FACULTÉ
DES
SCIENCES
R
ABATFaculté des Sciences, 4 Avenue Ibn Battouta B.P. 1014 RP, Rabat – Maroc Tel +212 (0) 37 77 18 34/35/38, Fax : +212 (0) 37 77 42 61, http://www.fsr.ac.ma
N° d’ordre : 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.
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 encour-agements 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 élabo-ration, 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 rap-porter 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 Fac-ulté 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 pro-fonde 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.
fesseur Bhagwati Prashad Duggal m’ont beaucoup apporté et sa contri-bution 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 Fontion-nelle de la Faculté des Sciences de Rabat, groupe d’Analyse FonctionFontion-nelle 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 Mathé-matiques 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 Mathé-matiques 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.
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 P-Symmetric 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 P-Symmetric 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 ele-mentary operator. We initiate the study of the class of Quasi-adjoint
opera-tors, 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
1 Preliminaries 9
1.1 P-symmetric operators . . . 9
1.2 Finite operators . . . 14
1.3 The essential spectrum . . . 16
1.4 The Riesz idempotent . . . 18
2 On the range kernel orthogonality and P-symmetric oper-ators 22 2.1 Introduction . . . 22
2.2 The range-kernel orthogonality . . . 24
2.3 P-symmetric operators . . . 29
3 On the range and the kernel of derivations 34 3.1 Introduction . . . 34
3.2 Main Results . . . 35
4 The P-symmetric operators and the range of a subnormal derivation 42 4.1 Introduction . . . 42
4.2 P-symmetric operators . . . 44
4.3 The range of a subnormal derivation . . . 47
5 On the range of the elementary operator X 7−→ AXA − X 52 5.1 Introduction . . . 52
5.2 The range of the elementary operator ∆A,B . . . 54
5.3 Quasi-adjoint operators . . . 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= δAet ∆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 δAdans 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 P-symétrique. 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
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 δAet 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
déri-vation 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
général-isée δA,B associé à deux opérateurs normaux A et B. Depuis, plusieurs
au-teurs 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
or-thogonal 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.
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 généralisa-tion 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 opéra-teurs 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 R(δA)
W
∩{A}0 est nul, avec R(δA) W
est la fermeture de R(δA) pour la
topolo-gie faible et {A}0 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 opéra-teurs P-symétriques.
En seconde partie, on a présenté une nouvelle classe d’opérateurs A ∈ L(H), satisfaisant R(δA) ∩ {A}
0
= {0}. Ainsi, on a montré que si A est un
opéra-teur sous-normal cyclique alors R(δA)∩{A}
0
= {0}. Comme conséquence on a prouvé que si P (A) est est normal, isométrique, co-isométrique ou sous-normal cyclique pour un certain polynôme P , alors tout opérateur dans R(δA) ∩ {A}
0
est nilpotent. Nous trouvons également de nouvelles classes
celui de J.A. Stampfli dans [34]. Le Théorème de Weber [38] affirme que tout
opérateur compact dansR(δA)
W
∩ {A}0 est quasi-nilpotent, où R(δA) W
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 R(δA)
W
∩ {A}0 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
quasi-adjoints (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 opéra-teurs 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−→+∞kAnk = 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
L(H), converges weakly to 0, denoted by Aα
W
−→ 0 weakly, if and only if, < Aαx, y >
α
−→ 0 for all x, y ∈ H. Let E be a subspace of L(H), we denote
by Ew 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(TiX)| < ε, i = 1, 2, · · · n}.
If E is a subspace of L(H), then Ew
∗
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
singular values of A, i.e. the eigenvalues of |A| = (A∗A)12 arranged in their
decreasing order. The operator A is said to belong to the Schatten p-class
Cp(H) if kAkp = ∞ X j=1 sj(A)p 1p =tr(A)p 1 p < ∞ , 1 ≤ p < ∞,
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 =
P∞
n=1kAenk2
12 < ∞,
where (en)n is an orthonormal basis for H.
(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)nin H such that kxnk = 1 and k(A−λI)xnk −→ 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
(15) Let A ∈ L(H), the commutant of A, {A}0 is defined by
{A}0 = {B ∈ L(H) : AB = BA}
(16) Let A ∈ L(H), the bicommutant of A, {A}00 is defined by
{A}00 = {C ∈ L(H) : CB = BC for all B ∈ {A}0}
(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
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∗kAxk ≤ kAk+1xk, 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.
