R E S E A R C H
Open Access
On operators satisfying an inequality
Salah Mecheri
**Correspondence:
[email protected] Department of Mathematics, College of Science, Taibah University, P.O. Box 30002, Al-Madinah-Al-Munawarah, Saudi Arabia
Abstract
An operatorTis said to bek-quasi-∗-classAifT∗k(|T2|–|T∗|2)Tk≥0, wherekis a
natural number. LetdA,B∈B(B(H)) denote either the generalized derivation
δ
A,B=LA–RBor the elementary operatorA,B=LARB–I, whereLAandRBare the left
and right multiplication operators defined onB(H) byLA=AXandRB=XB
respectively. This article concerns some spectral properties ofk-quasi-∗-classA
operators in a Hilbert space, as the property of being hereditarily polaroid. We also establish Weyl-type theorems forTanddA,B, whereTis ak-quasi-∗-classAoperator
andA,B∗are alsok-quasi-∗-classAoperators.
MSC: Primary 47B47; 47A30; 47B20; secondary 47B10
Keywords: k-quasi-∗-classA;∗-paranormal operator; Weyl’s spectrum
1 Introduction and preliminaries
LetB(H) be the algebra of all bounded linear operators acting on an infinite dimensional separable complex Hilbert spaceH. As an easy extension of normal operators, hyponormal operators have been studied by many mathematicians. Although there are many unsolved interesting problems for hyponormal operators (e.g., the invariant subspace problem), one of recent trends in operator theory is studying natural extensions of hyponormal opera-tors. So, we introduce some of these non-hyponormal operaopera-tors. Recall [, ] thatT∈B(H) is called hyponormal ifT∗T≥TT∗, and T is called paranormal (resp.,∗-paranormal) if Tx ≥ Tx (resp.Tx ≥ T∗x) for all unit vectorx∈H. Following [] and [], we say thatT∈B(H) belongs to the classAif|T| ≥ |T|. Recently Jeon and Kim [] have considered the following new class of operators: we say that an operatorT∈B(H) belongs to the∗-classAif|T| ≥ |T∗|.
For brevity, we shall denote the classes of hyponormal operators, paranormal operators, ∗-paranormal operators, classAoperators, and∗-classAoperators byH,PN,PN∗,A andA∗respectively. From [] and [], it is well known that
H⊂A⊂PN and H⊂A∗⊂PN∗.
Recently in [], the authors have extended∗-classAoperators to quasi-∗-classA oper-ators. An operatorT ∈B(H) is said to be quasi-∗-classAifT∗|T|T ≥T∗|T∗|T. If we denote this class of operators byQA∗, then
H⊂A∗⊂QA∗.
As a further generalization of both∗-classAoperators and quasi-∗-classAoperators, the author in [] introducedk-quasi-∗-classAoperators. An operatorT is calledk
-quasi-∗-classAif
T∗k|T|–|T∗|Tk≥,
wherekis a natural number. LetKQA∗be the class ofk-quasi-∗-classAoperators. Thus,
H⊂A∗⊂QA∗⊂KQA∗.
The spectral properties of quasi-classAand quasi-∗-classAoperators have been inves-tigated by many authors in the recent years (a useful survey on the spectral properties of these operators may also be found in []); see also []. In this paper we extend tok -quasi-∗-classAoperators some of these results, for instance the property of being hereditarily polaroid already observed for∗-paranormal operators and∗-classAoperators defined on Hilbert spaces [].
The fine structure of the spectrum of paranormal operators for classAoperators or∗ -paranormal operators has been studied by several authors, in particular, for these classes of operators, it has been proved that they satisfy Weyl’s theorem; see for instance [, ] for paranormal operators, [] for algebraically classAoperators, in [] for quasi-∗-class Aoperators. In this paper we extend these results by proving that some other variants of Weyl’s theorem hold fork-quasi-∗-classAoperators; for instance, the so-called property (w) introduced by Rakočević in [] and studied in [] and []. All Weyl-type theorems are established forT and fordA,B;T is ak-quasi-∗-classAoperator andA,B∗ are also k-quasi-∗-classAoperators.
