UNIVERSITÉ MOHAMMED V – AGDAL FACULTÉ DES

UNIVERSITÉ MOHAMMED V – AGDAL
FACULTÉ DES SCIENCES
Rabat
N° d’ordre :2473
THÉSE 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 ).
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
Avant-propos
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.
Je voudrais exprimer mes vifs remerciements à Monsieur le professeur Ahmed
Intissar de la Faculté des sciences de Rabat de s’être intéressé à mon travail, et
pour l’honneur qu’il me fait en acceptant de participer à ce jury. Je voudrais le
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.
Résumé
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
)
(
R δ f ( A )f ( B ) = R (δ AB ) . On déduit l'orthogonalité de l'image au noyau de la
dérivation
δ AB / C p si ( f ( A ) , f ( B ) )
admet la propriété de Fuglede- Putnam dans
C p pour p > 1 .
Au troisième chapitre on considère Les paires d'opérateurs
(A ,B )
telles que
R (δ AB ) 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,
si
( A , B ) est D*-symétrique si R (δ AB ) = R (δ A B
*
*
) . On a prouvé que :
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
W g ( A ) = {< Ax , x >, x ≤ 1} .
1
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
References………………………………………………………………….27
2
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
3
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 bornés sur H, on sait que toute dérivation est une dérivation intérieure (voir [19; 20; 22] ),
c’est à dire, de la forme δA avec
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 numérique 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 condition suffisante pour que R(δf (A)f (B) ) = R(δAB ).
Dans la deuxième partie, nous généralisons le résultat principal de [4], concernant l’orthogonalité 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 .
w
Nous généralisons le résultat principal de Weber [27] en montrons que si B ∈ R(δA ) ∩
w
{A}0 ( R(δA ) la fermeture de R(δA ) pour la topologie faible des opérateurs), et il existe
4
un entier n ≥ 1 tel que B n est 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 D-symétrique. On en déduit que l’ensemble des paires d’opérateurs Dsymétriques 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 propriétés et une description de ces ensembles. Nous obtenons C(A, B) = {0} et I(A, B) =
{A}0 ∩ {B}0 , si R(δAB ) 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)δB ∗ (B) ⊆ R(δAB ) ∩ R(δA∗ B ∗ ) ;
(3). A∗ R(δAB ) + R(δAB )B ∗ ⊆ R(δAB ) et AR(δA∗ B ∗ ) + R(δA∗ B ∗ )B ⊆ R(δA∗ 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 ;
5
(3). c. ([A], [B]) est essentiellement D*-symétrique, et
U
U
d. R(δAB ) = R(δA∗ B ∗ ) .
Comme conséquence nous déduisons que : si ils existent deux éléments f et g non nuls
de H et λ ∈ C,
I 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 numérique 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 ∈ C,
I kA − λk = ρ(A − λ) ( ρ(A) le
rayon spectral de A ) et kB − λk = ρ(B − λ) , alors l’image numérique généralisée
Wog (δAB ) = co(σ(δAB ) ∪ {0}).
6
R ÉFÉRENCES .
[1] J. H. A NDERSON, J. W. B UNCE, J. A. D EDDENS and J. P. W ILLIAMS , C*-algebras
and derivation ranges, Acta Sci. Math. (Szeged), 40(1978), 211-227.
[2] C. A POSTOL and L. F IALKOW, Structural properties of elementary operators, Canadian Journal of Mathematics, 38 (1986), 1485-524.
[3] G. D. BARRA, J. R. G ILES and B. S IMS , On the numerical range of compact operators on Hilbert spaces. J. London Math. Soc. (2), 5(1972), 704-706.
[4] M. B ENLARBI, S. B OUALI and S. C HERKI, Une remarque sur l’orthogonalité de
l’image au noyau d’une dérivation généralisée, Proc. Amer. Math. Soc, 126 (1998), 16771.
[5] F. F. B ONSALL and J. D UCAN, 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. B OUALI, and S. C HERKI, Approximation by generalised commutators, Acta Sci.
Math. (Szeged) 63 (1997), 273-8.
[7] S. B OUALI, et J. C HARLES , Extension de la notion d’opérateurs d-symétriques I, Acta
Sci. Math. (Szeged), 58(1993), 517-525.
[8] S. B OUALI, et J. C HARLES , Extension de la notion d’opérateurs d-symétriques II,
Linear Algebra And Its Applications, 225(1995), 175-185.
[9] L. A. F IALKOW AND R. L OBEL , Elementary mapping into ideals operators, Illinois
J. Math., 28 (1984).
[10] L. A. F IALKOW, Essential spectra of elementary operators, Transaction of the American Mathematical Society 267 (1981), 157-74.
[11] L. A. F IALKOW, Spectral properties of elementary operators, Acta Scientiarum Mathematicarum (Szeged) 46 (1983), 269-82.
[12] L. A. F IALKOW, Spectral properties of elementary operators II, Journal of the American Mathematical Society 290 (1985), 415-29.
[13] L. A. F IALKOW, The range inclusion problem for elementary operators, Michigan
Mathematical Journal 34 (1987), 451-9.
