On the Range and the Kernel of Elementary Operators.

Transcription

UNIVERSITÉ MOHAMMED V – AGDAL
FACULTÉ DES SCIENCES
RABAT
N° d’ordre : 2450
THÈSE DE DOCTORAT
Présentée par :
BOUHAFSI YOUSSEF
Discipline : Mathématiques.
Spécialité : Analyse Fonctionnelle.
Titre
:
On the Range and the Kernel of Elementary Operators.
Soutenue le Vendredi 5 Juin 2009 ,
Devant le Jury :
Président :
INTISSAR AHMED, P.E.S., Faculté des Sciences de Rabat.
Examinateurs :
BOUALI SAID, P.E.S., Faculté des Sciences de Kénitra.
ZEROUALI EL HASSAN, P.E.S., Faculté des Sciences de Rabat.
BOUSSEJRA ABDELHAMID, P.E.S., Faculté des Sciences de Kénitra.
ELKHADIRI ABDELHAFED, P.E.S., Faculté des Sciences de Kénitra.
BENLARBI DELAI M’HAMMED, P.E.S., Faculté des Sciences de 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 mémoire de thèse 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.
Ce travail a été effectué sous la direction du Professeur Said Bouali, de la
Faculté des Sciences de Kénitra. Sa disponibilité, son soutien, ses encouragements et sa patience m’ont permis d’achever ce travail. Je tiens de lui
exprimer ma reconnaissance et ma profonde gratitude.
Je lui suis très reconnaissant à la fois pour la qualité de sujet de recherche
et pour l’efficacité exceptionnelle de son encadrement. Je n’oubli jamais
l’attention sans faille qu’il a su porter à cette thèse au cours de son élaboration, toujours avec une grande humanité. Je le remercie sincèrement pour
ses commentaires trés pertinents.
J’adresse mes sincères remerciements à Monsieur le Professeur El Hassan
Zerouali, de la Faculté des Sciences de Rabat pour l’intérêt avec lequel il a
suivi mon 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.
Je voudrais exprimer mes vifs remerciements à Monsieur le Professeur Ahmed
Intissar, de s’être intéressé à mon travail, et pour l’honneur qu’il me fait en
acceptant de présider le Jury de cette thèse. Je voudrais le remercier pour
la qualité de ses suggestions et de son écoute. Qu’il trouve ici l’expression
de ma gratitude.
Je tiens à remercier chaleureusement Monsieur le Professeur Abdelhamid
Boussejra de la Faculté des Sciences de Kénitra d’avoir accepté de rapporter les résultats 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
trés attentive du manuscrit.
Je remercie vivement Monsieur le Professeur Abdelhafid Elkhadiri de la Faculté des Sciences de Kénitra d’avoir examiné ce travail et me faire partager
son grand intérêt pour la recherche. Qu’il trouve ici l’expression de ma profonde reconnaissance.
Je tiens à remercier Monsieur le Professeur M’Hammed Benlarbi Delai de
la de la Faculté des Sciences de Rabat pour l’intérêt qu’il a porté à mon
travail et pour l’honneur qu’il me fait en acceptant de participer dans le
Jury de cette thèse. Je le remercie aussi pour ses suggestions.
Les différentes et stimulantes discussions que j’eues avec Monsieur le Professeur Bhagwati Prashad Duggal m’ont beaucoup apporté et sa contribution dans ce travail est certaine. Je voudrais lui adresser mes plus vifs
remerciements.
Je suis trés reconnaissant à Monsieurs les Professeurs Omar El Fallah de la
Faculté des Sciences de Rabat et Mostafa Mbekhta de l’Université de Lille
I, pour leurs soutien et encouragement permanant.
Je doit beaucoup remercier tous les membres de, groupe d’Analyse Fontionnelle de la Faculté des Sciences de Rabat, groupe d’Analyse Fonctionnelle de
la Faculté des Sciences de Kénitra et le groupe des Équations aux dérivées
partielles et Géométrie Spectrale et l’Analyse Harmonique de la Faculté des
Sciences de Rabat, dont les travaux sont une source d’inspiration toujours
renouvlée, ayant largement participé à mon éveil scientifique.
Je suis heureux de remercier particulièrement les Professeurs, B. Aqzzouz,
S. Asserda, T. Belghiti, M. Yahyai et A. Kondah de la Faculté des Sciences
de Kénitra, M. Elkadiri de la Faculté des Sciences de Rabat, A. Jellal de la
Faculté des Sciences d’El Jadida , B. Magajna et A. Turnšek de l’Université
de Ljubljana Slovenia, Lawrence A. Fialkow de l’Université de New York,
M. Mathieu de l’Université de Belfast Irlande, T. Andô de l’Université de
Hokusei Gakuen Japon, P. Rosenthal de l’Université de Toronto Canada,
Pei Yuan Wu de l’Université de Chiao-Tung Taiwan, Helena King of the
Academy’s Mathematical proccedings, C. Ray Rosentrater de Westmont
College California, pour la documentation, le soutien et l’encouragement.
Un grand merci à Mme Rabiaa la Secrétaire de Département de Mathématiques et Informatique, Mme Meryem la Secrétaire de Vice Doyen, Mme
Nabila la Secrétaire de 3ème cycle de la Faculté des Sciences de Rabat ainsi
que Mme Laaziza, et Mme Amina la Secrétaire de Département de Mathématiques et Informatique de la Faculté des Sciences de Kénitra.
Toute ma gratitude va également à mes chers amis H. Amal, A. Ghanmi,
A. Hajji, N. Aboudi, M. Ech-had et K. Elhachimi de la Faculté des Sciences
de Rabat, A. Srhir, A. Essadiq, K. Hajioui, R. Nouira, M. Akkach, et L.
Zraoula de la Faculté des Sciences de Kénitra, M. Bertit et T. Benbouziane
pour l’affectueuse amitié dont ils ont toujours fait preuve. Je remercie aussi
les personnels de Lycée Moulay Youssef de Taroudant pour le soutien.
Je ne pourrais jamais oublier le soutien et l’aide de ma merveilleuse famille.
Je réserve une reconnaissance particulière et un remerciement chaleureux
à ma mère, mes soeurs et mes frères. Je remercie également la famille de
Lagbouri et la famille de Goupillière pour le soutien et l’encouragement.
À la mémoire de mon père
Abstract
The first chapter is essentially a survey and synthesis of what is known
about the properties of P-Symmetric operators and Finite operators.
In the second chapter, we establish the orthogonality of the range and the
kernel of a derivation δA induced by a cyclic subnormal operator A, in
the usual operator norm. We provide another proof of a principal result of
F.Wening and J.Guo Xing. We give a characterization of the class of PSymmetric operators. We characterize also operators A such that the pair
(A, A) satisfy the Putnam-Fuglede property in Cp (H), where Cp (H) denotes
the Von Newmann-Schatten class for p > 1.
In the third chapter, we wish to consider the class of Finite operators. We
use new techniques and approachs to generalize and develop some properties
of Finite operators.
In the following chapter, we give some properties concerning the class of PSymmetric operators. We turn our attention to commutant and derivation
ranges. We obtain the new results concerning the intersection of the kernel
and the closure of the range of an inner derivation. We obtain new classes
of operators A such that I 6∈ R(δA ), where R(δA ) is the norm closure of the
range of δA , (δA (X) = AX − XA).
The last chapter represents some properties which enjoy the range of an elementary operator. We initiate the study of the class of Quasi-adjoint operators, i.e. operators A for which R(∆A ) = R(∆A∗ ), where R(∆A ) denotes the
norm closure of the range of the elementary operator ∆A (X) = AXA − X.
We give a characterization and some basic properties concerning this class
of operators.
Contents
Introduction
1 Preliminaries
1.1 P-symmetric operators
1.2 Finite operators . . . .
1.3 The essential spectrum
1.4 The Riesz idempotent
2
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On the range kernel orthogonality and P-symmetric
ators
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . .
2.2 The range-kernel orthogonality . . . . . . . . . . . . .
2.3 P-symmetric operators . . . . . . . . . . . . . . . . . .
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9
9
14
16
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oper. . .
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22
22
24
29
3 On the range and the kernel of derivations
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . .
34
34
35
4 The P-symmetric operators and the range of a subnormal
derivation
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 P-symmetric operators . . . . . . . . . . . . . . . . . . . . .
4.3 The range of a subnormal derivation . . . . . . . . . . . . .
42
42
44
47
5 On
5.1
5.2
5.3
the range of the elementary operator X 7−→ AXA − X
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . .
The range of the elementary operator ∆A,B . . . . . . . . .
Quasi-adjoint operators . . . . . . . . . . . . . . . . . . . . .
5
52
52
54
56
Introduction
Soient H un espace de Hilbert complexe et L(H) l’algèbre de Banach des
opérateurs linéaires bornés sur H, L(H) est munie de la norme usuelle
d’opérateurs k.k. Pour A, B ∈ L(H), on définit la dérivation généralisée
δA,B sur L(H) par δA,B (X) = AX − XB, et l’opérateur élémentaire ∆A,B
par ∆A,B (X) = AXB −X, notons simplement que δA,A = δA et ∆A,A = ∆A .
L’opérateur δA est une dérivation intérieure sur L(H), et de manière assez
remarquable, toutes les dérivations sur L(H) sont de cette forme (voir [20],
[21] et [31]). Les propriétés des dérivations intérieures, leur spectre [25],
norme [33] et image [19],[34] et [40] ont été examinés minutieusement.
L’étude des opérateurs élémentaires δA,B et ∆A,B a engendré de nombreux
travaux, certaines de ces propriétés métriques et spectrales sont établies et
largement développées ces dernières années ([10],[12],[13],[15],[26],[28]), et
plusieurs problèmes concernant leurs images restent encore sans réponses[13].
Il est prouvé par J.H.Anderson, J.W.Bunce, J.A. Deddens et J.P. Williams
[3], que si A est un opérateur D-symétrique i.e. R(δA ) = R(δA∗ ), avec R(δA )
est la fermeture en norme de l’image R(δA ) de δA dans L(H), alors AT = T A
implique AT ∗ = T ∗ A pour tout opérateur T ∈ C1 (H), où C1 (H) désigne
l’ensemble des opérateurs de classe trace, tel opérateur A est appelé Psymétrique. La classe des opérateurs P-symétriques est introduite par S.
Bouali et J.Charles dans [5] et [6].
Vu son importance tout au long de ce travail on a tenu à présenter quelques
propriétés de base concernant les opérateurs P-symétriques. On a aussi
donné quelques propriétés fondamentales relatives aux opérateurs finis, i.e.
les opérateurs A ∈ L(H) tel que dist(I, R(δA )) = 1. La notion d’opérateur
fini est introduite par J.P. Williams [39], largement développée depuis par
de nombreux auteurs.
On a précisé le strict nécessaire relatif aux propriétés fondamentales dont
1
Introduction
2
jouit le spectre essentiel d’un opérateur et l’idempotent de Riesz qui nous
serons utiles pour la suite de notre travail. Ceci fait l’objet du premier
chapitre.
Dans le second chapitre on s’est intéressé à l’étude de l’orthogonalité de
l’image au noyau de la dérivation δA et à la classe des opérateurs P-symétriques.
J.Anderson [1] a montré que si A est un opérateur normal alors pour tout
T ∈ ker(δA ) et pour tout X ∈ L(H) on a
kδA (X) + T k ≥ kT k.
Ceci signifie l’orthogonalité de l’image R(δA ) au noyau ker(δA ) de la dérivation δA au sens de l’espace de Banach L(H) ( voir [4]). Ces résultats
seront généralisés par J.Anderson et C.Foias [2] à une dérivation généralisée δA,B associé à deux opérateurs normaux A et B. Depuis, plusieurs auteurs se sont intéressés à l’étude de l’orthogonalité de l’image au noyau des
opérateurs élémentaires δA,B et ∆A,B dans L(H), pour plus de détail citons
[8],[9],[17],[18],[22],[23],[24],[27],[36] et [37].
Dans la première section de ce chapitre nous montrons que si A est un
opérateur sous-normal cyclique, alors pour tout T ∈ ker(δA ) et pour tout
X ∈ L(H) on a kδA (X) + T k ≥ kT k. Ainsi, on a obtenu l’orthogonalité de
l’image au noyau de la dérivation δA induite par un opérateur sous-normal
cyclique A au sens de la norme usuelle d’opérateurs. Nous donnons une
condition suffisante à un opérateur sous-normal A pour que R(δA ) soit orthogonal au ker(δA ). Puis on en a tiré quelques résultats.
Soit F un idéal bilatère de L(H), nous dirons que la paire d’opérateurs
(A, B) admet la propriété (F P )F si AT = T B et T ∈ F implique A∗ T =
T B ∗ [7]. Dans la deuxième section on se propose de montrer que l’ensemble
Σ(F) = {A ∈ L(H) : (A, A) admet la propriété (F P )F }
n’est pas fermé pour la norme dans L(H) pour tout idéal bilatère F de L(H).
Nous donnons une caractérisation des opérateurs A tel que la paire (A, A)
admet la propriété (F P )Cp (H) pour p > 1, avec Cp (H) est la classe de Von
Neumann-Schatten. Comme conséquence nous obtenons une autre preuve
élémentaire et directe du résultat principal de F.Wening et J.Guo Xing [11].
On en déduit une autre caractérisation des opérateurs P-symétriques.
Introduction
3
Au troisième chapitre, on considère la classe des opérateurs finis [39], c’est
à dire les opérateurs A ∈ L(H) vérifiant
(∗∗)
kδA (X) + Ik ≥ 1,
pour tout X ∈ L(H). Dans un premier temps, on a montré que si A est un
opérateur n-multicyclique hyponormal et T est un opérateur hyponormal
tel que AT = T A alors pour tout X ∈ L(H) on a
kδA (X) + T k ≥ kT k.
Comme application on a déduit la même inégalité si A est un opérateur
quasi-normal. Dans un autre temps, on a tenu à donner une généralisation naturelle de l’inégalité (∗∗). En utilisant le théorème d’extension de
Berberian [41], on a montré aussi que si A est un opérateur fini et T est
un opérateur normal dans le commutant de A alors kδA (X) + T k ≥ kT k,
pour tout X ∈ L(H). Ayant adopté des démarches différentes et simples on
a établi certains résultats rencontrés dans la littérature.
Au chapitre suivant, nous nous intéressons à l’étude de la classe des opérateurs P-symétriques et à l’intersection des commutants et des fermetures
faibles et en norme des images de dérivations.
En première partie, nous considérons la classe des opérateurs P-symétriques,
nous donnons quelques propriétés concernant cette classe. On a montré que
si A est un opérateur algébrique alors tout opérateur P-symétrique dans
W
W
0
R(δA ) ∩{A} est nul, avec R(δA ) est la fermeture de R(δA ) pour la topolo0
gie faible et {A} est le commutant de A. Nous appliquons les propriétés des
opérateurs P-symétriques pour étudier les ensembles C◦ (A), I◦ (A) et B◦ (A)
introduit dans [6]. Puis nous donnons une description de ces ensembles pour
certaines classes d’opérateurs. On a obtenu une caractérisation des opérateurs P-symétriques.