We begin by explaining the relevant terminology. LetXbe a complex Banach space. For a bounded linear operatorT∈B(X), letN(T) denote the null space andranTdenote the range ofT. Letp:=p(T) be theascentof an operatorT. (I.e., the smallest non-negative integerpsuch thatN(Tp) =N(Tp+). If such integer does not exist, we putp(T) =∞.) Analogously, letq:=q(T) be thedescentof an operatorT;i.e., the smallest non-negative integerqsuch thatranTq=ranTq+, and if such integer does not exist, we putq(T) =∞. It is well known that ifp(T) andq(T) are both finite, thenp(T) =q(T) [, Proposition .]. Moreover, <p(λI–T) =q(λI–T) <∞precisely whenλis a pole of the resolvent ofT; see Proposition . of Heuser []. A bounded operatorT ∈B(X) is said to bepolaroid
if every isolated point of the spectrum is a pole of the resolvent.T ∈B(X) is said to be hereditarily polaroid if the restriction ofT to any closed invariant subspace is polaroid. Letσa(T) denote the classicalapproximate point spectrum.T is said to bea-polaroidif everyλ∈isoσa(T) is a pole of the resolvent ofT. Obviously,
Ta-polaroid ⇒ Tpolaroid.
if, for every open subsetGofCand any analytic functionf :G→Xsuch that (T–z)f(z)≡ onG, we havef(z)≡ onG.
We also have
p(λI–T) <∞ ⇒ Thas SVEP atλ, ()
and dually, ifT denotes the dual ofT,
q(λI–T) <∞ ⇒ T has SVEP atλ; ()
see [, Theorem .]. In the case of Hilbert space operators, the last implication is still true if we replaceT with the Hilbert adjointT∗. A bounded operatorT∈B(X) is said to haveBishop’s property(β) if for every open subsetGofCand every sequencefn:G→H
ofH-valued analytic functions such that (T–z)fn(z) converges uniformly to in norm on compact subsets ofG,fn(z) converges uniformly to in norm on compact subsets ofG. It is known that the property (β) forTentails thatThas SVEP; see [] for details.
2 Main results
We begin by the following lemma which is the essence of this paper and it is a structure theorem of ak-quasi-∗-classAoperatorT.
Lemma .[] Let T∈B(H)be a k-quasi-∗-class A operator,the range of Tkbe not dense and
T=
T T
T
on H=ranTk⊕kerT∗k.
Then Tis a∗-class A operator,Tk= andσ(T) =σ(T)∪ {}.
As a consequence, we obtain the following corollary.
Corollary . Let T∈B(H)be a k-quasi-∗-class A operator.If Tis invertible,then T is
similar to a direct sum of a∗-class A operator and a nilpotent operator.
Proof Since by assumption /∈σ(T) we haveσ(T)∩σ(T) =∅, then there exists an op-eratorSsuch thatTS–ST=T[]. Hence,
T=
T T
T
=
I S
I
–
T
T
I S
I
.
Corollary . Let T be a k-quasi-∗-class A operator.If T is quasinilpotent,then it must be a nilpotent operator.
Proof Invoking Lemma ., we findσ(T) = . Since T is∗-class A, we conclude that
T= []. SinceTk= , a computation shows that
Tk+=
TTk Tk+
Lemma .[] Let M be a closed T -invariant subspace of H.Then the restriction T|Mof a k-quasi-∗-class A operator T to M is a k-quasi-∗-class A operator.
Theorem .[] Let T∈B(H)be k-quasi-∗-class A.Then T satisfies Bishop’s property
(β),the single valued extension property and the Dunford property(C).
Lemma . Let T∈B(H)be an algebraically k-quasi-∗-class A operator,andσ(T) ={μ},
then T–μis nilpotent.
Proof Assumep(T) isk-quasi-∗-classA for some nonconstant polynomialp(z). Since σ(p(T)) =p(σ(T)) ={p(μ)}, the operatorp(T) –p(μ) is nilpotent by Corollary .. Let
p(z) –p(μ) =a(z–μ)k(z–μ)k(z–μt)kt,
whereμj=μsforj=s. Then
=p(T) –p(μ)
m
=am(T–μ)mk(T–μ)mk(T–μt)mkt
and hence (T–μ)mk= .
In the following theorem, we will prove that an algebraicallyk-quasi-∗-classAoperator is polaroid.
Theorem . Let T be an algebraically k-quasi-∗-class A operator.Then T is polaroid.