[14] C. K. F ONG and A.R. S OUROUR, On the operator identity ΣAk XBk , Canadian
Journal of Mathematics 31 (1979), 845-57.
[15] Z. G ENKAI, On the operator δA,B : X 7−→ AX − XB and τA,B : X 7−→ AXB − X,
7
Journal of Fudan University 23 (1989), 148-56.
[16] K. G USTAFSON and D. R AO, Numerical range, Springer 1996.on, N. J. (1967).
[17] P. R. H ALMOS, Hilbert space problem book, New York, 1970.
[18] D. A. H ERRERO , Approximation of Hilbert space operators. I, Pitman, Advanced
publishing program, Boston - Melbourne 1982.
[19] R. V. K ADISON, Derivations on operators algebras, Ann. of Math., 83 (1966), 280293.
[20] I. K APLANSKY , Modules over operators algebras, Ann. of Math., 27 (1959), 839-859.
[21] M. M ATHIEU, Spectral theory for multiplication operators on C*-algebras, Proceedings of the Royal Irish Academy 83A (1983), 231-49.
[22] S. S AKAI, Derivation on W* algebras, Ann. of Math., 83 (1966), 273-279.
[23] J. G. S TAMPFLI and J. P. W ILLIAMS, Growth conditions and the numerical range in
a Banach algebra, Tohoku Math. J. 20(1968), 417-424.
[24] J. G. S TAMPFLI , On self-adjoint derivation ranges, Pacific J. Math., 82 (1979), 25777.
[25] J. G. S TAMPFLI , The norm of a derivation, Pacific J. Math., 33 (1970), 737-47.
[26] R. E. W EBER, Analytic functions, ideals, and derivation ranges, Proc. Amer. Math
soc. 40 (1973), 492-6.
[27] R. E. W EBER, Derivation and the trace-class operators, Proc. Amer. Math soc., 1
(1979), 79-82.
[28] J. P. W ILLIAMS , Derivation ranges : open problems, Topics in modern operator
theory, Birkhauser-Verlag, 1981, 319-28.
[29] J. P. W ILLIAMS , On the range of a derivation, Pacific J. Math., 38 (1971), 273-9.
8
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 )n d’opérateurs converge en norme
vers 0, ce qu’on symbolise par Tn −→ 0, si kTn k −→ 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
w
vers 0, ce qu’on note Tn −→ 0, si < Tn x, y >−→ 0 quand n −→ ∞, pour tout x, y ∈ H.
w
La fermeture faible d’un sous-ensemble E de L(H) est E .
(iii). Topologie ultrfaible des opérateurs. une suite (Tn )n d’opérateurs converge ultrafaiblement vers 0, lorsque f (Tn ) −→ 0 quand n −→ ∞, pour toute forme linéaire f sur
L(H).
U
Si E est un sous-ensemble de L(H), E est la fermeture ultrafaible de E.
3. Le spectre de l’opérateurs A, noté σ(A), est l’ensemble des scalaires λ ∈ C
I 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 )n dans H ;
P
4.3. de classe trace si
| < 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 ;
4.11. une isométrie si A∗ A = I.
9
5. K(H) désigne l’espace des opérateurs compact.
6. Cp est la classe de Von Neumann-Schatten, et k.kp sa 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 δA est 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 commutent 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), LA et RA sont 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 −
X, X ∈ L(H).
10
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 containing σ(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(δA 21 ) =
R(δA ).
Given subspaces E and F of a Banach space B with norm k.k, E is said to be orthogonal 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
for all X ∈ L(H).
11
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
implies A∗ 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 all X ∈ 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 all X ∈ 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
all X ∈ L(H)
kδA (X) + T k22 = kδA (X)k22 + kT k22 .
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 k22 .
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].
12
Définition 2.1. [1] Let A ∈ L(H). If R(δA ) = R(δA∗ ) (i.e. R(δA ) is self-adjoint), we say
that A 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)δA∗ (A) ⊆ R(δA ) ;
(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
operator T 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 bilaterial ) with no point spectrum. Then A is D-symmetric.
Théorème 2.5. [11] Let A, B ∈ L(H). If A and B are D-symmetric operators with
disjoint spectra, then A ⊕ B is D-symmetric.
Théorème 2.6. [9] An essentially normal weighted shift S ( Sen = wn en+1 ) is DP
symmetric if and only if it satisfies the total products condition, that is, k wk .wk+1 .....wk+n−1 =
∞.
Théorème 2.7. [9] A hyponormal ( in particular subnormal ) weighted shift Sen =
wn en+1 is D-symmetric.
13
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
of A.
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 of A.
Théorème 3.4. [14] If A ∈ L(H) such that kA − λk = ρ(A − λ), for all λ in C,
I 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 C,
I then Wo (δAB ) = co(σ(δAB )).
R EFERENCES .
[1] J. H. A NDERSON, J. W. B UNCE, J. A. D EDDENS and J. P. W ILLIAMS, C*-algebras
and derivation ranges, Acta Sci. Math. (Szeged), 40(1978), 211-227.