En seconde partie, on a présenté une nouvelle classe d’opérateurs A ∈ L(H),
0
satisfaisant R(δA ) ∩ {A} = {0}. Ainsi, on a montré que si A est un opéra0
teur sous-normal cyclique alors R(δA )∩{A} = {0}. Comme conséquence on
a prouvé que si P (A) est est normal, isométrique, co-isométrique ou sousnormal cyclique pour un certain polynôme P , alors tout opérateur dans
0
R(δA ) ∩ {A} est nilpotent. Nous trouvons également de nouvelles classes
d’opérateurs A ∈ L(H) tel que I 6∈ R(δA ). Ce dernier résultat généralise
Introduction
4
celui de J.A. Stampfli dans [34]. Le Théorème de Weber [38] affirme que tout
W
W
0
opérateur compact dans R(δA ) ∩ {A} est quasi-nilpotent, où R(δA ) est
la fermeture faible de R(δA ). On a obtenu une nouvelle version de théorème
de Weber, ainsi on a montré que si A est un opérateur normal, isométrie,
co-isométrie ou sous-normal cyclique alors tout opérateur compact dans
W
0
R(δA ) ∩ {A} est nul.
Dans le dernier chapitre on a présenté des propriétés de l’image de l’opérateur
élémentaire ∆A,B , puis on a introduit la notion de classe d’opérateurs quasiadjoints (i.e. opérateur A ∈ L(H) pour lequel R(∆A ) = R(∆A∗ ), avec
R(∆A ) est la fermeture de R(∆A ) relative à la topologie de la norme.).
Dans [16] Z. Genkai a caractérisé les opérateurs A, B ∈ L(H) pour lesquels
R(∆A,B ), l’image de ∆A,B , est dense dans L(H) pour la topologie de la
norme. Dans le premier paragraphe on s’est intéressé à l’image des opérateurs élémentaires. On a déduit des propriétés dont jouit l’image de l’opérateur
∆A,B . On a caractérisé les opérateurs A, B ∈ L(H) tels que R(∆A,B ) est
dense dans L(H) pour la topologie faible et ultra-faible des opérateurs.
Dans le deuxième paragraphe on va initier l’étude sur la classe d’opérateurs
quasi-adjoints. A ces opérateurs on a donné une caractérisation. On a tenu à
démontrer quelques propriétés de base concernant cette classe d’opérateurs.
Les notations sont précisées au cours de chaque chapitre et chaque chapitre
possède sa propre bibliographie.
Notations and definitions
(1) Let H be a complex separable Hilbert space, and let L(H) denote the
algebra of all bounded linear operators on H into itself. The range and
the kernel of an operator A ∈ L(H), are denoted by R(A) and ker(A)
respectively.
(2) Norm topology (Uniform topology): a sequence of operators (An )n in
L(H) , converge uniformly to A ∈ L(H) if limn−→+∞ kAn k = 0, where
kAk = sup{kAxk : x ∈ H, kxk = 1}. If E is a subspace of L(H), we will
denote the norm closure of E by E.
(3) Weak operator topology: a generalized sequence of operators (Aα )α in
W
L(H), converges weakly to 0, denoted by Aα −→ 0 weakly, if and only if,
α
< Aα x, y >−→ 0 for all x, y ∈ H. Let E be a subspace of L(H), we denote
w
by E the closure of E in the weak operator topology.
(4) Ultra-weak operator topology: Let C1 (H) be the ideal of trace class
operators. Given n elements T1 , T2 , · · · Tn of C1 (H) and ε is non-negative
real number, a base of neighborhoods of zero is collection of the following
sets
Wε,T1 ,T2 ,···Tn = {X ∈ L(H) : |tr(Ti X)| < ε, i = 1, 2, · · · n}.
If E is a subspace of L(H), then E
w∗
denote the ultra-weak closure of E.
(5) An operator A ∈ L(H) has finite rank if R(A) is finite dimensional. The
ideal of continuous finite rank operators is denoted by B(H).
(6) An operator A ∈ L(H) is said to be compact if < Axn , xn >−→ 0, for
every orthonormal sequence (xn )n in H. The ideal of all compact operators
is denoted by K(H).
(7) Let A ∈ L(H) be compact, and let s1 (A) ≥ s2 (A) ≥ · · · ≥ 0 denote the
1
singular values of A, i.e. the eigenvalues of |A| = (A∗ A) 2 arranged in their
decreasing order. The operator A is said to belong to the Schatten p-class
5
6
Cp (H) if
kAkp =
∞
X
sj (A)p
p1
1
= tr(A)p p < ∞ ,
1 ≤ p < ∞,
j=1
where tr denotes the trace functional. Hence, we denotes C1 (H) the trace
class and C2 (H) the Hilbert-Schmidt class. Hence,
(i) A is said to be of trace class if kAk1 =
P∞
n=1
< |A|en , en >< ∞.
(ii) A is called a Hilbert-Schmidt operator if kAk2 =
where (en )n is an orthonormal basis for H.
P∞
n=1
kAen k2
12
< ∞,
(8) Let A ∈ L(H) and E be subspace of H. we say that E is an invariant
subspace for A, if Ah ∈ E whenever h ∈ E. In other words, AE ⊂ E.
We say that E is a reducing subspace for A if AE ⊂ E and AE ⊥ ⊂ E ⊥ .
(9) Let A ∈ L(H) the spectrum of A, denoted by σ(A), is defined by
σ(A) = {λ ∈ C : A − λI is not invertible}.
(10) If A ∈ L(H) the point spectrum of A, denoted by σp (A), is defined by
σp (A) = {λ ∈ C : ker(A − λI) 6= {0}}.
(11) The approximate point spectrum of A, denoted by σap (A), is defined
by
σap (A) = {λ ∈ C : ∃(xn )n in H such that kxn k = 1 and k(A−λI)xn k −→ 0}.
(12) The reducing point spectrum of A ∈ L(H), denoted by σpr (A), is
defined by
σpr (A) = {λ ∈ C : ∃x 6= 0 such that Ax = λx and A∗ x = λx}.
(13) The approximate reducing spectrum of A, denoted by σar (A), is the
set of scalars λ ∈ C for which there exists a sequence (xn )n of unit vectors
in H, such that
(A − λ)xn −→ 0,
and (A − λ)∗ xn −→ 0.
(14) The spectral radius of an operator A ∈ L(H), in symbols r(A), is
defined by
r(A) = sup{|λ| : λ ∈ σ(A)}.
7
0
(15) Let A ∈ L(H), the commutant of A, {A} is defined by
0
{A} = {B ∈ L(H) : AB = BA}
00
(16) Let A ∈ L(H), the bicommutant of A, {A} is defined by
00
0
{A} = {C ∈ L(H) : CB = BC for all B ∈ {A} }
(17) C ∈ L(H) is a commutator if there exists A, B ∈ L(H) such that
C = AB − BA.
(18) Let A ∈ L(H), recall that the operator A is said to be:
(i) diagonalizable if there exists an orthonormal basis for H consisting of
all eigenvectors of A.
(ii) algebraic operator if P (A) = 0 for some non trivial polynomial P .
(iii) nilpotent of order k if Ak = 0 and Ak−1 6= 0 for the positive integer k.
(iv) quasinilpotent if σ(A) = {0}, where σ(A) is the spectrum of A.
(v) positive if < Ax, x > ≥ 0 for all x ∈ H, in symbols this is denoted by
A ≥ 0.
(vi) self-adjoint if A∗ = A, where A∗ is the the operator adjoint of A.
(vii) normal if AA∗ = A∗ A.
(viii) unitary if AA∗ = A∗ A = I.
(ix) quasi-normal if A(A∗ A) = (A∗ A)A.
(x) subnormal if it has a normal extension. More precisely, an operator
A ∈ L(H) is subnormal if there exists a normal operator B on a Hilbert
space K such that H is a subspace of K, the subspace H is invariant under
the operator B and the restriction of B to H coincides with A.
(xi) isometry if A∗ A = I.
(xii) partial isometry if A|(ker A)⊥ is an isometry.
(xiii) co-isometry if A∗ is an isometry.
(ixv) hyponormal if A∗ A − AA∗ is a positive operator.
(xv) p-hyponormal, 0 < p ≤ 1, if |A∗ |2p ≤ |A|2p .
(xvi) dominant operator if to each complex number λ there corresponds a
real number Mλ such that
k(A − λ)∗ xk ≤ Mλ k(A − λ)xk,
8
for all x ∈ H. Or, equivalently R[(A − λ)] ⊂ R[(A − λ)∗ ] for all λ ∈ σ(A).
(xvii) M -hyponormal If A is a dominant operator for which there exists a
constant M such that Mλ ≤ M for all λ.
(iixx) k-quasihyponormal, k ≥ 1, some integer, if kA∗ k Axk ≤ kAk+1 xk, for
all x ∈ H.
(ixx) normaloid if r(A) = kAk.
(xx) contraction if kAk ≤ 1.
(19) Let H1 and H2 be both two Hilbert spaces. A ∈ L(H1 ) and B ∈ L(H2 )
are called unitarily equivalent, if there exists a linear unitary map U of H1
into H2 such that A = U ∗ BU .
(20) Every operator A ∈ L(H) admits a unique decomposition, called the
polar decomposition of A defined by A = U P , where U is a partial isometry
and P is a positive operator with ker(U ) = ker(P ).
(21) Let A be a complex Banach algebra with identity . A linear mapping
δ : A −→ A is a derivation if δ(xy) = δ(x)y + xδ(y) for all x, y ∈ A.
(22) The inner derivation induced by an element A ∈ L(H) is the map δA
defined on L(H) by δA (X) = AX − XA.
(23) For operators A and B in L(H), define the generalized derivation δA,B
by
δA,B : L(H) −→ L(H)
X 7−→
δA,B (X) = AX − XB.
(24) Given operators A, B in L(H), the elementary operator ∆A,B is defined
as follows:
∆A : L(H) −→ L(H)
X 7−→
∆A,B (X) = AXB − X.
When A = B, we simply write ∆A for ∆A,A .
Chapter 1
Preliminaries
1.1
P-symmetric operators
Let H be a complex Hilbert space and let L(H) denote the algebra of
all bounded linear operators on H into itself. For A ∈ L(H), the inner
derivation induced by A is the mapping δA defined as follows:
δA : L(H) −→ L(H)
X 7−→
δA (X) = AX − XA.
The operator A ∈ L(H), is said to be D-symmetric if R(δA ) = R(δA∗ ),
where R(δA ) is the closure of the range R(δA ), of δA in the norm topology.
Obviously, A is D-symmetric if and only if R(δA ) is a self-adjoint subspace
of L(H). Examples of D-symmetric operators includes isometries, normal
and cyclic subnormal operators. The properties of D-symmetric operators
have been carried out in a number of papers (see [3],[29],[30] and [35]).
In this section, we give characterizations and some basic properties of the
class of P-symmetric operators, that is, operators A ∈ L(H) that have the
following property: AT = T A and T ∈ C1 (H) implies AT ∗ = T ∗ A.
In addition to the notation introduced already, we shall use the following
further notation.
(1) We shall denote by K(H) the ideal of all compact operators
on H, B(H)
be the class of all finite rank operators and let C(H) = L(H)K(H) be the
Calkin algebra. Let π denote the canonical homomorphism from L(H) into
the Calkin algebra C(H).
9
Chapter 1: Preliminaries
10
(2) Given A, B ∈ L(H), let δA,B : L(H) −→ L(H) denote the generalized
derivation δA,B (X) = AX − XB. We simply write δA for δA,A .
(3) we shall denote R(δA,B ) the range of the generalized derivation δA,B and
ker(δA,B ) the kernel of δA,B .
W
Let R(δA,B ) be the norm closure, R(δA,B )
will denote the weak closure,
W∗
and R(δA,B ) denote the ultra-weak closure of the range R(δA,B ).
(4) Let C1 (H) be the ideal of trace class operators. The ideal C1 (H) admits
a complex valued function tr(T ) which has the characteristic P
properties of
the trace of matrices. The trace function is defined by tr(T ) = n (T en , en ),
where (en ) is any complete orthonormal system in H.
As a Banach spaces, C1 (H) may be identified with the conjugate space of the
ideal K(H) of compact operators by means of the linear isometry T 7−→ fT ,
where fT (X) = tr(XT ). Moreover, L(H) is the dual of C1 (H). The ultraweak continuous linear functionals on L(H) are those of the form fT for
some T ∈ C1 (H), and the weak continuous linear functionals on L(H) are
those of the form fT where T ∈ B(H). If ϕ is a linear functional on L(H),
then ϕ∗ the adjoint of ϕ is defined by ϕ∗ (X) = ϕ(X ∗ ) for all X ∈ L(H).
Properties of P-symmetric operators
Definition 1.1.1. Let A ∈ L(H), the operator A is called D-symmetric if
R(δA ) = R(δA∗ ). We denote the class of D-symmetric operators by D(H).
Theorem 1.1.2. [3] If A ∈ L(H), then the following statements are equivalent
(1) A is D-symmetric.
(2) (i) [A], the corresponding element of the Calkin algebra is D-symmetric.
(ii) AT = T A implies AT ∗ = T ∗ A for all T ∈ C1 (H).
Definition 1.1.3. Let A ∈ L(H). If AT = T A implies AT ∗ = T ∗ A for
all T ∈ C1 (H), we say that A is P-symmetric. The set of P-symmetric
operators is denoted by P (H).
Theorem 1.1.4. [5] Let A ∈ L(H), then
W∗
(1) A is P-symmetric if and only if R(δA )
is self-adjoint.
(2) P (H) the set of P-symmetric operators is self-adjoint.
Lemma 1.1.5. [5] Let A ∈ L(H), have the property that there is some
λ ∈ C so that
(1) There exists x ∈ H, x =
6 0 with Ax = λx and A∗ x 6= λx.
11
(2) There exists y ∈ H, y 6= 0 with A∗ y = λy.
W∗
Then R(δA )
fails to be self-adjoint.
Example 1.1.6. Let (en )n≥1 be an orthonormal
basis of H. Let H◦ =
1 0
vect{e1 , e2 } and define A◦ =
∈ L(H◦ ). Consider the operator
1
1
A◦ 0
A=
on H = H◦ ⊕ H◦⊥ . Clearly, we have Ae2 = e2 , A∗ e2 6= e2
0 I
and Ae1 = e1 . It follows from the above Lemma that A is not P-symmetric.
The following Theorem is a slight generalization of the preceding Lemma,
we include it for completeness.
Theorem 1.1.7. Let A, B ∈ L(H), have the property that there is some
λ ∈ C so that
(1) There exists x ∈ H, x 6= 0 with Ax = λx, A∗ x 6= λx and By = λy.
(2) There exists y ∈ H, y 6= 0 with A∗ y = λy.
W∗
Then R(δA,B )
fails to be self-adjoint.