Proof IfTis an algebraicallyk-quasi-∗-classAoperator, thenp(T) is ak-quasi-∗-classA
operator for some nonconstant polynomialp. Letμ∈iso(σ(T)), and letEμbe the Riesz idempotent associated toμdefined by
E:= πi
∂D
(μI–T)–dμ,
whereDis a closed disk centered atμwhich contains no other points of the spectrum ofT. ThenTcan be represented as follows:
T
T
,
where σ(T) ={μ} andσ(T) =σ(T)\ {μ}. SinceT is algebraicallyk-quasi-∗-classA operator by Lemma . andσ(T) ={μ}, it follows from Lemma . thatT–λIis nilpo-tent. Therefore,T–μhas finite ascent and descent. On the other hand, sinceT–μIis invertible, it has finite ascent and descent. Therefore,T–μIhas finite ascent and descent. Therefore,μis a pole of the resolvent ofT. Now ifμ∈iso(σ(T)), thenμ∈π(T). Thus,
iso(σ(T))∈π(T), whereπ(T) denotes the set of poles of the resolvent ofT. Hence,T is
polaroid.
Recall that an operatorTis said to be hereditarily polaroid if every part of it is polaroid. Hence, it follows from Lemma . that ak-quasi-∗-classAoperator is hereditarily polaroid
3 Weyl-type theorems
LetXbe a complex Banach space. For everyT∈B(X), define
E(T) :=λ∈isoσ(T) : <α(λI–T) ,
and
Ea(T) :=λ∈isoσa(T) : <α(λI–T) .
Obviously,E(T)⊆E(T)⊆Ea(T) for everyT∈L(X). Define
π(T) :=
λ∈isoσ(T) : <α(λI–T) <∞ ,
and
πa (T) :=λ∈isoσa(T) : <α(λI–T) <∞ .
Letp(T) :=σ(T)\σb(T),i.e.,p(T) is the set of all poles of the resolvent ofT.
Definition . A bounded operatorT∈B(X) is said to satisfyWeyl’s theorem, in symbol (W), ifσ(T)\σw(T) =π(T). T is said to satisfya-Weyl’s theorem, in symbol (aW), if
σa(T)\σuw(T) =πa (T).T is said to satisfy the property (w), ifσa(T)\σuw(T) =π(T).
Eithera-Weyl’s theorem or the property (w) entails Weyl’s theorem. The property (w) anda-Weyl’s theorem are independent; see [].
The concept of semi-Fredholm operators has been generalized by Berkani [, ] in the following way: for everyT∈B(X) and a nonnegative integern, let us denote byT[n]the restriction ofTtoTn(X) viewed as a map from the spaceTn(X) into itself (we setT
[]=T).
T∈B(X) is said to besemi-B-Fredholm(resp.B-Fredholm,upper semi-B-Fredholm,lower semi-B-Fredholm,) if for some integern≥, the rangeTn(X) is closed andT
[n]is a semi-Fredholm operator (resp. semi-Fredholm, upper semi-semi-Fredholm, lower semi-semi-Fredholm). In this caseT[m]is a semi-Fredholm operator for allm≥n[]. This enables one to define the index of a semi-B-Fredholm asindT=indT[n]. A bounded operatorT∈B(X) is said to be
B-Weyl(respectively,upper semi-B-Weyl,lower semi-B-Weyl) if for some integern≥,
Tn(X) is closed andT
[n] is Weyl (respectively, upper semi-Weyl, lower semi-Weyl). In an obvious way, all the classes of operators generate spectra, for instance, the B-Weyl spectrumσbw(T) and theupper B-Weyl spectrum σubw(T). Analogously, a bounded op-eratorT∈B(X) is said to beB-Browder(respectively,upper semi-B-Browder,lower semi-B-Browder) if for some integern≥,Tn(X) is closed andT
[n]is Weyl (respectively, upper semi-Browder, lower semi-Browder). TheB-Browder spectrumis denoted byσbb(T), the
upper semi-B-Browder spectrumbyσubb(T).
The generalized versions of Weyl-type theorems are defined as follows.
Definition . A bounded operatorT∈B(X) is said to satisfygeneralized Weyl’s theorem, in symbol, (gW), ifσ(T)\σbw(T) =E(T).T∈B(X) is said to satisfygeneralized a-Weyl’s
theorem, in symbol, (gaW), ifσa(T)\σubw(T) =Ea(T).T∈L(X) is said to satisfythe
In the following diagrams, we resume the relationships between all Weyl-type theorems:
(gw) ⇒ (w) ⇒ (W),
(gaW) ⇒ (aW) ⇒ (W);
see [, Theorem .], [] and []. The generalized property (w) and generalizeda-Weyl’s theorem are also independent; see []. Furthermore,
(gw) ⇒ (gW) ⇒ (W),
(gaW) ⇒ (gW) ⇒ (W);
see [] and []. The converse of all these implications in general does not hold. Further-more, by [, Theorem .],
(W) holds forT ⇔Browder’s theorem holds forTandp(T) =π(T).
