Che Yuhong Dai Lei

(School of Mathematics and Statistics,Weinan Normal University,Weinan 714099,China)

Abstract In this paper we investigate the necessary and sufficient conditions such that both a-Weyl’s theorem and the property(WE)hold for a bounded linear operator,and study the stability of a-Weyl’s theorem and the property(WE)under quasi-nilpotent or compact perturbations.As an application,the relative stability of special operators is studied.

Key words A-Weyl’s theorem Property(WE) Compact perturbation Quasi-nilpotent perturbation

1 Introduction

In 1909,Weyl[1]examined the compact perturbations of some self-adjoint operators and found that the intersection of their spectrums consisted precisely of those points of the spectrums which were not isolated eigenvalues of finite multiplicity.Later,this observation was abstracted into “Weyl’s theorem”.In recent decades,combining with the relationships between different spectrum subsets,mathematicians put forward a series of variants of Weyl’s theorem.In[2]and[3],Harte and Lee and Rako?eviintroduced the definition of a-Weyl’s theorem?Berkani and Kachad[4]gave another variant-property(WE).In this paper,we investigate the necessary and sufficient conditions such that both a-Weyl’s theorem and the property (WE) hold for a bounded linear operator,and study the stability of a-Weyl’s theorem and the property(WE)under quasi-nilpotent or compact perturbations.

To begin with,we introduce some terminologies and notations.Throughout this paper,C(resp.N)denotes the set of all complex numbers(resp.nonnegative integers).Hdenotes an infinite-dimensional complex separable Hilbert space andB(H) (resp.K(H)) denotes the set of all bounded linear (resp.compact)operators onH.In addition,F(H)denotes the set of all finite rank operators inB(H).ForT ∈B(H),we useN(T),R(T)andσ(T)to denote the kernel,the range and the spectrum ofT,respectively.Letρ(T)=Cσ(T).IfR(T)is closed andn(T)＜∞(resp.d(T)＜∞),thenTis said to be an upper(resp.lower)semi-Fredholm operator,wheren(T)=dimN(T)andd(T)=codimR(T).Especially,ifTis an upper semi-Fredholm operator withn(T)=0,thenTis called a bounded below operator.Now,the index ofTis defined by ind(T)=n(T)-d(T).If ind(T)is finite,thenTis called a Fredholm operator.T ∈B(H)is called a Weyl operator ifTis a Fredholm operator with ind(T)=0.The ascent and descent ofTare closely related with the kernel and range of the power ofT,which are respectively defined by asc(T)=inf{n∈N:N(Tn)=N(Tn+1)} and des(T)=inf{n∈N:R(Tn)=R(Tn+1)}.If the infimum does not exist,then we write asc(T)=∞(resp.des(T)=∞).It is known from Proposition 38.3 in[5]that asc(T)=des(T)if they are finite simultaneously.Tis called a Browder operator ifTis a Fredholm operator with both finite ascent and finite descent.The Browder spectrumσb(T),the Weyl spectrumσw(T),the approximate point spectrumσa(T)are respectively defined by:σb(T)={λ∈C:T-λIis not a Browder operator},σw(T)={λ∈C:T-λIis not a Weyl operator}andσa(T)={λ∈C:T-λIis not a bounded below operator}.Letρea(T)={λ∈C:T-λIis an upper semi-Fredholm operator with ind(T-λI)≤0}andρab(T)={λ∈C:T-λIis an upper semi-Fredholm opreator with asc(T -λI)＜∞}.We callσea(T)=Cρea(T) andσab(T)=Cρab(T) the essential approximate point spectrum and Browder essential approximate point spectrum respectively.Besides,forE ?C,we use isoEand accEto denote the set of all isolated points and all accumulation points ofErespectively.

LetT ∈B(H).Tis said to satisfy a-Weyl’s theorem,if

whereE(T)={λ∈isoσ(T):0＜n(T -λI)}(Definition 2.1,[6]).

2 A-Weyl’s Theorem and the Property(W E)

Firstly,we present some observations on a-Weyl’s theorem and the property(WE).

(1)T∈(WE)can’t infer a-Weyl’s theorem holds forT.

For instance,letA,B ∈B(?2)be defined by

(2)T ∈B(H)satisfying a-Weyl’s theorem does not necessarily impliesT∈(WE).