Theorem 1.1.8. [5] Let A ∈ P (H). Then the following statements are
equivalent
0
(1) {A} ∩ C1 (H) 6= {0}.
(2) A ∈ ∪n≥1 Rn .
(3) σpr (A) 6= φ.
W
(4) K(H) 6⊂ R(δA ) .
Example 1.1.9. Let (en )n≥1 be orthonormal basis for H. Define the operator S as follows:
2,
k odd
Sek = ωk ek+1 , where ωk =
−1
2 , k even
Then S satisfies Σ∞
k=1 ωk ωk+1 · · · ωk+n = ∞ for all n ≥ 1, it follows from [29]
0
that K(H) ⊂ R(δS ). Which is equivalent to {S} ∩ C1 (H) = {0}. Hence S
is P-symmetric, but it is easily seen that S is not D-symmetric.
This proves that D(H) is properly contained in P (H).
If A and B are P-symmetric operators how about A ⊕ B ? the following
example shows that some care must be exercised.
Example 1.1.10. [5] Let H1 = L2 (D), where D denotes the unit disc
D = {z ∈ C : |z| ≤ 1}.
12
Define an operator M ∈ L(H1 ) as follows :f 7−→ M f such that (M f )(z) =
zf (z) for all z ∈ D. Let H2 be a separable complex Hilbert space, (en )n≥1 an
orthonormal basis for H2 and S be the unilateral shift operator on H2 (i.e.
Sen = en+1 , n ≥ 1).
Define J : H2 −→ H1 byJen = z n χ
Dα , where
Dα = {z ∈ C : |z| ≤ α < 1}.
M 0
0 J
If we set A =
and T =
, then we have
0 S
0 0
Z 2π Z α
Z
α2n
2n
2n
2
|z |dz =
r drdθ = 2πα
.
kJen k =
2n + 1
Dα
0
0
√
α2n
<∞
2n + 1
Thus T is of trace class. We show next that A = M ⊕ S is not P-symmetric.
It is easy to see that M J = JS, this implies that AT = T A. Suppose that
AT ∗ = T ∗ A, which is equivalent to SJ ∗ = J ∗ M . It follows from [19] the
equation SX = XM have only the trivial solution X = 0, and this is a
contradiction.
kT k1 ≤ Σ∞
n=1 kJen k =
2παΣ∞
n=1 √
Theorem 1.1.11. [5] Let A, B ∈ L(H) be P-symmetric operators. If σ(A)∩
σ(B) = {0}, then A ⊕ B is P-symmetric.
Remark 1.1.12. (1) If λ is an eigenvalue of A and λ is not an eigenvalue
of A∗ , then A ⊕ λI is not P-symmetric. In particular, if S denotes the
unilateral shift and |λ| < 1, the operator S ⊕ λI is not P-symmetric and if
|λ| ≥ 1, we observe that S ⊕ λI is P-symmetric.
(2) The set of D-symmetric operators and the class of P-symmetric operators
are not norm closed (see[5],[35]).
R
Theorem 1.1.13. [5] Let A = λdE(λ) be a normal operator, and B be a
P-symmetric operator. If E[σ(A) ∩ σ(B)] = 0, then A ⊕ B is P-symmetric.
Let (en )n∈N (respectively (en )n∈Z ) be an orthonormal basis for H, and let S
be the unilateral (respectively bilateral) weighted shift Sek = ωk ek+1 , k ∈
N (respectively k ∈ Z) with nonzero weights (ωk ). By taking unitarily
weighted shift, we may assume that ωk =| ωk |.
Corollary 1.1.14. [6] let S be the unilateral (bilateral) weighted shift Sek =
ωk ek+1 , k ∈ N(Z). Then S is P-symmetric if and only if
X
ωk ωk+1 · · · ωk+n = ∞,
k
for all n ∈ N.
13
Properties and descriptions of C◦ (A), I◦ (A) and B◦ (A)
As noted before, P-symmetry of an operator is equivalent to the self-adjointness
of the ultra-weak closure of the range of a derivation. Hence, for A ∈ P (H)
it is natural to introduce the following subalgebras:
W∗
C◦ (A) = {C ∈ L(H) : CL(H) + L(H)C ⊂ R(δA )
},
W∗
I◦ (A) = {Z ∈ L(H) : ZR(δA ) + R(δA )Z ⊂ R(δA )
W∗
B◦ (A) = {B ∈ L(H) : R(δB ) ⊂ R(δA )
},
}.
Theorem 1.1.15. [6] If A is a P-symmetric, then
(1) C◦ (A), I◦ (A) and B◦ (A) are C ∗ -algebras, ultraweakly closed in L(H).
(2) C◦ (A) is two sided ideal I◦ (A).
W∗
(3) R(δB ) ⊂ R(δA )
for all B ∈ C ∗ (A), where C ∗ (A) is the C ∗ -algebra
generated by A and the identity operator.
Theorem 1.1.16. [6] For A ∈ L(H). The following assertions are equivalent:
(1) A is P-symmetric .
(2) A∗ A − AA∗ ∈ C◦ (A).
(3) A∗ ∈ I◦ (A).
Corollary 1.1.17. [6] Let A be P-symmetric and let X ∈ L(H).
If AX − XA ∈ C◦ (A) then AX ∗ − X ∗ A ∈ C◦ (A).
We consider the C ∗ -algebras C◦ (A), B◦ (A) and I◦ (A) for special P-symmetric
operators.
Remark 1.1.18. If H is finite dimensional Hilbert space, the result of J.
0
P. Williams [40] guarantees that C◦ (A) = {0}, I◦ (A) = {A} and B◦ (A) =
00
{A} .
W∗
When H is infinite dimensional if A ∈ L(H) such that R(δA )
= L(H),
then C◦ (A) = I◦ (A) = B◦ (A) = L(H).
W∗
In [6] S. Bouali and J. Charles proved that if R(δA ) contains no nonzero
0
positive operator, then C◦ (A) = {0} and I◦ (A) = {A} .
Corollary 1.1.19. [6] Let A be P-symmetric operator with countable spectrum. Then the following assertions are equivalent:
(1) C◦ (A) = {0}.
(2) A is a diagonal operator.
0
(3) I◦ (A) = {A} .
Theorem 1.1.20. [6] if A is a normal operator with finite spectrum, then
0
00
C◦ (A) = {0}, I◦ (A) = {A} and B◦ (A) = {A} .
1.2
14
Finite operators
Let L(H) be the algebra of all bounded linear operators on an infinite
dimensional complex and separable Hilbert space H. An operator A ∈ L(H)
is called finite if kAX − XA − Ik ≥ 1 for each X ∈ L(H). In [39] J.
P. Williams initiated the study of the class F (H) of finite operators. The
class F (H) is uniformly closed, contains every normal operator, all compact
operators, all operators having a direct summand of finite rank, and the
C ∗ -algebra generated by each of its members.
In the following section we present several properties of the class of finite
operators. The class F (H) is not invariant under similarity but it is invariant
under compact perturbation.
For each integer n ≥ 1, let Rn the set of operators on H that have an ndimensional reducing subspace. It is well known that Rn ⊂ F (H) for n ≥ 1,
where the bar indicates the norm closure of Rn . We give some sufficient
conditions for an operator A to belong in R1 . We will also see that every
dominant operator is finite.
Definition 1.2.1. Let A ∈ L(H). The operator A is said to be finite if
kAX − XA − Ik ≥ 1 for all X ∈ L(H). We denote by F (H) the set of all
finite operators.
Theorem 1.2.2. [39] Let B be a complex Banach algebra with identity I.
If A ∈ B such that kAk = r(A) then kAX − XA − Ik ≥ 1 for all X ∈ B.
Remark 1.2.3. The class of finite operators F (H), contains every hyponormal operator, all compact operators, every operator of the form finite+compact.
Remark 1.2.4. (1) F (H) is invariant under unitary equivalence.
(2) An operator A ∈ L(H), is said to be normaloid if kAk = r(A). Hence,
it follows from [39], that every normaloid is a finite operator.
(3) We have the inclusions relating the following class of operators:
hyponormal ⊂ p − hyponormal ⊂ paranormal ⊂ normaloid.
It result that F (H) contains every hyponormal, p-hyponormal, paranormal
operator.
15
Remark 1.2.5. (1) Let A ∈ L(H). If σar (A) 6= φ, then A is a finite operator.
(2) It was shown in [39] that
R1 = {A ∈ L(H) : σar (A) 6= φ}
(3) Each of the following conditions is a sufficient for an operator A to
belong to R1 :
(i) kA − λk = r(A − λ) for some complex number λ.
(ii) A = H + K, where H is hyponormal and K is compact.
(iii) A = T + K, where T is a Toeplitz operator and K is compact.
(iv) dominant operator.
Definition 1.2.6. Let A be a complex Banach algebra with identity e. The
set of normalized positive functionals (states) on A is defined by
0
P (A) = {f ∈ A : f (e) = kf k = 1}.
The numerical range of an element a ∈ A is the set
ω◦ (a) = {f (a) : f ∈ P (A)}.
Let A be a complex Banach with identity e, Let A ∈ A, then 0 ∈ ω◦ (A)
if and only if
|λ| ≤ kA − λk
(∗),
for all λ ∈ C (see[39]). Write AX − XA instead of A in (1) and choose
λ = 1, that is A is finite if and only if 0 ∈ ω◦ (AX − XA). This fact shows
that there is a relation between finite operators and numerical range of a
derivation. In [39] J. P. Williams establish the following result.
Theorem 1.2.7. Let A be a complex Banach with identity e. the following
statements are equivalent:
(1) 0 ∈ ω◦ (ax − xa), for all x ∈ A.
(2) kax − xa − ek ≥ 1 for all x ∈ A .
(3) There exists a state f ∈ P (A) such that f (ax) = f (xa) for all x ∈ A.
Corollary 1.2.8. [39] The class F (H) of finite operators is norm closed
in L(H). Moreover, if A ∈ F (H) then the C ∗ -algebra generated by A is
contained in F (H).
Theorem 1.2.9. [39] Rn ⊂ F (H) for all n ≥ 1.
1.3
16
The essential spectrum
An operator A ∈ L(H) is called left semi-Fredholm if R(A) is closed and
dim ker(A) < ∞. Analogously, A is right semi-Fredholm if R(A) is closed
and dimH|R(A) < ∞.
A is a semi-Fredholm operator if it is either left or right semi-Fredholm and
A is a Fredholm operator if it is both left and right semi-Fredholm.
Let Φ+ (H) be the set of left semi-Fredholm and Φ− (H) the set of right semiFredholm on the Hilbert space H. The set Fredholm operators is denoted
by Φ(H) = Φ+ (H) ∩ Φ− (H).
If A is a semi-Fredholm operator, we may define the Fredholm index of A
as follows:
ind(A) = dim ker(A) − dim[R(A)]⊥
= dim ker(A) − dim ker(A∗ )
Definition 1.3.1. If A ∈ L(H), the essential spectrum of A, denoted by
σe (A), is the spectrum of π(A) in the Calkin algebra L(H)|K(H). Similarly, the left and the right essential spectrum of A are defined by σle (A) =
σl (π(A)) and σre (A) = σr (π(A)).
The proof of the next proposition is an application of the general properties
of the various spectra in an arbitrary Banach algebra.
Proposition 1.3.2. Let A ∈ L(H).
(1) σe (A) = σle (A) ∪ σre (A).
(2) σle (A) = σre (A∗ )∗ .
(3) σle (A) ⊂ σl (A), σre (A) ⊂ σr (A) and σe (A) ⊂ σ(A).
(4) σle (A), σre (A) and σe (A) are compact sets.
(5) If K is a compact operator, then σle (A + K) = σle (A),σre (A + K) =
σre (A) and σe (A + K) = σe (A).
Remark 1.3.3. Let A ∈ L(H), then we have:
σle (A) = {λ ∈ C : A − λI 6∈ Φ+ (H)}
σre (A) = {λ ∈ C : A − λI 6∈ Φ− (H)}
σe (A) = {λ ∈ C : A − λI 6∈ Φ(H)}
17
Proposition 1.3.4. Let A ∈ L(H).
(1) λ ∈ σle (A) if and only if dim ker(A − λI) = ∞ or R(A − λI) is not
closed.
(2) λ ∈ σre (A) if and only if dim[R(A − λI)]⊥ = ∞ or R(A − λI) is not
closed.
Theorem 1.3.5. [14]
∩K∈K(H) σ(A + K) = σe (A) ∪ {λ : A − λ ∈ Φ(H) and ind(A − λ) 6= 0}.
Proof. If K is a compact operator, then σe (A) = σ(π(A)) = σ(π(A + K)) ⊂
σ(A + K). Moreover, if A − λ is Fredholm with nonzero index, then so is
A + K − λ for any compact operator K. In particular, A + K − λ is not
invertible. This proves that the set on the right in the theorem is contained
in the set on the left.
On the other hand, if λ does not belong to the set on the right, then A − λ
is Fredholm with index 0. This implies that A − λ is of the form B + K,
where B is invertible and K compact. Thus, λ 6∈ σ(A − K) and hence, λ
does not belong to the set on the left of the theorem.
Theorem 1.3.6. [14]
σ(A) = ∩K∈K(H) σ(A + K) ∪ σp (A).
Proof. The set in the right is clearly contained in σ(A). Suppose that λ ∈
σ(A) and λ 6∈ σ(A + K) for some compact K. Then
(A + K − λ) I − (A + K − λ)−1 K = A − λ,
is not invertible, so that I − (A + K − λ)−1 K is not invertible. Hence, 1 is
an eigenvalue of the compact operator (A + K − λ)−1 K. But if (A + K −
λ)−1 Kx = x with x 6= 0, then Kx = (A + K − λ)x, and so (A − λ)x = 0.
It other words, λ ∈ σp (A). This completes the proof.
Theorem 1.3.7. [14]
σ(A) = σe (A) ∪ σp (A) ∪ σp (A∗ )− ,
where the bar denotes complex conjugate.
¯ Then A−λ is injective
Proof. Suppose that λ ∈ σ(A) and λ 6∈ σp (A)∪ σp (A).
with dense range. Since A − λ is not invertible, it follows that R(A − λ) is
not closed. Therefore A − λ is not Fredholm so that λ ∈ σe (A).
1.4
18
The Riesz idempotent
Definition 1.4.1. Let A ∈ L(H) and let λ ∈ σ(A) be an isolated point of
σ(A). If there exists a closed disk Dλ centered at λ that satisfies Dλ ∩σ(A) =
{λ}, then the operator
Z
1
Eλ =
(z − A)−1 dz,
2πi ∂Dλ
is called the Riesz idempotent with respect to λ, which has the properties
that Eλ2 = Eλ ; AEλ = Eλ A ; ker(A − λ) ⊂ Eλ H and σ(A|Eλ H) = {λ}.
Remark 1.4.2. J. Stampfli proved in [32] that If A is hyponormal and
λ ∈ σ(A) is isolated, then the Riesz idempotent Eλ with respect to λ is
self-adjoint and satisfies Eλ H = ker(A − λ) = ker(A − λ)∗ .