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 ifR(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).
(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.
Let R(δA,B) be the norm closure, R(δA,B)
W
will denote the weak closure, and R(δA,B)
W∗
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 properties of
the trace of matrices. The trace function is defined by tr(T ) =P
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).
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 equiv-alent
(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
(1) A is P-symmetric if and only if R(δA)
W∗
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
(2) There exists y ∈ H, y 6= 0 with A∗y = λy.
Then R(δA)
W∗
fails to be self-adjoint.
Example 1.1.6. Let (en)n≥1 be an orthonormal basis of H. Let H◦ =
vect{e1, e2} and define A◦ =
1 0 1 1
∈ L(H◦). Consider the operator
A = A◦ 0
0 I
on H = H◦⊕ H◦⊥. Clearly, we have Ae2 = e2, A∗e2 6= e2
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.
Then R(δA,B)
W∗
fails to be self-adjoint.
Theorem 1.1.8. [5] Let A ∈ P (H). Then the following statements are equivalent (1) {A}0∩ C1(H) 6= {0}. (2) A ∈ ∪n≥1Rn. (3) σpr(A) 6= φ. (4) K(H) 6⊂ R(δA) W .
Example 1.1.9. Let (en)n≥1 be orthonormal basis for H. Define the
oper-ator S as follows:
Sek = ωkek+1 , where ωk=
2, k odd
2−1, k even
Then S satisfies Σ∞k=1ωkωk+1· · · ωk+n= ∞ for all n ≥ 1, it follows from [29]
that K(H) ⊂ R(δS). Which is equivalent to {S}
0
∩ 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
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 by J en= znχDα, where Dα = {z ∈ C : |z| ≤ α < 1}. If we set A = M 0 0 S and T = 0 J 0 0 , then we have kJenk2 = Z Dα |z2n|dz = Z 2π 0 Z α 0 r2ndrdθ = 2πα α 2n 2n + 1. kT k1 ≤ Σ∞n=1kJenk = √ 2παΣ∞n=1 α 2n √ 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 = J S, 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.
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]).
Theorem 1.1.13. [5] Let A =R 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 = ωkek+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 =
ωkek+1, k ∈ N(Z). Then S is P-symmetric if and only if
X
k
ωkωk+1· · · ωk+n= ∞,
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:
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) W∗ }. 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).
(3) R(δB) ⊂ R(δA) W∗
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 equiva-lent:
(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.
P. Williams [40] guarantees that C◦(A) = {0}, I◦(A) = {A}
0
and B◦(A) =
{A}00.
When H is infinite dimensional if A ∈ L(H) such that R(δA)
W∗
= L(H), then C◦(A) = I◦(A) = B◦(A) = L(H).
In [6] S. Bouali and J. Charles proved that if R(δA)
W∗
contains no nonzero
positive operator, then C◦(A) = {0} and I◦(A) = {A}
0
.
Corollary 1.1.19. [6] Let A be P-symmetric operator with countable spec-trum. Then the following assertions are equivalent:
(1) C◦(A) = {0}.
(2) A is a diagonal operator.
(3) I◦(A) = {A}
0
.
Theorem 1.1.20. [6] if A is a normal operator with finite spectrum, then C◦(A) = {0}, I◦(A) = {A}
0
and B◦(A) = {A}
00
1.2
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
n-dimensional 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 hy-ponormal operator, all compact operators, every operator of the form fi-nite+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.
Remark 1.2.5. (1) Let A ∈ L(H). If σar(A) 6= φ, then A is a finite
opera-tor.
(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
P (A) = {f ∈ A0 : 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).
1.3
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
semi-Fredholm on the Hilbert space H. The set semi-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).
Simi-larly, 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)}
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)−1K = A − λ,
is not invertible, so that I − (A + K − λ)−1K is not invertible. Hence, 1 is
an eigenvalue of the compact operator (A + K − λ)−1K. But if (A + K −
λ)−1Kx = 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.
