), the spectrum, the point spectrum and the spectral radius of X will be denoted by σ(X), σp(X) and ρ(X)

respectively.

13

Properties of Generalized Numerical Range.

Lemme 13.1. If A ∈ L(H) ; then Wog(A) = Wg(A).

Proof. λ ∈ Wog(A) is equivalent to ; there exists f ∈ Pg such that _{kf k}λ ∈ Wo(A) =

W (A). This occurs if and only if λ ∈ Wg(A). ♦

Théorème 13.1. If A ∈ L(H), then Wg(A) is convex.

Proof. Let < Ax, x > and < Ay, y > two elements in Wg(A), x1 = _kxkx , y1 = _kyky

and α = ptkxk2_{+ (1 − t)kyk}2 _{where t ∈ [0, 1]. Then < Ax}

1, x1 > and < Ay1, y1 >∈

W (A). Since W (A) is a convex set [5, solution 166, p. 317], there exists u ∈ H ; such that kuk = 1 and

t(kxk α ) 2 _{< Ax} 1, x1 > +(1 − t)( kyk α ) 2 _{< Ay} 1, y1 >=< Au, u > .

Hence

t < Ax, x > +(1 − t) < Ay, y >=< A(αu), αu >, and kαuk = |α| ≤ 1. Thus Wg(A) is convex. ♦

Corollaire 13.1. For A ∈ L(H), Wog(A) is convex, compact and contains the spectrum

ofA.

Remarques 13.1.

(1) Remark that For all A ∈ L(H) ; 0 ∈ Wg(A).

(2) It is clear that Wg(I) = [0, 1] and W (I) = {1}. Thus W (A) is properly contained in

Wg(A) in general.

(3) If T = 1 0

0 2 !

, then a simple calculation shows thatWg(T ) = [0, 2] and W (T ) =

[1, 2]. Thus W (T ) is a segment which does not contain 0.

Théorème 13.2. If A ∈ L(H), then the following assertions are equivalent : (1) Wg(A) = W (A);

(2) 0 ∈ W (A).

Proof. Since 0 ∈ Wg(A), it is sufficient to show that (2) ⇒ (1). Assume that 0 ∈

W (A). Let λ =< Ay, y >∈ Wg(A). Note that y = tx with |t| ≤ 1 and kxk = 1. Since

W (A) is convex [5, solution 166, p. 317] and

λ = t2 < Ax, x > +(1 − t2)0,

it follows that λ ∈ W (A). ♦

Corollaire 13.2. Let A ∈ L(H). If there exists λ ∈ σp(A) and r ∈ IR−, such that

rλ ∈ σp(A), then Wg(A) = W (A).

Proof. Suppose that there exists λ ∈ σp(A) and r ∈ IR−, such that rλ ∈ σp(A). Let

t = _1−r−r ∈ [0, 1]. A simple calculation shows that :

0 = tλ + (1 − t)rλ.

Since W (A) is convex [5, solution 166, p. 317], 0 ∈ W (A). Thus Wg(A) = W (A). ♦

Proposition 13.1. Let A ∈ L(H) and λ ∈ IC, such that |λ| = kAk. If λ ∈ Wg(A), then

Proof. Suppose that λ =< Ay, y > where kyk ≤ 1. Then we have : kAk = |λ| ≤ | < Ay, y > | ≤ kAykkyk ≤ kAk.

It follows that | < Ay, y > | = kAykkyk. Hence there exists µ ∈ IC such that Ay = µy. Consequently λ =< Ay, y >= µkyk2, which implies that y 6= 0. Thus µ ∈ σp(A). ♦

14

Generalized Numerical Range of Compact Operators.

Théorème 14.1. If A ∈ K(H), then Wg(A) is closed.

Proof. Let λ ∈ Wg(A), then there exists a sequence (< Axn, xn>)n, where kxnk ≤ 1

for all n, converging to λ. Since the unit ball is weakly compact, there exists a subsequence (xnk)kwhich is weakly convergent to an x where kxk ≤ 1. Since A is a compact operator,

(A(xnk))kis strongly convergent to Ax.

However,

| < Axnk, xnk > − < Ax, x > | ≤ kxnkkkAxnk− Axk + | < xnk, Ax > − < x, Ax > |.

Therefore (< Axnk, xnk >)kconverge to < Ax, x >. Thus λ =< Ax, x >∈ Wg(A). ♦

Remarque 14.1.

(1) Let (en)n≥0 be an orthonormal basis for H, and S the unilateral shift defined by

Sen = en+1. It is known that W (S) = D = {z ∈ IC / |z| < 1}. Since 0 ∈ W (S),

it follows from theorem 13.2 that Wg(S) = W (S) = D. This shows that Wg(A) is not

closed in general.

(2) Wg(I) = [0, 1] is closed but I is not compact. Thus the condition A ∈ K(H) is not

necessary.

Théorème 14.2. For a compact normal operator A on H, Wg(A) = co(σp(A) ∪ {0}).

Proof. Clearly co(σp(A) ∪ {0}) ⊂ Wg(A), so it is sufficient to show that Wg(A) ⊂

co(σp(A) ∪ {0}). Let λ =< Ax, x >6= 0 where kxk ≤ 1. If y = _kxkx , then

< Ay, y >∈ W (A) = co(σp(A)) [1, theorem 2].

Thus

λ = kxk2 < Ay, y > +(1 − kxk2)0 ∈ co(σp(A) ∪ {0}).