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J. Math. Anal. Appl. 203(1997), 868-873.
[25] G. Lumer and M. Rosenblum, Linear operator equations, Proc. Amer.
Math. Soc. 10(1959),32-49.
[26] B. Magajna, The norm of asymmetric elementary operator, Proc.
Amer. Math. Soc. 132(2003), 1747-1754.
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21
[27] P.J. Maher, Commutator Approximants, Proc. Amer. Math. Soc.
115(1992),995-1000.
[28] P. Rosenblum, On the operator equation BX − XA = Q, Duke. Math.
J. 23(1956), 263-269.
[29] C.R. Rosentrater, Not every d-symmetric operator is GCR, Proc.
Amer. Math. Soc. 81(1981), 443-446.
[30] C.R. Rosentrater, Compact operators and derivations induced by
weighted shifts, Pacific. J. Math. 104(1983), 465-470.
[31] S. Sakai, Derivation on W ∗ -algebras, Ann. Math. 83(1966), 273-279.
[32] J.G. Stampfli , Hyponormal operators, Pacific. J. Math. 12(1962),
1453-1458.
[33] J.G. Stampfli , The norm of a derivation, Pacific. J. Math. 33(1970),
737-747.
[34] J.G. Stampfli , Derivations on B(H): the range, Ill. J. Math. 17(1973),
518-524.
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349-354.
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[41] D. Xia, Spectral theory of hyponormal operators, Birkhauser Verlag
(Basel), 1983.
Chapter 2
On the range kernel
orthogonality and P-symmetric
operators
1
Abstract: Let H be a separable infinite dimensional complex Hilbert
space, and let L(H) denote the algebra of all bounded linear operators
on H. For given A ∈ L(H), we define the derivation δA : L(H) −→ L(H)
by δA (X) = AX − XA. In this paper we establish the orthogonality of
the range R(δA ) and the kernel ker(δA ) of a derivation δA induced by a
cyclic subnormal operator A, in the usual sense. We give a version of the
Fuglede-Putnam Theorem. We establish a short proof of the principal
Result of F. Wenying and J. Guo Xing in [10]. Related results for Psymmetric operators are also given.
2.1
Introduction
Let H be an infinite dimensional complex Hilbert space, and let L(H) denote
the algebra of all bounded linear operators acting on H. If A ∈ L(H), then
the inner derivation induced by A is the operator δA defined by
δA : L(H) −→ L(H)
X 7−→
1
Math. Inequal. Appl. 9(2006), no. 3, 511-519.
22
chapter 2: Range-Kernel orthogonality and P-symmetric operators 23
Given subspaces M and N of a Banach space V with norm k.k, M is said
to be orthogonal to N if km + nk ≥ knk for all m ∈ M and n ∈ N . This
definition generalizes the idea of orthogonality in Hilbert space.
Let A be a normal operator, J. Anderson [1] has shown that if S is in the
0
commutant {A} of A (i.e. [A, S] = AS − SA = 0), then for all X ∈ L(H)
we have
kδA (X) + Sk ≥ kSk
Where k.k is the usual operator norm. The above inequality says that the
range R(δA ) of the derivation δA is orthogonal to the kernel ker(δA ) of δA .
The study of the range-kernel orthogonality of derivations has been considered in a number of papers ([15], [9], [16] and [17] · · · ), and much attention
has been given to its investigations with respect to different norms (see [9],
[15], [16] and [18]).
It has been shown in Theorem 4 [14] that if A ∈ L(H) is a cyclic subnormal
0
operator and if S ∈ C2 (H)∩{A} , where C2 (H) is the Hilbert-Schmidt class
associated with the norm k.k2 , then for all X ∈ L(H) we have
kδA (X) + Sk22 = kδA (X)k22 + kSk22
In the same direction, it should be noted that F.Kittaneh remarked that
the Theorem 2 in [15], can be modified to insure that if A ∈ L(H) is a
0
cyclic subnormal operator and S ∈ J ∩ {A} , such that J is the norm ideal
associated with the unitarily invariant norm k.kJ , then for all X ∈ L(H)
we have also
kδA (X) + SkJ ≥ kSkJ .
The purpose of the first section is to prove the orthogonality of the range and
the kernel of a inner derivation induced by a cyclic subnormal operator in the
usual operator norm (i.e. on the whole space L(H)). Moreover, we give an
example showing that the cyclicity assumption on a subnormal operator A is
sufficient for the range-kernel orthogonality to be hold. Finally, it is natural
to ask if this range-kernel orthogonality result has a ∆A analogue, where
∆A is the elementary operator defined on L(H) by ∆A (X) = AXA − X
and A is a cyclic subnormal operator.
In the second section we give a version of the Fuglede-Putnam Theorem.
Given A, B ∈ L(H) and let F be a two sided ideal of L(H). The pair (A, B)
is said to possess the Fuglede-Putnam commutativity Theorem (F P )F if
AT = T B and T ∈ F implies A∗ T = T B ∗ . We show that the set
Σ(F) = {A ∈ L(H) : (A, A) has property (F P )F }
is not norm closed. This result allow us to give a characterization of operators A such that the pair (A, A) has the property (F P )Cp , where Cp denote
the Von Newmann-Schatten class for p > 1. Consequently, we obtain a short
proof of the principal Result of F.wenying and J.Guo Xing in [10]. We conclude this section with some notations.
Notations. 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).
Let C1 (H) be the idealP
of trace class operators, the trace function is defined
on C1 (H) by tr(T ) = n (T en , en ), where (en )n is any complete orthonormal sequence in H. For 1 < p < ∞ we denote CP (H) the Von NeumannSchatten class and k.kp its associated norm. R(δA /CP ) is the norm closure
of the range of δA /CP . The annhilateur of R(δA /CP ) is denoted by
0
R(δA /CP )◦ = {f ∈ (Cp (H)) : f (AX − XA) = 0 f or all X ∈ CP (H)}.
In addition to the notation already introduced, we shall use the following
further notation. Given X ∈ L(H), we shall denote the kernel, the orthogonal complement of the kernel and the range of X by ker X, (ker X)⊥ and
R(X) respectively. The spectrum, the essential spectrum, the point spectrum and the spectral radius of X will be denoted by σ(X), σe (X) , σp (X)
and r(X) respectively. Any other notation will be explained as and when
required.
2.2
The range-kernel orthogonality
Definition 2.2.1. A vector e◦ ∈ H is cyclic for for A ∈ L(H) if H is the
smallest invariant subspace for A that contains e◦ . the operator A is said to
be cyclic if it has a cyclic vector.
Definition 2.2.2. Let A ∈ L(H). The operator A is said to be subnormal
if there exists a normal operator B on a Hilbert space K such that H is a
subspace of K, the subspace H is invariant under the operator B, and the
restriction of B to H coincides with A.
The basic tools in the main result of this section is to use other technics
that this used , stated below as a Proposition and a Remark.
Proposition 2.2.3. Let a be a normal element of a C ∗ -algebra A. Then
for every element c ∈ A satisfying ac = ca, we have
kax − xa + ck ≥ kck
for all x ∈ A.
Proof. It is well known that there exists an ∗-isometric isomorphism ψ and
a Hilbert space H such that ψ : A −→ L(H) preserving the order [13].
It follows that ψ(a) is a normal operator and commutes with ψ(c). Then
combining the Anderson’s Result for normal operators [1] and the isometric
isomorphism , we get the related inequality
kax − xa + ck ≥ kck
for all x ∈ A.
Remark 2.2.4. ( [11, p.187]) A coset [A] has norm equal to its spectral
radius in each of the following cases:
(i) [A] is hyponormal.
(ii) [A] contains a Toeplitz operator.
(iii) A has norm equal to its spectral radius and A has no isolated eigenvalues
of finite multiplicity.
Theorem 2.2.5. Let A ∈ L(H) be a cyclic subnormal operator. For every
bounded linear operator T such that AT = T A, we have
kAX − XA + T k ≥ kT k
for all X ∈ L(H).
Proof. Let T be in L(H) such that AT = T A. Since A is a cyclic subnormal
operator, then it follows from Yoshino’s Result [20] that T is subnormal.
This implies that r(T ) = kT k. Hence it is enough to prove that
kAX − XA + T k ≥ |λ|
for all X ∈ L(H) and all λ ∈ σ(T ). Furthermore, since T is a subnormal
operator, then it is well known that σ(T ) = σp (T ) ∪ σe (T ) ( see [11]).
Let λ ∈ σ(T ). We consider the following cases for the location of λ:
Case 1. Assume that λ ∈ σp (T ). We shall divide this cases into two different
steps.
(i) if λ ∈ σp (T ) such that dim ker(T − λ) < ∞ . Let us denote Eλ the
subspace ker(T −λ). Since AT = T A and T is subnormal , then the subspace
Eλ is invariant by T and A. Moreover A/Eλ is normal, then Eλ reduces A
[18, p.514]. Hence for A and T we get the following decomposition
B 0
λ 0
A=
, T =
.
0 C
0 ∗
For an operator X =
Y Z
R S
, We have
AX − XA + T =
BY − Y B + λ ∗
∗
∗
.
Recall that the norm of an operator matrix is always greater than or equal
to the norm of the operator matrix consisting of its diagonal entries only
[8, p.82], applying this twice, we have from the above equality that
kAX − XA + T k ≥ kBY − Y B + λk
A is subnormal, then A is a finite operator [19], and therefore B thus. Then
we obtain
kBY − Y B + λk ≥ |λ|
Consequently, we have
for all X ∈ L(H), and all λ ∈ σp (T ) such that dim ker(T − λ) < ∞.
(ii) If λ ∈ σp (T ) such that dim ker(T − λ) = ∞. Since T is a subnormal
operator then dim ker(T − λ)∗ = ∞. It follows that T − λ is not a Fredholm
operator which is equivalent to λ ∈ σe (T ) (see the case 2.).
Case 2. If λ ∈ σe (T ). For this case we may distinguish two steps.
(i) T has no isolated eigenvalues of finite multiplicity.
The condition AT = T A implies that [A][T ] = [T ][A]. Since A is a cyclic
subnormal operator then [A] is a normal operator according to Shaw and
Berger’s Result[4]. Using the preceding Proposition 2.2.3 we obtain that
R(δ[A] ) is orthogonal to ker(δ[A] ). From this it follows that
kAX − XA + T k ≥ k[A][X] − [X][A] + [T ]k ≥ k[T ]k
For all X ∈ L(H). Since T is subnormal and has no isolated eigenvalues of
finite multiplicity, then by Remark 2.2.4 we have k[T ]k = r([T ]). Hence by
a standard argument we have
k[A][X] − [X][A] + [T ]k ≥ |λ|
for all X ∈ L(H). It follows that
For all X ∈ L(H).
(ii) If T hasWisolated eigenvalues of finite multiplicity. We consider the subspace E = µ∈β(T ) ker(T − µ), where β(T ) is the set of all isolated eigenvalues of T with finite multiplicity. The condition AT = T A implies that
T is subnormal. Since T /E is a normal operator then E reduces T . With
respect to the decomposition H = E ⊕ E ⊥ , we have
T1 0
T =
.
0 T2
Applying Proposition 2.2.3 to the Calkin algebra, it is easily seen that
kAX − XA + T k ≥ k[A][X] − [X][A] + [T ]k ≥ k[T ]k.
On the other hand it is clear to check that T is a Fredholm operator if and
only if T2 is a Fredholm operator (see [7, p.352]). It follows that λ ∈ σe (T ) if
and only if λ ∈ σe (T2 ). Consequently, We get σe (T ) = σe (T2 ). By hypothesis
we have λ ∈ σe (T ) = σe (T2 ) and T = T1 ⊕ T2 . Using [8, p. 82] one obtains
K2
K1 + T1
inf , K1 , K2 , K3 , K4 compacts
K3
T2 + K4
≥ inf {kT2 + K4 k, K4 compact }
Then it follows immediately that
k[T ]k ≥ k[T2 ]k
Since T2 has no isolated eigenvalues of finite multiplicity, then by the Remark
2.2.4 we have kAX − XA + T k ≥ |λ|. This implies that
for all X ∈ L(H) and all λ ∈ σe (T ).
Finally, we conclude that
For all X ∈ L(H) and all λ ∈ σ(T ). Then
0
For all X ∈ L(H) and all T ∈ {A} .
Example 2.2.6. Let U be the unilateral shift
operator
of multiplicity
one.
U 0
0 0
On H ⊕ H, we consider the operators A =
,T =
0
0
P
0
0 0
and X =
, where P = 1 − U U ∗ and Q = P U ∗ . Then A
Q 0
0
is a noncyclic subnormal operator and T ∈ {A} . It is easy to see that
δA (X) + T = 0 but kT k = 1; and so R(δA ) is not orthogonal to ker(δA ).
According to the preceding Theorem, this Example indicates that the cyclicity assumption on A is sufficient for the range-kernel orthogonality of δA to
hold. It has been used earlier in [15].
Remark 2.2.7. There exist subnormal operator A and operators X such
that AX = XA and A∗ X 6= XA∗ . Hence the Fuglede-Putnam commutativity
Theorem cannot be extended to subnormal operators.
Definition 2.2.8. An operator A ∈ L(H) is said to be paranormal, if
kAxk2 ≤ kA2 xkkxk, for all x ∈ H.
Remark 2.2.9. The inclusions relating some class of operators containing
strictly hyponormal operators and listed above are as follows:
hyponormal ⊂ p − hyponormal ⊂ paranormal ⊂ normaloid.
Proposition 2.2.10. Let A be cyclic subnormal operator, then
kT k ≤ dist(T, R(∆A )),
for all paranormal operator T in ker(∆A ).
Proof. Suppose that T is a paranormal operator in ker(∆A ). The condition
AT A = T implies that AT 2 = T 2 A. Applying the Theorem 2.2.5, we get
kT 2 k ≤ kAY − Y A + T 2 k
for all Y ∈ L(H). Since T is paranormal operator, then we have kT k2 ≤
kT 2 k. Replacing Y by XAT and using the fact that kT 2 k = kT k2 , one
obtains easily that
kT k ≤ kAXA − X + T k
for all X ∈ L(H). Which completes the proof.
Remark 2.2.11. If A is a cyclic subnormal operator, then we deduce from
0
the Theorem 2.2.5 that R(δA ) is orthogonal to ker(δA ), hence R(δA )∩{A} =
0
{0}. J. Anderson proved that R(δA )∩{A} = {0} if A is normal or isometric
(see [1]).
Open problem. Let ∆A denote the elementary operator ∆A defined on
L(H) by ∆A (X) = AXA − X. If A is a cyclic subnormal operator we ask if
we have the range-kernel orthogonality for ∆A ?.
2.3
Definition 2.3.1. Let A, B ∈ L(H) and F be a two sided ideal of L(H).
The pair (A, B) is said to possess the Fuglede-Putnam property (shortened
to (F P )F ) if AT = T B and T ∈ F implies A∗ T = T B ∗ .
The following Theorem is well known.
Theorem 2.3.2. Let A, B ∈ L(H) and F be a two-sided ideal of L(H).