For instance,letT ∈B(?2)be defined by

(3)There exists an operator which neither a-Weyl’s theorem nor the property(WE)holds.

For instance,letT ∈B(?2)be defined by

Based on the above observations,one may naturally ask: is it possible to find some conditions such that bothT∈(WE)and a-Weyl’s theorem hold?

We turn to a variant of the essential approximate point spectrum,involving a condition introduced by Saphar[7]and the zero jump condition of Kato[8].The new spectral set is defined as follows:

It is quite clear thatσ1(T)?σea(T)?σw(T)andσ1(T)?σab(T).Based on the setσ1(·),we will explore the conditions such that both a-Weyl’s theorem and the property(WE)hold for bounded linear operators.

Theorem 2.1LetT ∈B(H).Then both a-Weyl’s theorem and the property(WE)hold forTif and only ifσab(T)=σ1(T)∪{λ ∈σa(T):n(T -λI)=0}∪[accσ(T)∩{λ∈C:n(T -λI)=∞}].

ProofSufficiency.Since [σ(T)σw(T)∪E(T)]∩{σ1(T)∪{λ ∈σa(T):n(T -λI)=0}∪[accσ(T)∩{λ∈C:n(T -λI)=∞}]}=?,it followsσ(T)σw(T)∪E(T)?ρab(T).We getσ(T)σw(T)?σ0(T)andE(T)?σ0(T),whereσ0(T)=σ(T)σb(T).Thenσ(T)σw(T)=E(T),which meansT∈(WE).

Remark 2.1(1)In Theorem 2.1,ifT∈(WE)and a-Weyl’s theorem holds forT,the three parts of the decomposition form ofσab(T)are essential.We explain it with the following examples.

(i) LetT ∈B(?2)be defined by

Then both a-Weyl’s theorem and the property(WE)hold forT.Butσab(T){λ ∈σa(T):n(T-λI)=0}∪[accσ(T)∩{λ∈C:n(T -λI)=∞}].This meansσ1(T)cannot be removed.

(ii) LetT ∈B(?2)be defined by

Then both a-Weyl’s theorem and the property(WE)hold forT.Howeverσab(T)σ1(T)∪[accσ(T)∩{λ∈C:n(T -λI)=∞}].

(iii) LetA,B ∈B(?2)be defined by

(2)IfT∈(WE)and a-Weyl’s theorem holds forT,thenσab(T)?σ1(T)∪{λ ∈σa(T):n(T -λI)=0}∪accσ(T)andσab(T)=σ1(T)∪{λ ∈σa(T):n(T-λI)=0}∪{λ∈C:n(T-λI)=∞}.But the converse is not true.

For example,letA,B ∈B(?2)be defined by

But the property(WE)does not hold forT ∈B(H)ifσab(T)=σ1(T)∪{λ ∈σa(T):n(T-λI)=0}∪{λ∈C:n(T -λI)=∞}.Consider the following example.LetT ∈B(?2)be defined by

Thenσab(T)=σ1(T)∪{λ ∈σa(T):n(T -λI)=0}∪{λ∈C:n(T -λI)=∞}={0,1}.But sinceσ(T)=σw(T)={0,1}andE(T)={0,1},we see the property(WE)does not hold forT.

IfT∈(WE),we can prove accσ(T)=σ1(T)∪acc{λ∈C:n(T -λI)＜d(T -λI)}.By Theorem 2.1,the following fact holds.

Corollary 2.1LetT ∈B(H).Then both a-Weyl’s theorem and the property (WE) hold forTif and only ifσab(T)=σ1(T)∪{λ ∈σa(T):n(T -λI)=0} ∪[acc{λ∈C:n(T -λI)＜d(T -λI)}∩{λ∈C:n(T -λI)=∞}].

In Theorem 2.1 and Corollary 2.1,the set{λ ∈σa(T):n(T -λI)=0}cannot be replaced by the set{λ ∈σ(T):n(T -λI)=0}.

Since{λ ∈σ(T):n(T -λI)=0}?σb(T),we have the following corollary.

Letσs(T)={λ∈C:T -λIis not a surjective operator}.We have the following corollary.