[2] J. H. A NDERSON, On normal derivations, Proc. Amer. Math. Soc., 38(1973), 135-140.
[3] G. D. BARRA, J. R. G ILES and B. S IMS , On the numerical range of compact operators on Hilbert spaces. J. London Math. Soc. (2), 5(1972), 704-706.
[4] M. B ENLARBI, S. B OUALI and S. C HERKI, Une remarque sur l’orthogonalité de
l’image au noyau d’une dérivation généralisée, Proc. Amer. Math. Soc, 126 (1998), 16771.
14
[5] S. B OUALI, and J. C HARLES , Generalized derivation and numerical range, Acta Sci.
Math. (Szeged), 63(1997), 563-570.
[6] S. B OUALI, et J. C HARLES , Extension de la notion d’opérateurs d-symétriques I, Acta
Sci. Math. (Szeged), 58(1993), 517-525.
[7] F. K ITTANEH, Normal derivations in Hilbert-Schmidt type, Glasgow Math. J., 29(1987),
245-248.
[8] F. K ITTANEH, 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. S TAMPFLI and J. P. W ILLIAMS, Growth conditions and the numerical range in
a Banach algebra, Tohoku Math. J. 20(1968), 417-424.
[11] J. G. S TAMPFLI, On self-adjoint derivation ranges, Pacific J. Math., 82(1979), 257277.
[12] R. E. W EBER, Analytic functions, ideals, and derivation ranges, Proc. Amer. Math.
soc. 40 (1973), 492-6.
[13] R. E. W EBER, Derivation ranges, Thesis, Indiana University, Bloomington, Ind.,
1972.
[14] J. P. W ILLIAMS, Finite operators, Proc. Amer. Math. soc. 26 (1970), 129-136.
15
Chapter 2
Analytic Functions, Derivations and Orthogonality*
Abstract.
Let H be a separable infinite-dimensional complex Hilbert space. For A ∈ L(H), δA
1
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
w
w
to generalize the principal result in [4]. We also prove that ; if B ∈ {A}0 ∩ R(δA ) ( R(δA ) the
weak closure of R(δA ) ) and there exists a natural number n such that B n is 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 orthogonality.
* Sousmis au journal Linear Algebra and its Applications (Elsevier).
16
In the second section we generalize the principal result of [4] on the range-kernel orthogonality of the operator δAB/Cp , where Cp denote 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 ).
w
Finally we generalize the principal result of Weber [20]. We prove that ; if B ∈ R(δA ) ∩
w
{A}0 ( R(δA ) the weak closure of R(δA ) ) and there exists a natural number n such that
B n is 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.kp its associated norm. For A ∈ L(H), let LA and RA be 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 MA and 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
w
by R(δA ) .
5
Analytic Functions and Derivation Ranges
Théorème 5.1. Let A, B ∈ L(H) and f be an analytic function on an open set containing σ(A) ∪ σ(B). Then R(δf (A)f (B) ) ⊂ R(δAB ).
Proof. Let X ∈ L(H). Consider the operators on H ⊕ H
!
!
A 0
0 X
T =
and Y =
.
0 B
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
!
Z1 Z2
Z=
∈ L(H ⊕ H)
Z3 Z4
such that δf (T ) (Y ) = δT (Z). A simple calculation shows that δf (A)f (B) (X) = δAB (Z2 ).
Thus R(δf (A)f (B) ) ⊂ R(δAB ). ♦
17
Corollaire 5.1. Let A, B ∈ L(H) and f be a bi-analytic function on an open set containing σ(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(δe An e Bn ) for all n ≥ n0 .
Proof. Let D1 = { z ∈ C
I /z ∈
/ IR− } and D2 = { z ∈ C
I / 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( An ) ∪ σ( Bn ) = n1 K. Hence nz ∈ K, so that z ∈ D2 . Thus σ( An ) ∪ σ( Bn ) ⊂ D2
for each n ≥ n0 . It follows from corollary 5.1 that
R(δe An e Bn ) = R(δ A B ) = R(δAB ).
n n
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 variables f (z, w) analytic in a neighborhood of σ(a, b), an element f (a, b) ∈ A satisfying the following conditions :
(1) If f (z, w) = Σci dj z i wj is a polynomial, then f (a, b) = Σci dj ai bj .
(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 × C
I defined by f1 (z, w) = f (z), then f1 (a, b) = f (a), where f (a) is an analytic
function of the element a 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
R(τf (A)g(B) ) ⊂ R(τAB ),
18
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 , RB and
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 f1 an analytic function on U such that f (z) = zf1 (z) for all
z ∈ U . Since 0 ∈
/ σ(B), there exists an open set V 0 ∈ C
I such that 0 ∈
/ V 0 and σ(B) ⊂ V 0 .
Hence there exists g1 an analytic function on V ∩ V 0 such that g(z) = zg1 (z) for all
z ∈ V ∩ V 0 . Thus
f (z)g(w) = zwf1 (z)g1 (w)
for all (z, w) ∈ U × (V ∩ V 0 ). It follows from lemma 5.1 that
f (LA )g(RB ) = LA RB f1 (LA )g1 (RB ).