Then the following statements are equivalent.
(1) (A, B) has the property (F P )F .
(2) If AT = T B and T ∈ F, then R(T ) reduces A, ker(T )⊥ reduces B and
the restriction A|R(T ) and B|ker(T )⊥ are normal operators.
Remark 2.3.3. It is shown in Proposition 1 [4], that the pair (A, A) of
operators has the property (F P )F , where F is a two sided ideal of L(H),
under one of the following hypothesis:
(i) A is a normal operator.
(ii) A is an isometry.
(iii)A is a cyclic subnormal operator.
(iv) A is invertible such that kA−1 kkAk = 1.
Proposition 2.3.4. Let F be a two sided ideal of L(H). Then the set of
operators
Σ(F) = {A ∈ L(H) : (A, A) has the property (F P )F }
is not norm closed in L(H).
Proof. To see this, we define a sequence of operators (Sn )n and S as follows.
Let (ek )k≥0 be an orthonormal basis for H, we consider the operators
1
e , if ; k=0
n 1
Sn ek =
ek+1 , otherwise.
and
Sek =
0,
ek+1 ,
if; k=0
otherwise.
It is clear that kSn − Sk −→ 0. On the other hand , for all n ≥ 1 we have Sn
is a cyclic subnormal operator. Then from the preceding Remark, it follows
that Sn ∈ Σ(F) for all non-negative integer n.
Let us consider T = e◦ ⊗ e1 , the rank one operator defined by T x = (x, e1 )e◦
for all x ∈ H. Evidently T ∈ F and ST = T S. However, a simple calculation
show that S ∗ T 6= T S ∗ , which implies that S 6∈ Σ(F). This completes the
proof.
Remark 2.3.5. It is elementary to show that the weighted shift S defined
above is subnormal. Since S 6∈ Σ(F) for all two sided ideal F of L(H), it
follows from the Corollary 5 [4] that the range R(δS /C2 ) is not orthogonal
to the kernel ker(δS /C2 ).
Consequently, the cyclicity assumption on the subnormal operator S is fundamental for the orthogonality of R(δS /C2 ) and ker(δS /C2 ) to be hold. This
gives an affirmative answer to a question raised by F. Kittaneh in [14], and
treated by the authors F. Wenying and J. Guo Xing in [10].
Proposition 2.3.6. Let A ∈ L(H). For 1 < p < ∞ and if p1 + 1q = 1, then
the following statements are equivalent,
(i) (A, A) has the property (F P )Cp .
(ii) R(δA /Cq ) = R(δA∗ /Cq ).
(iii) If T ∈ ker(δA /Cp ), then R(T ) and ker(T )⊥ reduces A, and the restriction A/R(T ) and A/ker(T )⊥ are normal operators.
Proof. (i)⇐⇒ (ii) A simple calculation show that
R(δA /Cq ) = R(δA∗ /Cq )
if and only if, whenever f ∈ R(δA /Cq )◦ implies f ∗ ∈ R(δA∗ /Cq )◦ , where we
have f ∗ (X) = f (X ∗ ) for all X ∈ Cq . Therefore, it suffices to show that
0
R(δA /Cq )◦ ∼
= {A} ∩ Cp
It is convenient to note that
0
(Cq ) = {fT : T ∈ Cp } ∼
= Cp
for all p and q such that p1 + 1q = 1.
Consequently, if fT ∈ R(δA /Cq )◦ for some operator T ∈ Cp we get
fT (A(x ⊗ y)) = fT ((x ⊗ y)A)
for all x and y in H. From where
tr(T Ax ⊗ y) = tr(T x ⊗ A∗ y)
But since tr(u ⊗ v) = (u, v), then we obtain (T Ax, y) = (T x, A∗ y), hence
AT = T A.
0
Conversely, suppose that T ∈ {A} ∩ Cp . From the above computation, it
results easily that
fT (A(x ⊗ y)) = fT ((x ⊗ y)A)
for all x and y in H. Since the class of all finite rank operators is dense in
Cq for all q ≥ 1, then the desired result follows immediately .
(iii)⇐⇒ (i) Is a easy consequence from Lemma 2.3 [3].
Application. Let Ωp (A), Λp (A) and ∆p (A) the Banach subalgebras of Cp
associated with A defined as follows
Ωp (A) = {C ∈ Cp : CCp + Cp C ⊂ R(δA /Cp )}
Λp (A) = {Z ∈ Cp : ZR(δA /Cp ) + R(δA /Cp )Z ⊂ R(δA /Cp )}
∆p (A) = {B ∈ Cp : R(δB /Cp ) ⊂ R(δA /Cp )}
In the finite dimensional case, Ωp (A), Λp (A) and ∆p (A) coincides with the
subalgebras introduced in [2]. Consequently, we get Ωp (A) = {0}, Λp (A) =
0
00
{A} the commutant of A and ∆p (A) = {A} the bicommutant of A. By
considering the Fuglede-Putnam Theorem it follows that Ωp (A), Λp (A) and
∆p (A) are C ∗ - subalgebras if and only if A is normal.
In the infinite dimensional case, by using the Theorem 2.3.2 ones obtain
that Ωp (A), Λp (A) and ∆p (A) are C ∗ - subalgebras if A satisfy one of the
conditions of the previous Remark 2.3.3 .
Remark 2.3.7. The class of operators A ∈ L(H) such that the pair (A, A)
has the property (F P )C1 , is called the class of P-symmetric operators. For
a good accounts see ([5];[6]).
Bibliography
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[4] C. A. Berger and B. I. Shaw, Self-commutator of multicyclic hyponormal operators are always trace class, Bull. Amer. Math. Soc.
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[5] S. Bouali et J. Charles, Extension de la notion d’opérateur dsymétrique I, Acta. Sci. Math. 58(1993) 517-525.
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[7] J. B. Conway , A course in functional analysis, Springer Verlag, New
York, Berlin, Heidelberg ,(1990).
[8] B. P. Duggal, On
106(1988),139-148.
intertwining
operators,
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Math.
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[10] W. Feng and G. Ji, A counter example in the theory of derivations,
Glasgow Math. J. 31(1989),161-163.
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[12] I. C. Gohberg et M. G. Krein, Introduction to the theory of linear
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Chapter 3
On the range and the kernel of
derivations
1
Abstract: Let H be a separable infinite dimensional complex Hilbert
space and let L(H) denote the algebra of all bounded linear operators
on H into itself. Given A ∈ L(H), the derivation δA : L(H) −→ L(H)
is defined by δA (X) = AX − XA. In this paper we prove that if A is
an n-multicyclic hyponormal operator and T is hyponormal such that
AT = T A, then kδA (X) + T k ≥ kT k for all X ∈ L(H). We establish the
same inequality if A is a finite operator and commutes with normal
operator T . Some related results are also given.
3.1
Introduction
Let H be an infinite dimensional complex Hilbert space and let L(H) denote
the algebra of all bounded linear operators acting on H. If A ∈ L(H), then
the inner derivation induced by A is the operator δA defined on L(H) by
By finite operator we shall mean a bounded linear operator A on H such
that
kδA (X) + Ik ≥ 1
(1)
for every X ∈ L(H). As stated in [12] J.P. Williams proved that the class
of finite operators contains every normaloid (i.e.,operators A ∈ L(H) such
that the spectral radius r(A) of A equals the norm of A), every operator
1
Serdica. Math. J. 32(2006), no. 1, 31-38.
34
Chapter 3 : On the range and the kernel of derivations
35
with a compact direct summand, and the entire C ∗ - algebra generated by
each of its members. The purpose of this paper is to investigate this class
of operators to give natural generalizations of the norm inequality (1). The
basic tools in the main results is to use Anderson’s inequality for normal
operators [1], and the Berberian extension Theorem [13] .
The present paper is organized as follows. In Theorem 3.2.5, we initiate
a new approach to extend this results to certain intertwining nonnormal
operators A and T where A is an n-multicyclic hyponormal operator and
requiring that T is hyponormal . The point of view about finite operators
is developed in Theorem 3.2.11, in which we give a natural generalization
of the inequality (1). Using a very simple argument we show in Theorem
3.2.13, that if A satisfies a quadratic polynomial, then A is a finite operator
W
W
and that A∗ 6∈ R(δA ) , where R(δA ) is the weak closure of the range
R(δA ) of δA .
In addition to the notation already introduced, we shall use the following
further notation. Let K(H) be the ideal of all compact operators in L(H)
and let C(H) denote the Calkin algebra L(H)/K(H). For X ∈ L(H), let
[X] denote the projection of L(H) onto the Calkin algebra. We shall denote the kernel, the orthogonal complement of the kernel, the range of X
by ker X, (ker X)⊥ and R(X) respectively. The spectrum, the approximate
point spectrum and the point spectrum of X will be denoted by σ(X),
σap (X) and σp (X) , and the restriction of X to an invariant subspace M
will be denoted by X|M .
Given A ∈ L(H), there exists a Hilbert space H ◦ ⊃ H and an isometric
∗-isomorphism A −→ A◦ such that σ(A) = σ(A◦ ) and σap (A) = σap (A◦ ) =
σp (A◦ ). This is the Berberian extension Theorem [13].
3.2
Main Results
Definition 3.2.1. [12] Let A ∈ L(H). We say that A is a finite operator if,
kAX − XA + Ik ≥ 1,
for all X ∈ L(H). We denote the class of finite operators by F (H).
Example 3.2.2. The class F (H) of finite operators contains all hyponormal
operators, all compact operators all operators having a direct summand of
finite rank and the entire C ∗ -algebra generated by each of its members.
36
Definition 3.2.3. Let A ∈ L(H). The operator A is said to be n-multicyclic
if there
n vectors x1 , x2 , · · · , xn ∈ H, called generating vectors, such
Pexists
n
that { i=1 fi (A)xi : f1 , f2 , · · · fn ∈ Rat(σ(A))} has span dense in H, where
Rat(σ(A)) denotes the rational functions with poles off σ(A).
Theorem 3.2.4. [2] If A is an n-multicyclic hyponormal operator, then
[A∗ , A] is in trace class, and tr([A∗ , A]) ≤ πn ω(σ(A)), where ω is planar
Lebesgue measure.
Theorem 3.2.5. Let A ∈ L(H). If A is an n-multicyclic hyponormal operator, then for every hyponormal operator T such that AT = T A, we have
for all X ∈ L(H).
Proof. We omit the proof, which may be based entirely on the proof of
Theorem 2.2.5 and Theorem 3.2.4 and Remark [5, p.187].
Definition 3.2.6. An operator A ∈ L(H), is called quasi-normal if A commutes with A∗ A.
Example 3.2.7.
1) Every normal operator is quasi-normal.
2) Every isometry is quasi-normal.
Remark 3.2.8.
(1) An operator A with polar decomposition A = U P is quasi-normal if and
only if U P = P U .
(2) Any quasi-normal operator is subnormal, that is, it has a normal extension.
As a special case we get the following Corollary.
Corollary 3.2.9. Let A, T ∈ L(H), such that A quasi-normal operator , T
hyponormal and AT = T A. Then
for all X ∈ L(H).
Proof. Since A is a quasi-normal operator, it follows from [6] that A =
N + K, where N is normal and K is compact. Hence, by using the same
argument as in the above Theorem, we get the desired inequality.
37
Corollary 3.2.10. Let A and B be operators in L(H). If B is invertible
such that kAkkB −1 k ≤ 1, then
kAX − XB + T k ≥ kT k
for all X ∈ L(H), and all T ∈ L(H) such that AT = T B.
Proof. Let T ∈ L(H), such that AT = T B. This implies that AT B −1 = T .
Since kAkkB −1 k ≤ 1, it follows from [11, Theorem 1.1] that
kAY B −1 − Y + T k ≥ kT k
for all Y ∈ L(H). If we set X = Y B −1 , then we obtain that
kAX − XB + T k ≥ kT k.
Theorem 3.2.11. Let A and T be commuting operators such that A is a
finite operator and T is normal. Then
for all X ∈ L(H).
Proof. Let λ ∈ σp (T ) and let E be the subspace E = ker(T − λ). Since
A commutes with T it follows from Fuglede-Putnam Theorem [8] that E
reduces A and T simultaneously. Hence, with respect to the decomposition
H = E ⊕ E ⊥ , we have
B 0
λ 0
A=
and
T =
.
0 ∗
0 ∗
Y ∗
Let X ∈ L(H) have the representation X =
∈ L(H). Then,
∗ ∗
kAX − XA + T k = k
BY − Y B + λ ∗
∗
∗
k ≥ kBY − Y B + λk.
Since B is a finite operator, it follows that
kAX − XA + T k ≥ |λ|,
for all X ∈ L(H) and all λ ∈ σp (T ).
Using the Berberian extension Theorem , we have that A◦ is finite , T ◦ is
38
normal and A◦ T ◦ = T ◦ A◦ . Since, σp (T ◦ ) = σap (T ◦ ) = σap (T ) = σ(T ), it
follows from the first part that
kAX − XA + T k = kA◦ X ◦ − X ◦ A◦ + T ◦ k ≥ |λ|,
for all X ∈ L(H) and all λ ∈ σ(T ). Hence,
kAX − XA + T k ≥ r(T ) = kT k,
for all X ∈ L(H). This completes the proof.
Corollary 3.2.12. Let A be a finite operator and T a normal operator such
that AT A = T . Then
kAXA − X + T k ≥ kT k,
for all X ∈ L(H).
Proof. Let T a normal operator such that AT A = T . It is easy to see that
AT AT A = T 2 A, that is, AT 2 = T 2 A. Hence, it follows from the preceding
Theorem that
kAY − Y A + T 2 k ≥ kT 2 k,
for all Y ∈ L(H). If we choose Y = XAT for an arbitrary X ∈ L(H), then
we get
kAXAT − XAT A + T 2 k ≥ kT 2 k = kT k2 .
Then
kAXA − X + T k ≥ kT k.
This completes the proof
Theorem 3.2.13. Let A ∈ L(H). If A satisfies some quadratic polynomial,
W
then A is a finite operator and A∗ 6∈ R(δA ) .
Proof. Suppose that A satisfies A2 −2αA+β = 0, hence (A−α)2 is a normal
operator. Then, by Putnam’s
result[10]
we may write A−α = N ⊕M , where
B C
N is normal and M =
, with B normal and C is an injective
0 −C
positive operator such that BC= CB. Therefore,
A = (N +αI)⊕(M +αI).
Y Z
Then for linear operator X =
we have
R S
NY − Y N + I ∗
AX − XA + I =
.
∗
∗
39
Since the norm of operator matrix always dominates the norm of its diagonal
part [7, p.82] one obtains
kAX − XA + Ik ≥ kN Y − Y N + Ik.
Hence it follows from Williams’s Result[12]on normal operators that
kAX − XA + Ik ≥ 1
W
for all X ∈ L(H). Let us assume that A∗ ∈ R(δA ) . An easy calculation
W
W
leads us to A∗ A ∈ R(δA ) . Since R(δA ) contains nonzero positive operator
[3], it follows that A = 0.