Corollary 2.3LetT ∈B(H).Then the following statements are equivalent:

From Theorem 2.1,ifσab(T)=σ1(T),then both a-Weyl’s theorem and the property(WE)hold forT.But the converse is not true.For example,letT ∈B(?2)be defined by

We callT ∈B(H) finite-a-isoloid if isoσa(T)?{λ∈C: 0＜n(T -λI)＜∞},and callT ∈B(H)a-isoloid if isoσa(T)?{λ∈C:0＜n(T -λI)}.Clearly,finite-a-isoloid implies a-isoloid.

Ifσab(T)=σ1(T),thenTis finite-a-isoloid.In fact,letλ0∈isoσa(T),thenλ0σ1(T).Byσab(T)=σ1(T),we haveλ0σab(T).Thenλ0∈[ρab(T)∩σa(T)].This impliesλ0∈{λ∈C:0＜n(T -λI)＜∞}.

Corollary 2.4LetT ∈B(H),thenTis finite-a-isoloid and both a-Weyl’s theorem and the property(WE)hold forTif and only ifσab(T)=σ1(T).

In Corollary 2.4,if we change the condition “Tis finite-a-isoloid” to “Tis a-isoloid”,the result will not be true.For example,letA,B ∈B(?2)be defined by

(2)isoσa(T)={0}?{λ∈C:0＜n(T -λI)},soTis a-isoloid.But sinceσab(T)={0}∪{λ∈C:|λ|=1}andσ1(T)={λ∈C:|λ|=1},we getσab(T)1(T).

Corollary 2.5LetT ∈B(H).ThenTis a-isoloid and both a-Weyl’s theorem and the property(WE)hold forTif and only ifσab(T)=σ1(T)∪[accσ(T)∩{λ∈C:n(T -λI)=∞}.

ProofFrom Theorem 2.1,ifσab(T)=σ1(T)∪[accσ(T)∩{λ∈C:n(T -λI)=∞},then both a-Weyl’s theorem and the property(WE)hold forT.Since{λ∈isoσa(T):n(T-λI)=0}∩[σ1(T)∪[accσ(T)∩{λ∈C:n(T -λI)=∞}]=?,it follows{λ∈isoσa(T):n(T -λI)=0} ?ρab(T).Then{λ∈isoσa(T):n(T -λI)=0} ?ρa(T),thus{λ∈isoσa(T):n(T -λI)=0}=?,which meansTis a-isoloid.

For the converse,we supposeTis a-isoloid and both a-Weyl’s theorem and the property(WE)hold forT.By Theorem 2.1,σab(T)=σ1(T)∪{λ ∈σa(T):n(T -λI)=0}∪[accσ(T)∩{λ∈C:n(T -λI)=∞}].Since both a-Weyl’s theorem and the property (WE) hold forT,similar to the proof of Theorem 2.1,we can get accσa(T)?σ1(T).SinceTis a-isoloid,we know{λ∈isoσa(T):n(T -λI)=0}=?,then{λ ∈σa(T):n(T -λI)=0}={λ∈isoσa(T):n(T -λI)=0}∪{λ∈accσa(T):n(T -λI)=0}?σ1(T).So

The converse inclusion is clear.Thenσab(T)=σ1(T)∪[accσ(T)∩{λ∈C:n(T -λI)=∞}].

From Remark 2.1,we see the condition “Tis isoloid” is essential in Corollary 2.5.

ForT ∈B(H),letHol(σ(T))be the class of all complex-valued functions which are analytic in a neighborhood ofσ(T)and are not constant on any component ofσ(T).In the following,we will explore a-Weyl’s theorem and the property(WE)for the functions of operatorT.

Remark 2.2ForT ∈B(H),Tsatisfies both a-Weyl’s theorem and the property (WE) cannot implyf(T)satisfies both a-Weyl’s theorem and the property(WE)forf ∈Hol(σ(T)).

For instance,letA,B ∈B(?2)be defined by

Next,we consider a-Weyl’s theorem and the property(WE)under functional calculus.LetPab(T)=σa(T)σab(T)and={λ∈C:T -λIis an upper semi-Fredholm operator}.

Theorem 2.2LetT ∈B(H).Then both a-Weyl’s theorem and the property(WE)hold forf(T)for anyf ∈Hol(σ(T))if and only if the following conditions hold:

(1)Tsatisfies a-Weyl’s theorem and has the property(WE)?