Using [14, p. 33], we obtain
Lf (A) Rg(B) = LA RB f1 (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 ∈ C,
I 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.
19
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) ∈ C
I 2 / zw 6= 1 } = W,
then R(∆f (A)g(B) ) ⊂ R(∆AB ).
Proof. Let A be the maximal abelian subalgebra of L(L(H)) containing LA , RB and
the identity. We have
σ(LA , RB ) ⊂ σ(A) × σ(B) ⊂ (U × V ) ∩ W.
Let
h : (U × V ) ∩ W −→ C
I
f (z)g(w) − 1
(z, w) 7−→
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
(LA RB − 1)h(LA , RB ) = f (LA )g(RB ) − 1.
Using [14, p. 33], we obtain
(LA RB − 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
implies A∗ T = T B ∗ .
20
Théorème 6.1. Let A, B in L(H) and f be a bi-analytic function on an open set U
containing σ(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 kp for all X ∈ L(H) and T ∈ ker(δAB/Cp ).
n
) = ker(δAB/Cp ) for all n ≥ 1.
(2). ker(δAB/C
p
Proof. (1). let T be in ker(δAB/Cp ), then AT = T B. Hence
(λ − A)−1 T = 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
k.kp
R(δAB/Cp )
∩ ker(δAB/Cp ) = { 0 }.
Using lemma 2.3 in [4], we obtain
n
ker(δAB/C
) = ker(δAB/Cp ) for all n ≥ 1. ♦
p
w
Théorème 6.2. Let A be in L(H) and B ∈ R(δA ) ∩ {A}0 . If there exists n ∈ IN ∗ such
that B n is compact, then B is quasinilpotent.
w
Proof. If B ∈ R(δA ) ∩ {A}0 , then there exists a sequence {Xα }α ⊂ L(H) such that
w
AXα − Xα A −→ B.
Then
w
B n−1 AXα − B n−1 Xα A −→ B n ∈ K(H).
Hence
w
B n ∈ R(δA ) ∩ {A}0 ∩ K(H).
It follows from theorem 3 in [20] that σ(B n ) = { 0 }. Thus σ(B) = { 0 }. ♦
21
Remarque 6.1. The result ceases being true if, we replace the assumption B n 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 }n in L(H) such that
k.k
k.k
AXn − Xn A −→ I and BYn − Yn B −→ K.
We put
A1 =
A 0
0 B
!
, Zn =
Xn
0
0
Yn
!
and K1 =
I
0
0 K
!
.
k.k
A simple calculation shows that δA1 (Zn ) −→ K1 , that is, K1 ∈ R(δA1 ). It follows that
w
K1 ∈ R(δA1 ) ∩ {A1 }0 and K12 − K1 ∈ K(H ⊕ H).
But in this case I − K1 is not invertible, so that σ(K1 ) 6= { 0 }.
R EFERENCES .
[1] J. H. A NDERSON , Derivation ranges and the identity, Bull. Amer. Math. Soc., 79 (1973), 7059.
[2] C. A POSTOL and L. F IALKOW, Structural properties of elementary operators, Canadian Journal of Mathematics, 38 (1986), 1485-524.
[3] M. B ENLARBI, S. B OUALI and S. C HERKI, Une remarque sur l’orthogonalité de l’image au
noyau d’une dérivation généralisée, Proc. Amer. Math. Soc, 126 (1998), 167-71.
[4] S. B OUALI, and S. C HERKI, Approximation by generalised commutators, Acta Sci. Math.
(Szeged) 63 (1997), 273-8.
[5] L. A. F IALKOW AND R. L OBEL , Elementary mapping into ideals operators, Illinois J. Math.,
28 (1984).
[6] L. A. F IALKOW, Essential spectra of elementary operators, Transaction of the American Mathematical Society 267 (1981), 157-74.
[7] L. A. F IALKOW, Spectral properties of elementary operators, Acta Scientiarum Mathematicarum (Szeged) 46 (1983), 269-82.
[8] L. A. F IALKOW, Spectral properties of elementary operators II, Journal of the American Mathematical Society 290 (1985), 415-29.
22
[9] L. A. F IALKOW, The range inclusion problem for elementary operators, Michigan Mathematical Journal 34 (1987), 451-9.
[10] C. K. F ONG and A.R. S OUROUR, On the operator identity ΣAk XBk , Canadian Journal of
Mathematics 31 (1979), 845-57.
[11] F. W. G AMELIN, Uniform algebras, Prenlice-Hall, Englewood Cliffs, N. J., 1969.
[12] Z. G ENKAI, 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. H ERRERO , Approximation of Hilbert space operators. I, Pitman, Advanced publishing
program, Boston - Melbourne 1982.
[14] G. L UMER and M. ROSENBLUM, Linear operator equations, Proc. Amer. Math. Soc. 10
(1959), 32-41.
[15] M. M ATHIEU, Spectral theory for multiplication operators on C*-algebras, Proceedings of
the Royal Irish Academy 83A (1983), 231-49.
[16] J. G. S TAMPFLI , Derivations on B(H) : the range ., Illinois J. Math., 17 (1962), 518-24.