Remark 3.2.14. The above Theorem is due to J. P. Williams[12], however
we proved it by other method that this used.
Another interesting class of operators for which the Theorem 3.2.11 is satisfied is the class of operators A such that A∗ A and A + A∗ commute. It is
well known that this class has the property that r(A) = kAk(see[4]).
Bibliography
[1] J. H. Anderson, On normal derivations, Proc. Amer. Math. Soc.
38(1973), 135-140.
[2] C. A. Berger and B. I. Shaw, Selfcommutators of multicyclic hyponormal operators are always trace class, Bull. Amer. Math. Soc.79(1973),
1193-1199.
[3] S. Bouali and J. Charles, Extension de la notion d’opérateur dsymétrique II, Linear algebra and its applications, 225(1995)175-185.
[4] S. L. Campbell and R. Gellar, Linear operators for which T ∗ T and
T + T ∗ commute, Proc. Amer. Math. Soc. 60(1976), 197-202.
[5] P. A. Fillmore, J. G. Stampfli, and J. P. Williams, On the essential
numerical range, the essential spectrum, and a problem of Halmos,
Acta Sci. Math. 33(1972),179-192.
[6] M. I. Gil, Inner bound for the spectrum of quasi-normal operators,
Proc. Amer. Math. Soc. 131(2003), 3737-3746.
[7] I. C. Gohberg et M. G. Krein, Introduction to the theory of linear
nonself adjoint operators, transl. Math. Monographs, vol 18, A. Math.
Soc., Providence, R.I.(1969).
[8] P. R. Halmos, A Hilbert space problem book, Springer-Verlag, New
York, 1982.
[9] R. L. Moore and D. D. Rogers, Note on intertwining M-hyponormal
operators,Proc. Amer. Math. Soc. 83(1981), 514-516.
[10] C. H. Putnam, Range of normal and subnormal operators, Michigan
Math. J. 18(1971),33-36.
[11] A. Turnsěk, Elementary operators and orthogonality, Linear Algebra
Appl., 317(2000)207-216.
40
BIBLIOGRAPHY
41
[12] J. P. Williams, Finite operators, Proc. Amer. Math. Soc. 26(1970),
129-136.
[13] D. Xia, Spectral theory of hyponormal operators, Birkhauser Verlag(Basel), 1983.
Chapter 4
The P-symmetric operators and
the range of a subnormal
derivation
1
Abstract : The inner derivation δA implemented by an element A of the
algebra L(H) of all bounded linear operators on the separable complex
Hilbert space H into itself is the map X 7−→ AX − XA for X ∈ L(H).
In this paper, we are interested to the class of operators A ∈ L(H)
which satisfy the following property, AT = T A implies AT ∗ = T ∗ A for
all T ∈ C1 (H)(trace class operators). Such operators are termed Psymmetric. We establish some properties of this class. We also turn
our attention to commutants and derivation ranges. Hence, we obtain
the new results concerning the intersection of the kernel and the closure
of the range of an inner derivation.
4.1
Introduction
Let L(H) be the algebra of all bounded linear operators acting on a separable complex Hilbert space H. The derivation induced by A ∈ L(H), is
the mapping δA (X) = AX − XA, from L(H) into itself. A is said to be
D-symmetric if R(δA ), the norm closure of the range R(δA ) of δA , is closed
under taking adjoints. This concept of D-symmetry of an operator was intro1
Acta Sci. Math.(Szeged) 72(2006), no. 3-4, 701-708.
42
Chapter 4 :The P-symmetric operators and the range of a derivation
43
duced by J.H. Anderson, J.W. Bunce, J.A. Deddens and J.P. Williams in [2].
The properties of such operators have been much studied in [2],[4],[5],[9],[10]
and [11].
It is well known that if A is D-symmetric then AT = T A implies AT ∗ = T ∗ A
for all T ∈ C1 (H)(trace class operators).
As extension of D-symmetric operators it was introduced in [4] the class of
P-symmetric operators , i.e. the class of operators A that have the property
AT = T A implies AT ∗ = T ∗ A for all T ∈ C1 (H). Examples of such operators include the normal operators, isometries and the cyclic subnormal
operators. In the first section of this paper, we would like to consider the
class of P-symmetric operators. We show that if A is an algebraic operator,
W
W
0
then every P-symmetric operator in R(δA ) ∩ {A} vanishes, where R(δA )
0
is the weak closure of R(δA ) and {A} is the commutant of A. We establish
some basic results on the following subalgebras of L(H) associated with A:
W∗
C◦ (A) = {C ∈ L(H) : CL(H) + L(H)C ⊆ R(δA )
},
W∗
I◦ (A) = {Z ∈ L(H) : ZR(δA ) + R(δA )Z ⊆ R(δA )
and
W∗
B◦ (A) = {B ∈ L(H) : R(δB ) ⊆ R(δA )
}
},
W∗
where R(δA )
is the ultra-weak closure of R(δA ), these subalgebras were
initially introduced in [5].
The paper has been expanded to include in the second section some results
0
concerning the intersection of the closure of R(δA ) and {A} .
0
In finite dimensions, it is known that every operator in R(δA )∩{A} is nilpotent. In infinite dimensions, the Theorem of Kleinecke-Shirokov [8] shows
0
that R(δA ) ∩ {A} consists only of quasinilpotent operators. However, an
0
operator in R(δA ) ∩ {A} is not necessarily quasinilpotent. Also, in [1] An0
derson proved the remarkable Result that R(δA )∩{A} = {0} if A is normal
or isometric. We consider this problem here, and we prove that if A is cyclic
0
subnormal then R(δA ) ∩ {A} = {0}. As a consequence, every operator
0
in R(δA ) ∩ {A} is nilpotent if P (A) is cyclic subnormal for some polynomial P . In Theorem 3 [13] Weber shows that every compact operator in
W
0
R(δA ) ∩ {A} is quasinilpotent. We give another version of Weber’s Theorem, hence in Theorem 4.3.7, we show that if A is normal, isometric or cyclic
W
0
subnormal then R(δA ) ∩ {A} contains no nonzero compact operator.
4.2
44
Definition 4.2.1. Let A ∈ L(H). The operator A is said to be P-symmetric
if it satisfies the following property AT = T A implies AT ∗ = T ∗ A for every
T ∈ C1 (H).
Theorem 4.2.2. Let A ∈ L(H) be an algebraic operator. Then every PW
0
symmetric operator in R(δA ) ∩ {A} vanishes.
W
0
Proof. Suppose that T be a P-symmetric operator in R(δA ) ∩ {A} . Then
there exists a generalized sequence (Xα )α in L(H) such that
W
0
AXα − Xα A −→ T ∈ {A} .
By assumption, we can write P (A) = 0 for some polynomial P . If P (k)
denotes the k-th derivative of P , then
W
P (A)Xα − Xα P (A) −→ P (1) (A)T.
Which implies that P (1) (A)T = 0. Now we have
W
P (1) (A)Xα − Xα P (1) (A) −→ P (2) (A)T,
which gives
P (2) (A)T 3 = 0.
By repeating the same argument it follows that T k = 0 for a given integer
k, hence T is nilpotent.
Suppose that T 6= 0. We prove next that T is not P-symmetric.
Let n > 1 be the smallest integer such that T n = 0 and T n−1 6= 0. Let e be
in H such that T n−1 e 6= 0. If we set f = T n−1 e, then it follows immediately
that T f = 0 and T ∗ f 6= 0. Since T ∗ n−1 6= 0, hence we can choose k ∈ H such
that g = T ∗ n−1 k 6= 0. Consequently there exists nonzero vectors f, g ∈ H
such that T f = 0, T ∗ f 6= 0 and T ∗ g = 0. We conclude from Lemma 2.1[4]
that T is not P-symmetric.
Remark 4.2.3. It is well known [4] that A is P-symmetric if and only if
W∗
R(δA ) , the ultra-weak closure of the range R(δA ) of δA , is a self-adjoint
subspace of L(H).
45
Theorem 4.2.4. Let A be a P-symmetric operator, then we have the following statements:
0
(i) C◦ (A) ⊆ B◦ (A) ⊆ B◦ (A) + {A} ⊆ I◦ (A).
(ii) B◦ (A)/C◦ (A) is contained in the center of I◦ (A)/C◦ (A).
(iii) I◦ (A) = {Z ∈ L(H) : [Z, B◦ (A)] ⊆ C◦ (A)}
= {Z ∈ L(H) : [Z, A] ∈ C◦ (A)}.
(iv) If A is isometric, then Z ∈ I◦ (A) if and only if A∗ ZA − Z ∈ C◦ (A).
W∗
Proof. (i) If C ∈ C◦ (A), then we obtain CX − XC ∈ R(δA )
for all
X ∈ L(H). Which implies that C◦ (A) ⊆ B◦ (A). For the remainder of (i) it
suffices to show that B◦ (A) ⊆ I◦ (A). Indeed, let B ∈ B◦ (A), then
BδA (X) = δA (BX) + δB (AX) − AδB (X)
W∗
W∗
W∗
is in R(δA ) . Hence, BR(δA ) ⊆ R(δA ) . Similarly R(δA )B ⊆ R(δA )
Then B◦ (A) ⊆ I◦ (A).
(ii) Let D ∈ I◦ (A) , B ∈ B◦ (A) and X is any operator in L(H). Then
W∗
δD (B)X = DδB (X) − δB (DX) ∈ R(δA )
W∗
XδD (B) = δB (X)D − δB (XD) ∈ R(δA )
.
.
.
It follows that δD (B) ∈ C◦ (A).
(iii)It is clear that the inclusions
I◦ (A) ⊆ {Z ∈ L(H) : [Z, B◦ (A)] ⊆ C◦ (A)} ⊆ {Z ∈ L(H) : [Z, A] ∈ C◦ (A)}
are trivial. For the reverse inclusion, assume that δA (X) ∈ C◦ (A), where X
is an operator in L(H). It is easy to check that
XδA (Z) = δA (XZ) − δA (X)Z.
δA (Z)X = δA (ZX) − ZδA (X).
Hence, X ∈ I◦ (A). This proves (iii).
(iv) Let Z ∈ I◦ (A), then we set
A∗ ZA − Z = A∗ (ZA − AZ) = −A∗ δA (Z) ∈ C◦ (A).
Conversely, suppose that A∗ ZA − Z ∈ C◦ (A) for an operator Z in L(H).
Hence,
Z(AX − XA) = δA (ZX) + (I − AA∗ )ZAX.
46
W∗
Since I − AA∗ = [A∗ , A] ∈ C◦ (A), then Z(AX − XA) belongs to R(δA ) .
Using the fact that A∗ Z ∗ A − Z ∗ ∈ C◦ (A) and A is a P-symmetric operator
we obtain
W∗
W∗
W∗
Z ∗ R(δA∗ ) = Z ∗ R(δA ) ⊆ R(δA ) ,
W∗
and this implies R(δA )Z ⊆ R(δA )
under taking adjoints.
Corollary 4.2.5. Let A, B ∈ L(H). If B is similar to A i.e. SBS −1 =
A for some invertible operator S. Then A is P-symmetric if and only if
S −1 (A∗ A − AA∗ )S ∈ C◦ (B).
W∗
W∗
Proof. We have δA (SXS −1 ) = SδB (X)S −1 , then R(δA ) = SR(δB ) S −1 .
Consequently we obtain C◦ (B) = S −1 C◦ (A)S. From the Theorem 5.2[5], it
follows that A is P-symmetric if and only if A∗ A − AA∗ ∈ C◦ (A), which is
equivalent to S −1 (A∗ A − AA∗ )S ∈ C◦ (B) as required.
The inspiration for the following proposition comes from the Theorem 2.1[5],
which provide another characterization of the class of P-symmetric operators.
Proposition 4.2.6. Let A be an operator in L(H). A is P-symmetric if and
only if AT = T A implies δA (|T ∗ |) = 0 and δA (U ) = 0 for every T ∈ C1 (H),
where T has the polar decomposition T = U |T |.
Proof. Assume that A is P-symmetric. Let T ∈ C1 (H) has the polar decomposition T = U |T |, such that AT = T A. It follows that AT ∗ = T ∗ A.
Consequently we have
A|T ∗ |2 = (AT )T ∗ = T (AT ∗ ) = T T ∗ A = |T ∗ |2 A,
A|T |2 = (AT ∗ )T = (T ∗ A)T = T ∗ T A = |T |2 A.
Which implies that A|T ∗ | = |T ∗ |A and A|T | = |T |A. Also, since AT =
T A and
A|T |⊥= |T |A one obtains that (AU − U A)|T | = 0, then we get
δA (U )(ker T ) = 0. Noting A : ker T (= ker U ) −→ ker T , hence we conclude that δA (U ) = 0.
Conversely, Let T ∈ C1 (H) such that AT = T A. By virtue of hypothesis we
have δA (|T ∗ |) = 0 and δA (U ) = 0. From where, it results that A|T ∗ | = |T ∗ |A
R(T ) reduces A
T )⊥ reduces A, and
and A|T | = |T |A . Clearly,
and (ker
0
⊥
the restrictions A1 = AR(T ) and A1 = A(ker T ) are normal operators.
⊥
0
0
Let A = A1 ⊕ A2 on H = H1 = R(T ) ⊕ R(T ) , and A = A1 ⊕ A2 on
H = H2 = (ker T )⊥ ⊕ ker T . For the linear operator T from H2 into H1 , we
0
have T = T1 ⊕ 0. It follows from AT = T A that A1 T1 = T1 A1 . The Fuglede∗
0
Putnam theorem’s implies that A∗1 T1 = T1 A1 , and so also AT ∗ = T ∗ A.
Thus A is P-symmetric.
4.3
47
The range of a subnormal derivation
Definition 4.3.1. Let A ∈ L(H). The operator A is said to be subnormal
if there exists a normal operator B on a Hilbert space K such that H is a
subspace of K, the subspace H is invariant under the operator B, and the
restriction of B to H coincides with A.
Definition 4.3.2. Let A be in L(H). We say that A is a cyclic operator if
there is a vector e ∈ H such that {Ak e, k ≥ 0} span H. Such vector is said
to be a cyclic vector for A.
Theorem 4.3.3. Let A be in L(H). Then we have the following statements:
(1) If A is cyclic subnormal operator then R(δA ) ∩ {A}0 = {0}.
(2) If P (A) is cyclic subnormal operator for some polynomial P then every
operator in R(δA ) ∩ {A}0 is nilpotent. In particular, every normal operator
in R(δA ) ∩ {A}0 vanishes.
Proof. (1) Suppose that A is cyclic subnormal with cyclic vector and T ∈
R(δA )∩{A}0 . Noting [T ], element of the Calkin algebra, is in R(δ[T ] )∩{[T ]}0 .
Since [T ] is normal by a Result of Berger and Shaw [3], it follows from [1]
that T is compact. T commutes with A, then T is subnormal by Yoshino’s
Theorem [14]. But any compact subnormal operator is normal. Since T is
quasinilpotent from Theorem 3 [13] , we conclude that T = 0.