ProofNecessity.(1)is obvious.In the following,we will prove(2)and(3).

wherenis finite andmis finite or infinite.Ifmis finite,we putf(T)=(T -λ0I)n(T -μ0I)m? ifmis infinite,we putf(T)=(T -λ0I)(T -μ0I).Then 0∈σa(f(T))σea(f(T)).Since a-Weyl’s theorem holds forf(T),it follows thatf(T)is an upper semi-Fredholm operator with finite ascent.Then asc(T-μ0I)＜∞.This induces ind(T-μ0I)≤0.It is in contradiction to the fact ind(T-μ0I)=n ＞0.Then(2)is true.

Sufficiency.We will take the following two cases into consideration.

Case 1Pab(T)=?.

Case 2σab(T)=σ1(T)∪[accσ(T)∩{λ∈C:n(T -λI)=∞}].

By Corollary 2.5,we seeTis a-isoloid.

For anyf ∈Hol(σ(T)),takeμ0∈σ(f(T))σw(f(T)).Suppose

whereλiλjifijandg(T) is invertible.Then for eachi,T -λiIis a Fredholm operator and=0.By(2),we knowT-λiIis a Weyl opreator.Since the property(WE)holds forT,T -λiIis a Browder opreator.Thenf(T)-μ0Iis a Browder opeartor,which meansμ0∈E(f(T)).

Takeμ0∈σa(f(T))σea(f(T)) and assumef(T)-μ0Ihas the same decomposition as above.Sinceσea(·)satisfies the spectral mapping theorem(by(2)),it followsλi ∈ρea(T)(1≤i ≤t).ThenT-λiIis bounded below orλi ∈σa(T)σea(T).Since a-Weyl’s theorem holds forT,T-λiIhas finite ascent.Thenf(T)-μ0Ihas finite ascent.This showsμ0∈

For the converse,letμ0∈and assumef(T)-μ0Ihas the same decomposition as above.Thenn(T-λiI)＜∞.Without loss of generality,we supposeλi ∈σa(T).Thusλi∈isoσa(T).SinceTis a-isoloid,it followsλi∈ThenT-λiIis an upper operator with ind(T-λiI)≤0.Sof(T)-μ0Iis an upper semi-Fredholm operator with ind(f(T)-μ0I)≤0,that is,μ0∈σa(f(T))σea(f(T)).

Ifμ0∈E(f(T))andf(T)-μ0Ihas the same decomposition as above,without loss of generality,we supposeλi ∈σ(T),thusλi∈isoσ(T).SinceTis a-isoloid,it followsλi ∈E(T).ThenT -λiIis a Weyl operator,and sof(T)-μ0Iis also a Weyl operator,that is,μ0∈σ(f(T))σw(f(T)).

As showed above,we get both a-Weyl’s theorem and the property (WE) hold forf(T) for anyf ∈Hol(σ(T)).

3 A-Weyl’s theorem and the property(WE)under perturbations

An operatorR ∈B(H)is called a Riesz operator ifT -λIis a Fredholm operator for all nonzeroλ∈C.Evidently,quasi-nilpotent operators and compact operators are Riesz operators.It is known ifT ∈B(H),thenσb(T+R)=σb(T),σw(T+R)=σw(T)andσea(T+R)=σea(T)for every Riesz operatorRcommuting withT.

Remark 3.1(1)LetT ∈B(H)andK ∈K(H)(orK ∈F(H))withKT=TK.Even if both a-Weyl’s theorem and the property(WE)hold forT,we can not induceT+Ksatisfies a-Weyl’s theorem orT+K∈(WE).

For example,letA,B,C,P ∈B(?2)be defined by

(2)LetT ∈B(H)andQbe a quasi-nilpotent operator(or nilpotent operator)commuting withT.Even ifT∈(WE)and a-Weyl’s theorem holds forT,it does not follow that both a-Weyl’s theorem and the property(WE)hold forT+Q.

For instance,letA,Q ∈B(?2)be defined by

andQ=-T.ThenQis a quasi-nilpotent operator commuting withT,T∈(WE)and a-Weyl’s theorem holds forT.But we seeT+Q/∈(WE).

(3)IfF ∈F(H)commutes withT ∈B(H),then(see[4,11])

(i) isoσ(T+F)?isoσ(T)∪ρ(T)and accσa(T+F)=accσa(T)?

(ii)n(T+F -λI)＜∞?n(T -λI)＜∞;

(iii) IfT∈(WE)is a-isoloid,and isoσa(T)?σp(T)={λ∈C:n(T -λI)＞0},then for anyF ∈F(H)withTF=FT,the property(WE)holds forT+F.