[17] J. G. S TAMPFLI , On self-adjoint derivation ranges, Pacific J. Math., 82 (1979), 257-77.
[18] J. G. S TAMPFLI , The norm of a derivation, Pacific J. Math., 33 (1970), 737-47.
[19] R. E. W EBER, Analytic functions, ideals, and derivation ranges, Proc. Amer. Math soc. 40
(1973), 492-6.
[20] R. E. W EBER, Derivation and the trace-class operators, Proc. Amer. Math soc., 1 (1979),
79-82.
[21] J. P. W ILLIAMS , Derivation ranges : open problems, Topics in modern operator theory,
Birkhauser-Verlag, 1981, 319-28.
[22] J. P. W ILLIAMS , On the range of a derivation, Pacific J. Math., 38 (1971), 273-9.
23
Chapter 3
Generalized D-symmetric Operators I*
Abstract.
Let H be an infinite-dimensional complex Hilbert space and let A, B ∈ L(H), where
2
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(δAB ) 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 δAB associated 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 δA in the norm topology ) were studied by J. H. Anderson, 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
D-symmetric. In this work we extend the results of the D-symmetric operators to Dsymmetric pairs.
In the first part we give some properties and characterizations which concern the D2. 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.
24
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.
U
3. For A, B ∈ L(H), R(δAB ) 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 M o .
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) If R(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 by GD(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 in C(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 ;
(2). a. ([A], [B]) is D-symmetric in C(H), and
25
b. BT = T A implies BT ∗ = T ∗ A for all T ∈ C1 (H) ;
(3). c. ([A], [B]) is D-symmetric in C(H), and
U
U
d. R(δAB ) = R(δB ∗ A∗ ) .
U
Proof. Note that R(δAB ) is self-adjoint if and only if R(δAB )0 ∩ L(H)0U is selfadjoint. Using lemma 8.1 we have
R(δAB )0 ∩ L(H)0U ' ker (δBA ) ∩ C1 (H).
U
Consequently we obtain : R(δAB ) 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 ∈ C
I 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 ∈ C,
I 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, A2 h >= 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.
U
Suppose that B ∗ X − XA∗ ∈ R(δAB ) . Then there exists a net (Yα )α ⊂ L(H) such that,
for all x and y in H, we have :
< (AYα − Yα B)x, y >−→< (B ∗ X − XA∗ )x, y > .
So that,
0 =< (AYα − Yα B)f, g >−→< (B ∗ X − XA∗ )f, g >= −kgk2 .
U
It follows that g = 0 ; this proves that B ∗ X − XA∗ ∈
/ R(δAB ) . Consequently we obtain
that (A, B) is not D-symmetric by theorem 8.1. ♦
26
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≥1 as follows :
(
Sn (ek ) =
1
e,
n 2
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
C(A, B) and I(A, B) are C ∗ −algebras, C(A, B) is a two-sided ideal of I(A, B). ♦
27
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, then C(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 δZ (A) = δZ (B) = 0. It follows from lemma 9.1
that Z ∈ I(A, B). ♦
R EFERENCES .
[1] J. H. A NDERSON, J. W. B UNCE, J. A. D EDDENS and J. P. W ILLIAMS , C*-algebras
and derivation ranges, Acta Sci. Math. (Szeged), 40(1978), 211-227.
[2] J. H. A NDERSON and C. F OIAS, Properties which normal operators share with normal derivation and related operators, Pacific J. Math., 61(1976), 313-325.
[3] S. B OUALI, et J. C HARLES , Extension de la notion d’opérateurs d-symétriques I, Acta
Sci. Math. (Szeged), 58(1993), 517-525.
[4] S. B OUALI, et J. C HARLES , Extension de la notion d’opérateurs d-symétriques II,
Linear Algebra And Its Applications, 225(1995), 175-185.
[5] D. A. H ERRERO , Approximation of Hilbert space operators. I, Pitman, Advanced publishing program, Boston - Melbourne 1982.
28
[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. S TAMPFLI , On self-adjoint derivation ranges, Pacific J. Math., 82(1979), 257-277.
[9] J. G. S TAMPFLI , The norm of a derivation, Pacific J. Math., 33(1970), 737-747.
[10] J. P. W ILLIAMS , Derivation ranges : Open problems, Topics in Modern Operator
Theory, Birkhauser-Verlag, 1981, pp. 319-328.
[11] J. P. W ILLIAMS , On the range of a derivation, Pacific J. Math., 38(1971), 273-279.
29
Chapter 4
Generalized D-symmetric Operators II*
Abstract.
3
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 δAB associated 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 operators A, B ∈ L(H) that have R(δAB ) = R(δA∗ B ∗ ), where R(δAB ) 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).
30
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.
U
4. For A, B ∈ L(H), R(δAB ) 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 M o .
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
!
!