LetdA,B∈B(H) denote either the generalized derivationδA,B=LA–RBor the elementary
operatorA,B=LARB–I, whereLAandRBare the left and right multiplication operators
defined onB(H) byLA=AXandRB=XBrespectively. We will show that ifA,B∗ arek
-quasi-∗-classA, thendA,Bis polaroid and satisfies all Weyl-type theorems. For this we need
the following lemmas.
Lemma .[] Let A,B∈B(H).If A,B are polaroid operators,then dA,Bis polaroid.
Lemma . If A,B∗are k-quasi-∗-class A operators,then dA,Bis polaroid.
Proof It is known in a Hilbert space [] thatBis polaroid if and only ifB∗ is polaroid.
Hence, it suffices to apply the previous lemma.
Recall that an operatorT∈B(X) is said to have the property (δ) if for every open covering (U,V) ofC, we haveX=HT(U) +HT(V).
Lemma . Let A,B∈B(H).If A,B have the property(β),then dA,Bhas SVEP.
Proof It is known [, Theorem ..] thatBsatisfies the property (β) if and only ifB∗
satisfies the property (δ). SinceA,Bhave the property (β) by Theorem .,B∗satisfies the property (δ). Hence, it results from [, Corollary ..] that bothLAandRBsatisfy the Dunford property (C). SinceLAandRBcommute, henceLA–RBandLARBhave SVEP by
[, Theorem .. and Note ..]. Therefore,dA,Bsatisfies SVEP.
Corollary . Let A,B∈B(H).If A, B∗ are k-quasi-∗-class A operators, then dA,B has SVEP.
If a Banach space operatorThas SVEP (everywhere), the single-valued extension prop-erty, thenT andT∗ satisfy Browder’s (equivalently, generalized Browder’s) theorem and
generalized Weyl’s) theorem is thatT is polaroid. Now sinceT andT∗are polaroid op-erators whenT is algebraicallyk-quasi-∗-classA, then Weyl’s theorem and generalized Weyl’s theorem hold forTandT∗whenTis algebraicallyk-quasi-∗-classA. Now, observe for polaroid operatorsTsatisfying generalized Weyl’s theorem,
E(T) =π(T) =πT∗=ET∗,
whereπ(T) is the set of poles of the resolvent ofT. Hence, for a polaroid operatorT,T∗
satisfies generalized Weyl’s theorem if and only ifT satisfies generalized Weyl’s theorem if and only ifTsatisfies Weyl’s theorem if and only ifT∗satisfies Weyl’s theorem.
Theorem . Let T,A,B∈B(H).If T,A,B∗are algebraically k-quasi-∗-class A,then the following statements are equivalent.
(i) generalized Weyl’s theorem holds forT∗(resp.fordA∗,B∗). (ii) generalized Weyl’s theorem holds forT(resp.fordA,B).
(iii) Weyl’s theorem holds forT (resp.fordA,B).
Recall that a sufficient condition for an operator T satisfying Browder’s (generalized Browder’s) theorem to satisfy Weyl’s (resp. generalized Weyl’s) theorem is thatT is po-laroid. Observe that ifT ∈B(H) has SVEP, then σ(T) =σa(T∗). Hence, if T has SVEP
and is polaroid, thenT∗satisfies generalizeda-Weyl’s (so, alsoa-Weyl’s) theorem []. It follows from Theorem . thatk-quasi-∗-classAoperator has SVEP. Thus, we have the following theorem.
Theorem . Let T,A,B∈B(H).
(i) IfT is algebraicallyk-quasi-∗-classAandA,B∗arek-quasi-∗-classA,then generalizeda-Weyl’s theorem holds forT∗(resp.fordA∗,B∗).
(ii) IfT∗is algebraicallyk-quasi-∗-classAandA,B∗arek-quasi-∗-classA,then generalizeda-Weyl’s theorem holds forT (resp.fordA∗,B∗).
Recall [] that ifTis polaroid, thenTsatisfies generalized Weyl’s theorem (resp. gener-alizeda-Weyl’s theorem) if and only ifTsatisfies Weyl’s theorem (resp.a-Weyl’s theorem). Hence ifTis an algebraicallyk-quasi-∗-classAoperator, we have the following result.