(2) Assume that T ∈ R(δA ) ∩ {A}0 . Then there exists (Xn )n in L(H) such
that AXn − Xn A −→ T ∈ {A}0 . Therefore
P (A)Xn − Xn P (A) −→ P (1) (A)T,
which shows that P (1) (A)T belongs to R(δP (A) ) ∩ {P (A)}0 , hence it results
that P (1) (A)T = 0. On the other hand
P (1) (A)Xn − Xn P (1) (A) −→ P (2) (A)T,
which gives
0 = T P (1) (A)Xn T − T Xn P (1) (A)T −→ P (2) (A)T 3 ,
that is P (2) (A)T 3 = 0. By repeating the same argument, it follows that
P (m) (A)T m+1 = 0 where m is the degree of P , hence T is nilpotent. Consequently every normal operator in R(δA ) ∩ {A}0 vanishes.
Lemma 4.3.4. Let A ∈ L(H) and f be an analytic function on an open
0
0
set containing σ(A). If T ∈ R(δA ) ∩ {A} , then f (A)T ∈ R(δf (A) ).
48
Proof. Let γ be a Jordan system that lies entirely in the domain of regularity
of f and encloses σ(A). We have
Z
1
f (A) =
f (λ)(λ − A)−1 dλ,
2πi γ
Z
1
0
f (A) =
f (λ)(λ − A)−2 dλ.
2πi γ
Suppose that T ∈ R(δA ) ∩ {A}0 . Then there exists a sequence (Xn ) in L(H)
such that
kAXn − Xn A − T k −→ 0.
It follows that
0
f (A)Xn − Xn f (A) − f (A)T =
1
2πi
Z
f (λ)[(λ − A)−1 Xn − Xn (λ − A)−1 − (λ − A)−2 T ]dλ,
γ
0
Since T ∈ {A} , then (λ − A)−2 T = (λ − A)−1 T (λ − A)−1 . Consequently we
obtain
(λ−A)−1 Xn −Xn (λ−A)−1 −(λ−A)−2 T = (λ−A)−1 (AXn −Xn A−T )(λ−A)−1 .
This implies that
kf (λ)[(λ − A)−1 Xn − Xn (λ − A)−1 − (λ − A)−2 T ]k
≤ sup kf (λ)kk(λ − A)−1 k2 kAXn − Xn A − T k.
λ∈γ
Hence
0
kf (A)Xn − Xn f (A) − f (A)T k −→ 0,
and the proof is complete.
Corollary 4.3.5. Let A ∈ L(H). If f (A) is cyclic subnormal operator,
where f is an analytic function on an open set containing σ(A), such that
f 0 does not vanish on some neighborhood of σ(A), then I ∈
/ R(δA ).
Proof. Suppose that I ∈ R(δA ). Then there exists a sequence (Xn )n in L(H)
such that AXn − Xn A −→ I. It follows from lemma 4.3.4 that
f (A)Xn − Xn f (A) −→ f 0 (A).
Therefore f 0 (A) ∈ R(δf (A) )∩{f (A)}0 implies f 0 (A) = 0 from Theorem 4.3.3.
Hence, by the Theorem of the minimal equation [7], f 0 vanishes on some
neighborhood of σ(A), which is absurd. Then I ∈
/ R(δA ).
49
Remark 4.3.6. J.Anderson proved that R(δA ) ∩ {A}0 = {0} if A is normal
or isometric [1]. Hence the above Theorem 4.3.3 and Corollary 4.3.5 are
still hold with respect to such class of operators.
Theorem 4.3.7. Let A ∈ L(H). If A is normal, isometric or cyclic subW
normal then R(δA ) ∩ {A}0 contains no nonzero compact operator.
Proof. (1) Suppose that A is a normal operator and let T be compact operW
0
ator in R(δA ) ∩ {A} . It follows from the Fuglede-Putnam Theorem that
W
0
T ∗ T ∈ R(δA ) ∩ {A} . Hence there exists a generalized sequence (Xα )α of
operators in L(H) such that
W
AXα − Xα A −→ T ∗ T.
Take 0 6= λ ∈ σ(T ∗ T ) and let P be the orthogonal projection onto finite
dimensional subspace ker(T ∗ T − λ). Note that P commutes with A. Then
W
from AXα − Xα A −→ T ∗ T it follows that
W
(P AP )(P Xα P ) − (P Xα P )(P AP ) −→ P (T ∗ T )P.
Then, since the trace of the left side is 0, λ must be 0. Thus σ(T ∗ T ) = {0},
hence T = 0.
W
(2) Let T ∈ R(δA ) ∩ {A}0 , where A is isometric and T is compact. The
condition AT = T A implies A∗ T A = T . It follows from Theorem 2.2 [12]
W
that AT A∗ = T that is A∗ T = T A∗ . Then we have T ∗ T ∈ R(δA ) ∩ {A}0 .
By using Theorem 3 [13] we obtain that σ(T ∗ T ) = {0}, hence T = 0.
(3) We consider now the case where A is a cyclic subnormal operator. Let
W
T be a compact operator in R(δA ) ∩ {A}0 . Since A commutes with T ,
then it follows from Yoshino’s Result [14] that T is subnormal and therefore
normal. Hence we get from Theorem 3 [13] that T = 0.
Remark 4.3.8. The preceding Theorem gives another version of Weber’s
Theorem [13] for normal operators, isometries, co-isometries and cyclic subnormal operators.
An immediate consequence of Theorem 4.3.3 and Remark 4.3.6 is the following result.
Corollary 4.3.9. If A ∈ L(H) satisfies one of the following conditions,
(1) A∗ A − AA∗ is compact.
(2) A∗ A − I or AA∗ − I is compact.
(3) P (A) is normal, isometric or cyclic subnormal.
Then every hyponormal operator in R(δA ) ∩ {A}0 vanishes.
Bibliography
[1] J.H. Anderson, On normal derivations, Proc. Amer. Math. Soc.
38(1973) 135-140.
[2] J.H. Anderson, J.W. Bunce, J.A. Deddens and J.P. Williams, C ∗ algebras and derivation ranges, Acta. Sci. Math. 40(1978) 211-227.
[3] C. A. Berger, B.I. Shaw, Self-commutators of multicyclic hyponormal
operators are always trace class, Bull. Amer. Math. Soc. 79(1973) 11931199.
[4] S. Bouali and J. Charles, Extension de la notion d’opérateur dsymétrique. I, Acta. Sci. Math. (Szeged) 58(1993) 517-525.
[5] S. Bouali and J. Charles, Extension de la notion d’opérateur dsymétrique. II, Linear algebra and its applications. 225(1995) 175-185.
[6] B.P. Duggal, Putnam-Fuglede theorem and the range kernel orthogonality of derivations, IJMMS, 27(2001), 573-582.
[7] N. Dunford, J.T. Schwartz, Linear operators I, Pure and Appl. Math.
vol 7. Interscience, New York(1958).
[8] D.C. Kleinecke, On operator commutators, Proc. Amer. Math. Soc.
8(1957)535-536.
[9] C.R. Rosentrater, Not every d-symmetric operator is GCR, Proc.
Amer. Math. Soc. 81(1981) 443-446.
[10] C.R. Rosentrater, Compact operators and derivations induced by
weighted shifts, Pacific. J. Math. 104(1983) 465-470.
[11] J. G. Stampfli , On self-adjoint derivation ranges, Pacific. J. Math.
82(1979) 257-277.
[12] A. Turnsěk, Orthogonality in Cp classes, Monatsh. math. 132(2001)
349-354.
50
BIBLIOGRAPHY
51
[13] R.E. Weber, Derivations and the trace class operators, Proc. Amer.
Math. Soc. 73(1979) 79-82.
[14] T. Yoshino, Subnormal operators with a cyclic vector, Tohoku Math.
J. 21(1969) 47-55.
Chapter 5
On the range of the elementary
operator X 7−→ AXA − X
1
Abstract : Let L(H) denote the algebra off all bounded linear operators
on a separable infinite dimensional complex Hilbert space H into itself.
Given A ∈ L(H), we define the elementary operator ∆A : L(H) −→ L(H)
by ∆A (X) = AXA − X. In this paper, we initiate the study of the class of
operator A for which R(∆A ) = R(∆A∗ ), where R(∆A ) denotes the norm
closure of the range of ∆A . We called such operators quasi-adjoint. We
give a characterization and some basic results concerning this class of
operators.
5.1
Introduction
Let H be a separable infinite dimensional complex Hilbert space and let
L(H) denote the algebra of all bounded linear operators on H into itself.
Given A, B ∈ L(H), we define the elementary operator ∆A,B as follows
∆A,B : L(H) −→ L(H)
X 7−→
∆A,B (X) = AXB − X.
If A = B, we simply write ∆A for ∆A,A . The properties of elementary operators, their spectrum (see [6],[7],[10]), norm ([15],[18] and [19]) and ranges
([1], [2] ,[3], [5],[9], [11], [12],[13], [16], [17] and [20]) have been much studied,
1
Math. Proc. R. Ir. Acad. 108(2008), no. 1, 1-6.
52
Chapter 5 :On the range of the elementary operator X 7−→ AXA − X
53
and many of its problems remains also open [12].
L. Fialkow [8] and Z. Genkai [14] studied the problem of characterizing operators A, B ∈ L(H) for which R(∆A,B ), the range of ∆A,B , is dense in
L(H) for the norm topology.
Our aim in this paper is a modest one. In the first section, we provide a
characterization of the case when the range R(∆A,B ) is weakly and ultraweakly dense in L(H). Complementary results related to the range of the
elementary operator ∆A,B are also given.
In the second section, the particular classes which have drawn a lot of attention are those consisting of operators A for which R(∆A ) = R(∆A∗ ), where
R(∆A ) is the closure of the range R(∆A ) of ∆A in the norm topology. Such
operators are called quasi-adjoint. We give a characterization and some basic properties about this class of operators. Finally, we shall pose some open
questions suggested by our results.
Notations and definitions.
(1) Let L(H) be the algebra of all bounded linear operators acting on
a complex separable Hilbert space H, Let K(H) denote the ideal of all
compact operators
on H, B(H) the class of all finite rank operators and
C(H) = L(H)K(H) the Calkin algebra.
(2) Given A, B ∈ L(H), we shall denote R(∆A,B ) the range of the elementary operator ∆A,B and ker(∆A,B ) the kernel of ∆A,B .
W
Let R(∆A,B ) be the norm closure, R(∆A,B )
W∗
and R(∆A,B )
will denote the weak closure,
denote the ultra-weak closure of the range R(∆A,B ).
(3) Let C1 (H) be the ideal of trace class operators. The ideal C1 (H) admits
a complex valued function tr(T ) which has the characteristic P
properties of
the trace of matrices. The trace function is defined by tr(T ) = n (T en , en ),
where (en ) is any complete orthonormal system in H.
As a Banach spaces, C1 (H) may be identified with the conjugate space
of the ideal K(H) of compact operators by means of the linear isometry
T 7−→ fT , where fT (X) = tr(XT ). Moreover, L(H) is the dual of C1 (H).
The ultra-weak continuous linear functionals on L(H) are those of the form
fT for some T ∈ C1 (H), and the weak continuous linear functionals on L(H)
are those of the form fT where T ∈ B(H).
If ϕ is a linear functional on L(H), then ϕ∗ the adjoint of ϕ is defined by
ϕ∗ (X) = ϕ(X ∗ ) for all X ∈ L(H).
54
(4) Recall that for x, y ∈ H, the operator x ⊗ y ∈ L(H) is defined by
(x ⊗ y)z = (z, y)x for all z ∈ H.
(5) Let S be a subspace of L(H), we denote the polar of S by
0
S ◦ = {f ∈ L (H) : f (x) = 0 f or all x ∈ S}.
5.2
The range of the elementary operator ∆A,B
Lemma 5.2.1. Let S1 and S2 be two subspaces of L(H). Then S ◦ 1 ⊂ S ◦ 2
if and only if S2 ⊂ S1 .
Proof. Let S ◦ 1 ⊂ S ◦ 2 and suppose that S2 6⊂ S1 . Hence, there is an x ∈ S2
and x 6∈ S1 .
0
The Hahn Banach Theorem guarantees that there exists f ∈ L (H) such
that f |S1 = 0 and f (x) = dist(x, S1 ) > 0. Since S ◦ 1 ⊂ S ◦ 2 and f ∈ S ◦ 1 , it
follows that f ∈ S ◦ 2 , then we have f (x) = 0, which is absurd.
For the converse if S2 ⊂ S1 , then it is easily seen that S ◦ 1 ⊂ S ◦ 2 .
Theorem 5.2.2. Let A, B ∈ L(H), then
R(∆A,B )◦ ' R(∆A,B )◦ ∩ K(H)◦ ⊕ ker(∆B,A ) ∩ C1 (H)
Proof. Let Φ be a norm continuous linear functional that vanishes on R(∆A,B ).
The result of J. Dixmier [4] guarantees that Φ = ΦT + Φ◦ , where T ∈ C1 (H)
and Φ◦ ∈ K(H)◦ . Let x, y ∈ H then we have
Φ(A(x ⊗ y)B) = ΦT (A(x ⊗ y)B) = tr(T Ax ⊗ B ∗ y) =< T Ax, B ∗ y >
and
Φ(x ⊗ y) = ΦT (x ⊗ y) = tr(T (x ⊗ y)) =< T x, y > .
It follows that
< T Ax, B ∗ y >=< T x, y >,
for all x, y ∈ H, and hence
ΦT (AXB) = ΦT (X)
for all finite rank operators X. Since the class of finite rank operators is
dense in L(H) relatively to the ultra-weak operator topology, it follows that
ΦT ∈ R(∆A,B )◦ . This implies that
Φ◦ = Φ − ΦT ∈ R(∆A,B )◦ .
Conversely, the preceding computation shows that if BT A = T and T ∈
C1 (H), then ΦT ∈ R(∆A,B )◦ , and the proof is complete.
55
Theorem 5.2.3. Let A, B ∈ L(H). Then the following statements are
equivalent:
W∗
(1) R(∆A,B ) = L(H).
(2) K(H) ⊂ R(∆A,B ).
(3) ker(∆B,A ) ∩ C1 (H) = {0}.
Proof. The negation of (1) and (3) is equivalent to the fact that there exists
a nonzero ultraweakly continuous linear form ΦT , such that ΦT ∈ R(∆A,B )◦ .
By Theorem 5.2.2 this occurs if and only if R(∆A,B )◦ 6⊂ K(H)◦ . It follows
from the Lemma 5.2.1 , that the last condition is equivalent to K(H) 6⊂
R(∆A,B ).
Corollary 5.2.4. Let A, B ∈ L(H), then
W∗
R(∆A,B ) ∩ K(H) = R(∆A,B )
∩ K(H)
W∗
Proof. Let K ∈ R(∆A,B ) ∩ K(H). If Φ ∈ R(∆A,B )◦ , then it result from
Theorem 5.2.2 that Φ = ΦT + Φ◦ , where Φ◦ ∈ K(H)◦ ∩ R(∆A,B )◦ and
W∗
0
T ∈ C1 (H) ∩ ker(∆B,A ). Since ΦT ∈ R(∆A,B )◦ ∩ L (H) , it follows that
Φ(K) = ΦT (K) + Φ◦ (K) = 0.