0 X
0 Xn
M=
,
Mn =
,
0 0
0 0
T =
and
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 ⊂
k.k
L(H ⊕ H), such that δT ∗ (Nn ) −→ M . A simple calculation proves that there exists a
k.k
sequence (Yn )n ⊂ L(H), such that δA∗ B ∗ (Yn ) −→ 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
and B are D-symmetric operators by [4; Th 2.5]. Since σ(A) ∩ σ(B) = ∅, Theorem 11.1
implies that (A, B) is D*-symmetric.
31
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)δB ∗ (B) ⊆ R(δAB ) ∩ R(δA∗ B ∗ ) ;
(3). A∗ R(δAB ) + R(δAB )B ∗ ⊆ R(δAB ) and AR(δA∗ B ∗ ) + R(δA∗ B ∗ )B ⊆ R(δA∗ B ∗ ).
Proof. (1) ⇒ (2). For all X ∈ L(H) we have :
δA∗ (A)X = δA∗ B ∗ (AX) − AδA∗ B ∗ (X) and XδB ∗ (B) = δAB (X)B ∗ − δAB (XB ∗ ).
Since AR(δA∗ B ∗ ) ⊆ AR(δAB ) ⊆ R(δAB ) and R(δAB )B ∗ ⊆ R(δA∗ B ∗ )B ∗ ⊆ R(δAB ), it
follows that
δA∗ (A)L(H) + L(H)δB ∗ (B) ⊆ R(δAB ).
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 δB ∗ (B)
and
AδA∗ B ∗ (X) + δA∗ B ∗ (Y )B = δA∗ B ∗ (AX + Y B) − δA∗ (A)X − Y δB ∗ (B).
(3) ⇒ (1). Suppose that (3) holds. Then A∗n R(δAB ) ⊆ R(δAB ) for each n in IN . We
always have the inclusion Am R(δAB ) ⊆ R(δAB ) for each m in IN .
We shall prove that R(δAB )o = R(δA∗ B ∗ )o . Let f ∈ R(δAB )o and X ∈ L(H). Observe
that
A∗n AX − AA∗n X = A∗n δAB (X) − δAB (A∗n X)
for each n in IN . Hence A∗n AX −AA∗n X ∈ R(δAB ) for each n in IN . A similar argument
using mathematical induction on m shows that A∗n Am X − Am A∗n X ∈ R(δAB ) for each
n and m in IN . Thus f (A∗n Am X) = f (Am A∗n X) for each n and m in IN . It follows that
∗
∗
f (eαA eβA X) = f (eβA eαA X)
for all complex numbers α and β.
An induction argument shows that
f ((αA + βA∗ )n X) =
n
X
n
k=0
k
!
f ((αA)k (βA∗ )n−k X)
for each n in IN and for all complex numbers α and β. Hence
∗
∗
∗
f (eαA+βA X) = f (eαA eβA X) = f (eβA eαA X)
32
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 (XeαB eβB ) = 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 (XeαB ) or f (eαA Xe−αB ) = f (X) for all α ∈ C
I and X ∈ L(H).
These relations yield, for all λ ∈ C,
I the equations
∗
∗
∗
∗
f (eıλA Xe−ıλB ) = f (eıλA eıλA Xe−ıλB e−ıλB )
∗
= f (eı(λA+λA ) Xe−ı(λB
∗ +λB)
).
Define the function g on C
I 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 ∈ C,
I
|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) = δU V (−U ∗ XV ∗ ) − P XV ∗ .
Hence, to prove that R(δU ∗ V ∗ ) ⊆ R(δU V ), it suffices to show that P X ∈ R(δU V ) for all
X ∈ L(H). Let
n−1
X
k
Tn =
( − 1)U k P XV ∗k+1 , n ∈ IN ∗ ,
n
k=0
where IN ∗ = IN \{0}. A simple calculation shows that
n
1X k
δU V (Tn ) − P X = −
U P XV ∗k .
n k=1
33
Since < U j P x, U k P y >= 0 for j 6= k and x, y in H, then
(1)
k
n
X
∗k
k
2
U P XV xk =
k=1
n
X
kU k P XV ∗k xk2 ≤ nkP Xk2 kxk2 .
k=1
1
Thus kδU V (Tn ) − P Xk ≤ n− 2 kP Xk, that is, P X ∈ R(δU 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
( − 1)U k+1 P XV ∗k , n ∈ IN ∗ .
n
k=0
A simple calculation shows that
n
δU ∗ V ∗ (Sn ) + P X =
1X k
U P XV ∗k .
n k=1
1
It follows from (1) that kδU ∗ V ∗ (Sn ) + P Xk ≤ n− 2 kP Xk. Thus P X ∈ R(δU ∗ V ∗ ). Consider
n−1
X
k
( − 1)U k+1 XQV ∗k , n ∈ IN ∗ ,
Rn =
n
k=0
Then
n
δU ∗ V ∗ (Rn ) + XQ =
1X k
U XQV ∗k .
n k=1
Hence
n
(δU ∗ V ∗ (Rn ) + XQ)∗ =
1X k
V QX ∗ U ∗k .
n k=1
1
Thus kδU ∗ V ∗ (Rn ) + XQk ≤ n− 2 kQX ∗ k, and so XQ ∈ R(δU ∗ V ∗ ). Since
U δU ∗ V ∗ (X) = δU ∗ V ∗ (U X) − P X and δU ∗ V ∗ (X)V = δU ∗ V ∗ (XV ) − XQ,
then
U R(δU ∗ V ∗ ) + R(δU ∗ V ∗ )V ⊆ R(δU ∗ 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
implies A∗ T = T B ∗ .