Proof. Suppose that λ ∈ σ(A) and λ 6∈ σp(A)∪σp¯(A). Then A−λ is injective
with dense range. Since A − λ is not invertible, it follows that R(A − λ) is
1.4
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
Eλ = 1 2πi Z ∂Dλ (z − A)−1dz,
is called the Riesz idempotent with respect to λ, which has the properties that E2
λ = Eλ ; AEλ = EλA ; ker(A − λ) ⊂ EλH and σ(A|EλH) = {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
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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 P-symmetric 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−→ δA(X) = AX − XA.
1Math. Inequal. Appl. 9(2006), no. 3, 511-519.
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
commutant {A}0 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 consid-ered 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
operator and if S ∈ C2(H) ∩ {A}
0
, 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+ kSk 2 2
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
cyclic subnormal operator and S ∈ J ∩ {A}0, 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
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 con-clude 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 ideal of trace class operators, the trace function is definedon C1(H) by tr(T ) =
P
n(T en, en), where (en)n is any complete
orthonor-mal sequence in H. For 1 < p < ∞ we denote CP(H) the Von
Neumann-Schatten 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
R(δA/CP)◦ = {f ∈ (Cp(H))
0
: 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
orthog-onal 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
spec-trum 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
A = B 0 0 C , T = λ 0 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
kAX − XA + T k ≥ |λ|
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 has isolated eigenvalues of finite multiplicity. We consider the
sub-space E = W
µ∈β(T )ker(T − µ), where β(T ) is the set of all isolated
eigen-values 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
T = T1 0
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
inf K1+ T1 K2 K3 T2+ K4 , K1, K2, K3, K4 compacts ≥ inf {kT2+ K4k, K4 compact }
Then it follows immediately that
k[T ]k ≥ k[T2]k
Since T2has no isolated eigenvalues of finite multiplicity, then by the Remark
2.2.4 we have kAX − XA + T k ≥ |λ|. This implies that kAX − XA + T k ≥ |λ|
for all X ∈ L(H) and all λ ∈ σe(T ).
Finally, we conclude that
kAX − XA + T k ≥ |λ| For all X ∈ L(H) and all λ ∈ σ(T ). Then
kAX − XA + T k ≥ kT k
Example 2.2.6. Let U be the unilateral shift operator of multiplicity one.
On H ⊕ H, we consider the operators A = U 0
0 0 , T = 0 0 P 0 and X = 0 0 Q 0
, where P = 1 − U U∗ and Q = P U∗. Then A
is a noncyclic subnormal operator and T ∈ {A}0. 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
cyclic-ity assumption on A is sufficient for the range-kernel orthogonalcyclic-ity 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 ≤ kA2xkkxk, 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 AT2 = T2A. Applying the Theorem 2.2.5, we get
kT2k ≤ kAY − Y A + T2k
for all Y ∈ L(H). Since T is paranormal operator, then we have kT k2 ≤
kT2k. Replacing Y by XAT and using the fact that kT2k = 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
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
= {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
2.3
P-symmetric operators
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−1kkAk = 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
Snek = 1 ne1, if ; k=0 ek+1, otherwise. and Sek= 0, if; k=0 ek+1, 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
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 fun-damental 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 1p + 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
restric-tion 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 ∈ C
q. Therefore, it suffices to show that
R(δA/Cq)◦ ∼= {A}
0
∩ Cp
It is convenient to note that (Cq)
0
= {fT : T ∈ Cp} ∼= Cp
for all p and q such that 1p + 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.
Conversely, suppose that T ∈ {A}0 ∩ 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 .
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+ CpC ⊂ 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) =
{A}0 the commutant of A and ∆p(A) = {A}
00
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
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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
δA(X) = AX − XA.
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
1Serdica. Math. J. 32(2006), no. 1, 31-38.
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
and that A∗ 6∈ R(δA)
W
, where R(δA) W
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 de-note 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 σ
ap(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
Definition 3.2.3. Let A ∈ L(H). The operator A is said to be n-multicyclic if there exists n vectors x1, x2, · · · , xn ∈ H, called generating vectors, such
that {Pn
i=1fi(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]) ≤ nπω(σ(A)), where ω is planar
Lebesgue measure.
Theorem 3.2.5. Let A ∈ L(H). If A is an n-multicyclic hyponormal oper-ator, then for every hyponormal operator T such that AT = T A, we have
kAX − XA + T k ≥ kT k 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
com-mutes 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 exten-sion.
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
kAX − XA + T k ≥ kT k 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.