This implies that
K ∈ ∩{ker(Φ) : Φ ∈ R(∆A,B )◦ } = R(∆A,B ).
W
(1) Every finite rank operator in R(∆A,B ) ∩ ker(∆A∗ ,B ∗ ) vanishes.
W∗
(2) Every trace class operator in R(∆A,B )
∩ ker(∆A∗ ,B ∗ ) vanishes.
W
Proof. (1) Let T be a finite rank operator in R(∆A,B ) ∩ ker(∆A∗ ,B ∗ ), then
T ∗ ∈ ker(∆B,A ) ∩ B(H). It follows that ΦT ∗ vanishes on the range of ∆B,A .
Particularly, it results that ΦT ∗ (T ) = tr(T ∗ T ) = 0, that is T ∗ T = 0, thus
T = 0.
(2) It suffices to replace B(H) by C1 (H) in the above proof.
W
(1) R(∆A,B ) = L(H) if and only if ker(∆B,A ) ∩ B(H) = {0} .
W∗
(2) R(∆A,B )
= L(H) if and only if ker(∆B,A ) ∩ C1 (H) = {0} .
W
Proof. (1) Suppose that R(∆A,B )
W
results that T ∗ ∈ R(∆A,B )
= L(H) and T ∈ ker(∆B,A ) ∩ B(H). It
∩ ker(∆A∗ ,B ∗ ), hence T = 0 by Theorem 5.2.5.
W
Conversely, assume that there exists T ∈ L(H)R(∆A,B ) . It follows that
56
there is an operator S ∈ B(H) such that tr(ST ) 6= 0 and tr(SX) = 0
for each X ∈ R(∆A,B ). Hence, we obtain that S ∈ ker(∆B,A ) ∩ B(H) and
S 6= 0.
(2) It suffices to replace B(H) by C1 (H) in the preceding proof.
Remark 5.2.7. If A, B ∈ L(H) such that kAkkBk < 1, then Corollary
W
W∗
5.2.3 and Theorem 5.2.6 show that R(∆A,B ) = R(∆A,B ) = L(H).
W
W
1) R(∆B ) ⊂ R(∆A ) if and only if ker(∆A ) ∩ B(H) ⊂ ker(∆B ) ∩ B(H).
W∗
W∗
2) R(∆B ) ⊂ R(∆A ) if and only if ker(∆A )∩C1 (H) ⊂ ker(∆B )∩C1 (H).
Proof. (1) Assume that ker(∆A ) ∩ B(H) ⊂ ker(∆B ) ∩ B(H). Let ΦT be a
weakly continuous linear form that vanishes on R(∆A ). Then it is easy to
see that
ΦT (AXA − X) = tr[T (AXA − X)] = tr[(AT A − T )X] = 0,
for all X ∈ L(H), hence AT A = T and T ∈ ker(∆A ) ∩ B(H) ⊂ ker(∆B ) ∩
B(H). Observe that
ΦT (BXB − X) = tr[T (BXB − X)] = 0,
W
W
then ΦT annihilates R(∆B ). It follows that R(∆B ) ⊂ R(∆A ) .
For the opposite implication we reverse the above argument.
(2) It suffices to replace B(H) by C1 (H) in the preceding proof.
Remark 5.2.9. Let A = (A1 , A2 , · · · , An ) and B = (B1 , B2 , · · · , Bn ) be
n-tuples of operators in L(H),
Plet RA,B denote the elementary operator on
L(H) defined by RA,B (X) = ∞
i=1 Ai XBi . Note that the above results still
hold for the elementary operator RA,B .
5.3
Quasi-adjoint operators
Definition 5.3.1. Let A be a C ∗ -algebra and let a ∈ A. We say that a is
quasi-adjoint if R(∆a ) = R(∆a∗ ).
Remark 5.3.2. Let A ∈ L(H), then A is quasi-adjoint if and only if R(∆A )
is a self adjoint subspace of L(H). Equivalently, R(∆A )◦ the annihilator of
0
R(∆A ) is a self adjoint subspace of L (H) in the sense that f ∈ R(∆A )◦
implies f ∗ ∈ R(∆A )◦ .
57
Theorem 5.3.3. If A ∈ L(H) the following statements are equivalent
(1) A is quasi-adjoint.
(2) (i) the element [A] of the Calkin algebra is quasi-adjoint, and
(ii) for T ∈ C1 (H), AT A = T implies A∗ T A∗ = T .
W∗
W∗
(3) (i)R(∆A ) = R(∆A∗ ) .
(ii)[A] is quasi-adjoint.
Proof. (1) =⇒ (2). Suppose that A is quasi-adjoint. (i) Let ψ ∈ R(∆[A] )◦ .
We define the bounded linear functional f on L(H) by f (X) = ψ([X]). It is
clear that f ∈ R(∆A )◦ if and only if ψ ∈ R(∆[A] )◦ . Since A is quasi-adjoint,
it follows from the above remark that f ∗ ∈ R(∆A )◦ and consequently ψ ∗ ∈
R(∆[A] )◦ . Then [A] is quasi-adjoint.
(ii) If AT A = T and T ∈ C1 (H), then Theorem 5.2.2 implies that fT ∈
R(∆A )◦ . Since A is quasi-adjoint, it follows that (fT )∗ = fT ∗ ∈ R(∆A )◦ ,
from which we get A∗ T A∗ = T .
(2) =⇒ (1) Let f ∈ R(∆A )◦ . We can write f = f◦ +fT , where f◦ ∈ R(∆A )◦ ∩
K(H)◦ and T ∈ ker(∆A ) ∩ C1 (H). By using (ii) one obtains A∗ T A∗ = T ,
that is fT ∗ ∈ R(∆A )◦ . It remains to show that f◦∗ ∈ R(∆A )◦ . Let ϕ the
linear functional on the Calkin algebra defined by ϕ([X]) = f◦ (X). Since f◦
vanishes on K(H), it follows that ϕ is well defined . From(i) [A] is quasiadjoint, then ϕ ∈ R(∆[A] )◦ implies that ϕ∗ ∈ R(∆[A] )◦ that is f◦∗ ∈ R(∆A )◦ .
Thus we have shown that f ∗ = f◦∗ + fT ∗ ∈ R(∆A )◦ , consequently A is quasiadjoint.
W∗
W∗
0
∗
(2) ⇐⇒ (3) R(∆A ) = R(∆A∗ ) if and only if, f ∈ R(∆A )◦ ∩ L (H)W
0
∗
implies f ∗ ∈ R(∆A )◦ ∩ L (H)W . Then it follows from the Theorem 5.2.2
that
0
∗
R(∆A )◦ ∩ L (H)W ∼
= ker(∆A ) ∩ C1 (H)
This completes the proof.
Proposition 5.3.4. Let V ∈ L(H). If V is an isometry then V is quasiadjoint.
Proof. Let V be an isometry. We consider the operator P defined by P =
I − V V ∗ . It obvious that ∆V ∗ (X) = ∆V (−V ∗ XV ∗ ) − P X for all X in L(H).
Hence, It suffices to show that P L(H) ⊆ R(∆V ). Let (Tn )n a sequence of
operators defined by
Tn =
n−1
X
k−n
k=0
We have
n
V k P XV k .
n
1X k
∆V (Tn ) − P X = −
V P XV k .
n k=1
58
But it is easy to see that < V k P x, V j P y >= 0 for every x, y ∈ H and for
all k 6= j. Hence, it follows that
n
n
X
X
k
k 2
V P XV x =
kV k P XV k xk2 ≤ nkP Xk2 kxk2
k=1
k=1
For all x ∈ H. Which implies that
k∆V (Tn ) − P Xk ≤
√1 kP Xk
n
, conse-
quently we obtain P X ∈ R(∆V ). Then, V is quasi-adjoint.
Proposition 5.3.5. Let A and B be quasi-adjoint operators. If 1 6∈ σ(A)σ(B)
then A ⊕ B is quasi-adjoint.
Proof. Let X be an operator on H ⊕ H. It is obvious to check that
R(∆A ) R(∆A,B )
R(∆A⊕B ) =
R(∆B,A ) R(∆B )
Under the hypotheses 1 6∈ σ(A)σ(B), it follows from [6, Theorem 3.2]
that ∆A,B and ∆B,A are invertible. Hence,
R(∆A ) L(H)
R(∆A⊕B ) =
L(H) R(∆B )
Since A and B are quasi-adjoint,then we have X ∈ R(∆A⊕B ) implies X ∗ ∈
R(∆A⊕B ). Consequently A ⊕ B is quasi- adjoint.
Proposition 5.3.6. Let A ∈ L(H). If there exists α, β ∈ C with αβ = 1
and nonzero vectors f, g ∈ H such that ,
(i) Af = αf and kA∗ f k =
6 kαf k.
∗
(ii) A g = β̄g.
Then A is not quasi-adjoint.
Proof. We must show that R(∆A ) 6= R(∆A∗ ). Suppose first that A∗ f 6= 0.
Let us consider the operator T = g ⊗ A∗ f . It is clear that < (AY A −
Y )f, g >= 0 for all Y ∈ L(H).
On the other hand, we have
< (A∗ T A∗ − T )f, g >= β̄(kA∗ f k2 − kαf k2 )kgk2 .
W∗
If A∗ T A∗ − T ∈ R(∆A ) . Then there exists a sequence (Xn )n in L(H)
such that
AXn A − Xn −→ A∗ T A∗ − T.
59
Which gives
0 =< (AXn A−Xα )f, g >−→< (A∗ T A∗ −T )f, g >= β̄(kA∗ f k2 −kαf k2 )kgk2 .
It follows that β̄(kA∗ f k2 − kαf k2 )kgk2 = 0 which is absurd.
If A∗ f = 0 we consider the operator T = g ⊗ f . By repeating the same
argument we get the result.
Let (ek )k∈Z be an orthonormal basis for H and let S be the bilateral weighted
shift Sen = ωn en+1 , n ∈ Z, with nonzero weights ωn . By taking a unitarily
equivalent weighted shift, we may assume that ωn = |ωn | > 0.
Proposition 5.3.7. Let S be the bilateral shift SeP
i = ωi ei+1 , such that ωi ≥
0 for all i ∈ Z. Then, K(H) ⊆ R(∆S ) implies that j∈Z ωj−n ωj−n+1 · · · ωj+n−1 =
∞ for every n ∈ N.
Proof. Assume K(H) ⊆ R(∆S ). It follows from Theorem 5.2.3 that ker(∆S )∩
C1 (H) = {0}. Let X be a nonzero operator in ker(∆S ). Then
P SXS = X
n
n
implies that S XS = X for all n ∈ N. If we set Xej = k∈Z bk,j ek for
j ∈ Z, then a simple calculation shows that
X
Xej =
ωk−n · · · ωk−1 ωj · · · ωj+n−1 bk−n,j+n ek
k∈Z
From where we get
bk,j = ωk−n · · · ωk−1 ωj · · · ωj+n−1 bk−n,j+n
for all j, k ∈ Z and n ∈ N.
X
X
|bj,j | =
ωj−n · · · ωj+n−1 |bj−n,j+n |.
j
j
≤ kXk
X
ωj−n · · · ωj+n−1 .
j
Thus
P
j
ωj−n · · · ωj+n−1 = ∞ for every n ∈ N.
Open questions:
1)
P The above proposition suggest the following question:
for all n ∈ N if and only if K(H) ⊆
j∈Z ωj−n ωj−n+1 · · · ωj+n−1 = ∞
R(∆S )?.
2) Is every normal operator quasi-adjoint?.
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AXB − X, Journal of Fudan University 23(2)(1989), 148-156.
[15] B. Magajna, The norm of asymmetric elementary operator,
30(2)(1987), The Proceedings of the American Mathematical Society 132(2003),1747-1754.
[16] M. Mathieu, Rings of quotients of ultraprime Banach algebras with
applications to elementary operators, Proceedings of the Centre for
Mathematical Analysis, Australian National University 21(1989),297317.
[17] M. Mathieu, How to use primeness to describe properties of elementary operators, Proceedings of Symposia in Pure Mathematics
51(2)(1990),195-199.
[18] M. Mathieu, The norm problem for elementary operators, Recent
progress in functional analysis, ( Valencia 2000), 363-368, NorthHolland Math. Stud. 189, North-Holland, Amsterdam (2001).
[19] L. L. Stachò and B. Zalar, On the norm of Jordan elementary operators in standard operator algebra, Publications Mathematicae Debrecen 49(1996), 127-134.
[20] A. Turnšek, On the range of elementary operators, Publications
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UNIVERSITÉ MOHAMMED V – AGDAL
FACULTÉ DES SCIENCES
RABAT
La synthèse de divers travaux sur les opérateurs P- Symétriques et les opérateurs Finis
fait l'objet du premier chapitre.
Au second chapitre, on a établi l'orthogonalité de l'image au noyau d'une dérivation
δ A , induite par un opérateur sous-normal cyclique A au sens de la norme usuelle
d’opérateurs. On a obtenu une autre preuve du résultat principal de F. Wening et J. Guo
Xing. On a déduit une caractérisation des opérateurs P- Symétriques. On a caractérisé
aussi les opérateurs A tels que la paire ( A, A ) admet la propriété de Fuglede- Putnam
dans C p ( H ) pour p>1, où C p ( H ) est la classe de Von Newman- Schatten.
Au troisième chapitre, on a considéré la classe des opérateurs Finis. On a adopté des
démarches différentes et simples pour établir et généraliser certains résultats rencontrés
dans la littérature.
Au chapitre suivant, on s'est intéressé à l'étude de la classe des opérateurs PSymétriques et à l'intersection des commutants et des fermetures faibles et en norme
des images de dérivations. On a donné de nouvelles classes d’opérateurs A
vérifiant I ∉ R (δ A ) , où
(δ ( X ) = AX − XA) .
R (δ A ) est la fermeture en norme de l’image de δ A ,
A
Au dernier chapitre, on a présenté les propriétés dont jouit l'image d'un opérateur
élémentaire. On a initie l'étude sur la classe des opérateurs Quasi-adjoints i.e.
( )
opérateurs A tels que R ( Δ A ) = R Δ A∗ , où R ( Δ A )
l'image R ( Δ A ) de l’opérateur
est la fermeture en norme de
Δ A ( X ) = AXA − X . A ces opérateurs on a donné une
caractérisation. On a tenu à démontrer quelques propriétés de base concernant cette
classe d'opérateurs.
-----------------------------------------------------------------------------------------------------------------------Mots-clefs (5) : Opérateur Élémentaire, Orthogonalité Image-Noyau, Propriété de
Fuglede- Putnam, Opérateur P-Symétrique, Opérateur Fini, Technique d'Extension
de Berberian, Opérateur Quasi-adjoint.
-----------------------------------------------------------------------------------------------------------------------
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

On the Range and the Kernel of Elementary Operators.

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