34
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
U
U
d. R(δAB ) = R(δA∗ B ∗ ) .
U
U
Proof. Note that R(δAB ) = R(δA∗ B ∗ ) 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).
U
U
This proves that R(δAB ) = R(δA∗ B ∗ ) 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)o and T ∈ ker (δBA )∩C1 (H)
( where fT (X) = tr(XT ) for each X in L(H) ). Since R(δAB )
U
U
= R(δA∗ B ∗ ) , it
follows that T ∈ ker (δB ∗ A∗ ) ∩ C1 (H). Let Z ∈ R(δA∗ 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 )n in L(H) and a sequence (Kn )n of compact operators in K(H) such that
AXn − Xn B + Kn −→ Z.
But
f0 (AXn − Xn B + Kn ) = f0 (AXn − Xn B) + f0 (Kn ) = 0,
and thus f0 (Z) = 0. It follows that f0 ∈ R(δA∗ B ∗ )o ∩ K(H)o , and hence f ∈ R(δA∗ B ∗ )o .
Therefore, R(δAB )o ⊆ R(δA∗ B ∗ )o . We obtain the reverse inclusion by a similar argument.
♦
Corollaire 11.3. If U and V are two isometries, then (U, V ) has the property (F, P )C1 .
35
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 ∈ C,
I 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 ∈ C,
I 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.
U
Suppose that A∗ X − XB ∗ ∈ R(δAB ) . Then there exists a net (Yα )α in L(H) such that
for all x and y in H, we have :
< (AYα − Yα B)x, y >−→< (A∗ X − XB ∗ )x, y > .
So that
0 =< (AYα − Yα B)f, g >−→< (A∗ X − XB ∗ )f, g >= −kgk2 .
U
U
It follows that g = 0. This proves that A∗ X − XB ∗ ∈
/ R(δAB ) , that is, R(δAB )
6=
U
R(δA∗ B ∗ ) . Consequently we obtain that (A, B) is not D*-symmetric by Theorem 11.3.♦
R EFERENCES .
[1] J. H. A NDERSON, J. W. B UNCE, J. A. D EDDENS and J. P. W ILLIAMS, C*-algebras
and derivation ranges, Acta Sci. Math. (Szeged), 40(1978), 211-227.
36
[2] J. H. A NDERSON and C. F OIAS, Properties which normal operators share with normal derivation and related operators, Pacific J. Math., 61(1976), 313-325.
[3] M. B ENLARBI, S. B OUALI and S. C HERKI, Une remarque sur l’orthogonalité de
l’image au noyau d’une dérivation généralisée, Proc. Amer. Math. Soc, 126(1998), 167171.
[4] S. B OUALI, et J. C HARLES, Extension de la notion d’opérateurs d-symétriques I, Acta
Sci. Math. (Szeged), 58(1993), 517-525.
[5] S. B OUALI, et J. C HARLES, Extension de la notion d’opérateurs d-symétriques II, Linear Algebra And Its Applications, 225(1995), 175-185.
[6] D. A. H ERRERO , Approximation of Hilbert space operators. I, Pitman, Advanced publishing program, Boston - Melbourne 1982.
[7] M. A. ROSENBLUM, On the operator equation BX − XA = Q, Duke Math. J.,
23(1956), 263-269.
[8] J. G. S TAMPFLI, On self-adjoint derivation ranges, Pacific J. Math., 82(1979), 257-277.
[9] J. P. W ILLIAMS, Derivation ranges : Open problems, Topics in Modern Operator
Theory, Birkhauser-Verlag, 1981, pp. 319-328.
[10] J. P. W ILLIAMS, On the range of a derivation, Pacific J. Math., 38(1971), 273-279.
37
Chapter 5
Generalized Numerical Range*
Abstract.
In this paper we will introduce the generalized numerical range Wg (A) of an operator
4
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 generalized 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, hyponormal operator.
* Sousmis au journal Extracta Mathematicae (Espagne).
38
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 C,
I 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
λ
kf k
∈ 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.
y
x
Proof. Let < Ax, x > and < Ay, y > two elements in Wg (A), x1 = kxk
, y1 = kyk
p
and α = tkxk2 + (1 − t)kyk2 where t ∈ [0, 1]. Then < Ax1 , 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 2
kyk 2
) < Ax1 , x1 > +(1 − t)(
) < Ay1 , y1 >=< Au, u > .
α
α
39
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
of A.
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. !
1 0
(3) If T =
, then a simple calculation shows that Wg (T ) = [0, 2] and W (T ) =
0 2
[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=
−r
1−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 ∈ C,
I such that |λ| = kAk. If λ ∈ Wg (A), then
the point spectrum of A is not empty.