Theorem . Let T,A,B∈B(H).If T is algebraically k-quasi-∗-class A and A,B∗ are k-quasi-∗-class A,then
(i) Weyl’s theorem holds forT (resp.fordA,B)if and only if generalized Weyl’s theorem
holds forT(resp.fordA,B).
(ii) a-Weyl’s theorem holds forT∗(resp.fordA∗,B∗)if and only if generalizeda-Weyl’s theorem holds forT∗(resp.fordA∗,B∗).
Theorem . Let T,A,B∈B(H).
(i) IfT∗is algebraicallyk-quasi-∗-classAandA,B∗arek-quasi-∗-classA,then Weyl’s
theorem,a-Weyl’s theorem,generalized Weyl’s theorem and generalizeda-Weyl’s
theorem hold forT(resp.dA,B)and these are equivalent.
(ii) IfT is algebraicallyk-quasi-∗-classAandA,B∗arek-quasi-∗-classA,then Weyl’s
theorem,a-Weyl’s theorem,generalized Weyl’s theorem and generalizeda-Weyl’s
Proof Since Weyl’s theorem holds forT (resp. fordA,B). It suffices to show that Weyl’s
theorem is equivalent to each one of the other Weyl-type theorems forT (resp. fordA,B),
generalized or not. SinceT∗(resp.dA∗,B∗) has SVEP, Weyl’s theorem anda-Weyl’s theorem
hold forT(resp. fordA,B) and are equivalent by [, Theorem .]. Theorem .(i) implies
that Weyl’s theorem and generalized Weyl’s theorem hold forT (resp. fordA,B) and are
equivalent. Nowa-Weyl’s theorem and generalizeda-Weyl’s theorem hold forT(resp. for
dA,B) and are equivalent by Theorem .(ii).
Letf ∈Hol(σ(T)), whereHol(σ(T)) is the space of all functions that are analytic in an open neighborhoods ofσ(T). IfTis polaroid, thenf(T) is polaroid too []. Thus, we have
Theorem . Let T,A,B∈B(H).
(i) IfT∗is algebraicallyk-quasi-∗-classAandA,B∗arek-quasi-∗-classA,thenf(T∗)
(resp.f(dA∗,B∗))satisfies Weyl’s theorem,a-Weyl’s theorem,generalized Weyl’s
theorem and generalizeda-Weyl’s theorem.
(ii) IfT is algebraicallyk-quasi-∗-classAandA,B∗arek-quasi-∗-classA,thenf(T)
(resp.f(dA,B))satisfies Weyl’s theorem,a-Weyl’s theorem,generalized Weyl’s theorem
and generalizeda-Weyl’s theorem.
Proof
(i) IfT∗is algebraicallyk-quasi-∗-classAandA,B∗arek-quasi-∗-classA, thenT∗
(resp.dA∗,B∗) is polaroid []. SinceT∗(resp.dA∗,B∗) is polaroid, the result holds by [, Theorem .]
(ii) IfTis algebraicallyk-quasi-∗-classAandA,B∗arek-quasi-∗-classA, thenf(T)
(resp.f(dA,B)) is polaroid. SinceT(resp.f(dA,B)) is polaroid, the result holds by [,
Theorem .].
According to [, Theorem .] Theorem . may be extended as follows.
Theorem . Let T,A,B∈B(H).
(i) IfT∗is algebraicallyk-quasi-∗-classAandA,B∗arek-quasi-∗-classA,then Weyl’s
theorem,a-Weyl’s theorem,generalized Weyl’s theorem and generalizeda-Weyl’s
theorem hold forf(T)(resp.f(dA,B))and these are equivalent.
(ii) IfT is algebraicallyk-quasi-∗-classAandA,B∗arek-quasi-∗-classA,then Weyl’s
theorem,a-Weyl’s theorem,generalized Weyl’s theorem and generalizeda-Weyl’s
theorem hold forf(T∗)(resp.f(dA∗,B∗))and these are equivalent.
Remark . According to [], the previous results on a Weyl-type theorem still true for the property (w).
Competing interests
The author declares that they have no competing interests.
Acknowledgements
The author would like to thank the referee for his good reading of the paper and his comments. This paper is supported by Taibah University Research Center Project (1433-808).
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doi:10.1186/1029-242X-2012-244