40
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 ∈ C
I 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 kxn k ≤ 1
for all n, converging to λ. Since the unit ball is weakly compact, there exists a subsequence
(xnk )k which is weakly convergent to an x where kxk ≤ 1. Since A is a compact operator,
(A(xnk ))k is strongly convergent to Ax.
However,
| < Axnk , xnk > − < Ax, x > | ≤ kxnk kkAxnk − Axk + | < xnk , Ax > − < x, Ax > |.
Therefore (< Axnk , xnk >)k converge 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 ∈ C
I / |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 =
x
,
kxk
then
< Ay, y >∈ W (A) = co(σp (A)) [1, theorem 2].
Thus
λ = kxk2 < Ay, y > +(1 − kxk2 )0 ∈ co(σp (A) ∪ {0}).
Therefore, Wg (A) ⊂ co(σp (A){0}). ♦
41
Remarque 14.2.
It is clear that Wg (I) = [0, 1] = co(σp (I) ∪ {0}), but I is not compact. Thus the condition
A ∈ K(H) is not necessary.
15
Generalized Numerical Range of Derivation.
Théorème 15.1. If A ∈ L(H) such that kA − λk = ρ(A − λ), for all λ in C,
I then
Wog (A) = co(σ(A) ∪ {0}).
Proof. Let λ = f (A), where f ∈ Pg . If f = kf kg, we can write g(A) ∈ Wo (A) =
co(σ(A)) [3, p. 564]. Thus
λ = kf kg(A) + (1 − kf k)0 ∈ co(σ(A) ∪ {0}).
Therefore, Wog (A) ⊂ co(σ(A)∪{0}). Using corollary 13.1, we obtain Wog (A) = co(σ(A)∪
{0}). ♦
Corollaire 15.1. For a hyponormal operator A on H, Wog (A) = co(σ(A) ∪ {0}).
Proof. Using theorem 1 [6] we have kA − λk = ρ(A − λ), for all λ in C.
I It follows
from theorem 15.1 that Wog (A) = co(σ(A) ∪ {0}). ♦
Théorème 15.2. If A, B ∈ L(H) such that kA−λk = ρ(A−λ) and kB −λk = ρ(B −λ)
for all λ in C,
I then Wog (δAB ) = co(σ(δAB ) ∪ {0}).
Proof. Let λ = f (δAB ), where f ∈ Pg . If we write f = tg with t ∈ [0, 1] and kgk = 1,
then g(δAB ) ∈ Wo (δAB ) = co(σ(δAB )) [3, p. 565]. Thus
λ = tg(δAB ) + (1 − t)0 ∈ co(σ(δAB ) ∪ {0}).
Therefore, Wog (δAB ) ⊂ co(σ(δAB ) ∪ {0}). Since Wog (δAB ) is convex and contains the
spectrum of δAB , it follows that Wog (δAB ) = co(σ(δAB ) ∪ {0}). ♦
Corollaire 15.2. For hyponormal operators A and B on H, Wog (δAB ) = co(σ(δAB ) ∪
{0}).
Proof. Using theorem 1 [6] we have kA − λk = ρ(A − λ) and kB − λk = ρ(B − λ),
for all λ in C.
I It follows from theorem 15.2 that Wog (δAB ) = co(σ(δAB ) ∪ {0}). ♦
42
R EFERENCES .
[1] G. D. BARRA, J. R. G ILES and B. S IMS , On the numerical range of compact operators on Hilbert spaces. J. London Math. Soc. (2), 5(1972), 704-706.
[2] F. F. B ONSALL and J. D UCAN, Numerical ranges of operators on normed spaces and
elements of normed algebras. London Math. Soc. Lecture Note Series. Cambridge Univ.
Press. Cambridge. 1971.
[3] S. B OUALI, and J. C HARLES , Generalized derivation and numerical range, Acta Sci.
Math. (Szeged), 63(1997), 563-570.
[4] K. G USTAFSON and D. R AO, Numerical range, Springer 1996.on, N. J. (1967).
[5] P. R. H ALMOS, Hilbert space problem book, New York, 1970.
[6] J. G. S TAMPFLI , Hyponormal operators, Pacific J. Math., 12(1962), 1453-1458.
[7] J. G. S TAMPFLI and J. P. W ILLIAMS, Growth conditions and the numerical range in
a Banach algebra, Tohoku Math. J. 20(1968), 417-424.
UNIVERSITÉ MOHAMMED V – AGDAL
FACULTÉ DES SCIENCES
RABAT
Résumé
________________________________________________________________
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
)
(
R δ f ( A )f ( B ) = R (δ AB ) . On déduit l'orthogonalité de l'image au noyau de la dérivation
δ AB /C p si ( f ( A ) , f ( B ) )
admet la propriété de Fuglede- Putnam dans C p pour p
Au troisième chapitre on considère Les paires d'opérateurs
(A ,B )
telles que
> 1.
R (δ A B )
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
W g ( A ) = {< Ax , x >, x ≤ 1} .
Mots-clefs :
Putnam,
compact.
Dérivation Généralisée, Orthogonalité Image Noyau, Propriété de FugledeOpérateur D-symétrique, Opérateur Normal, Image Numérique, Opérateur
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