(4)IfTis a-isoloid and a-Weyl’s theorem holds forT,then?F ∈F(H)withTF=FT,a-Weyl’s theorem holds forT+F(see[12]).

(5)SupposeTis a-isoloid andF ∈F(H)commutes withT.ThenT+F∈(WE)and satisfies a-Weyl’s theorem can not induce a-Weyl’s theorem holds forTandT∈(WE).

For example,letA,B,C,P ∈B(?2)be defined by

Theorem 3.1LetT ∈B(H) andF ∈F(H) withFT=TF,T+Fis a-isoloid.ThenTis a-isoloid andTsatisfies a-Weyl’s theorem andT∈(WE)if and only if the following statements hold:

(1)T+Fsatisfies a-Weyl’s theorem and has the property(WE)?

(2)T+Fis a-isoloid.

From Theorem 2.1,ifσb(T)=σ1(T),thenTsatisfies a-Weyl’s theorem and has the property(WE).

Corollary 3.1LetT ∈B(H)andσb(T)=σ1(T).IfF ∈F(H)withFT=TF,thenT+Fsatisfies a-Weyl’s theorem and has the property(WE).

However,the converse of Corollary 3.1 does not hold in general.For example,letT,F ∈B(?2)be defined by

It’s more interesting to find the conditions such that the converse of Corollary 3.1 holds.

Corollary 3.2LetT ∈B(H)andF ∈F(H)withFT=TF.Thenσb(T)=σ1(T)if and only ifT+Fis a-isoloid,σa(T+F)=σ(T+F),andT+Fsatisfies a-Weyl’s theorem and has the property(WE).

In the following,we will explore a-Weyl’s theorem and the property (WE) under quasi-nilpotent perturbation(or nilpotent perturbation).

Lemma 3.1LetT ∈B(H).IfN ∈B(H)is a nilpotent operator withTN=NT,then

(1)0＜n(T+N)＜∞if and only if 0＜n(T)＜∞;

(2)n(T+N)=0 if and only ifn(T)=0.

(2)It is a immediate result from(1).

Theorem 3.2LetT ∈B(H)andNbe a nilpotent operator withNT=TN.ThenT∈(WE)and satisfies a-Weyl’s theorem if and only ifT+Nsatisfies a-Weyl’s theorem and has the property(WE).

Corollary 3.3LetT ∈B(H)withσb(T)=σ1(T).IfNis a nilpotent operator withNT=TN,thenT+Nsatisfies a-Weyl’s theorem and has the property(WE).

The converse of Corollary 3.3 is not true.For example,letA,P ∈B(?2)be defined by

Corollary 3.4LetT ∈B(H).IfNis a nilpotent operator withNT=TN,thenσb(T)=σ1(T)if and only if

(1)T+Nsatisfies a-Weyl’s theorem and has the property(WE)?

(2)σa(T+N)=σ(T+N).

ForT ∈B(H),ifQis a quasi-nilpotent operator withQT=TQ,thenσa(T+Q)=σa(T)andσ(T+Q)=σ(T)(Theorem 4.9 and Lemma 2.75,[13]).

From Remark 3.1,a-Weyl’s theorem and the property (WE) can not transmit to the operator with quasi-nilpotent perturbations.The result of Theorem 3.2 may be strongly improved if we consider injective quasi-nilpotent perturbations.

Lemma 3.2SupposeT ∈B(H).IfQis an injective quasi-nilpotent operator withQT=TQ,thenn(T)=0 whenn(T)＜∞.

From Lemma 3.2,we know ifQis an injective quasi-nilpotent operator withQT=TQ,thenσea(T)=σa(T),σ(T)=σw(T),σea(T+Q)=σa(T+Q) andσ(T+Q)=σw(T+Q).IfQis an injective quasi-nilpotent operator withQT=TQand ifT∈(WE)and satisfies a-Weyl’s theorem,thenThus we have the following corollary.

Corollary 3.5LetT ∈B(H)andQbe an injective quasi-nilpotent operator withQT=TQ.IfT∈(WE)and satisfies a-Weyl’s theorem,thenT+Qsatisfies a-Weyl’s theorem and has the property(WE)if and only ifE(T+Q)=?.

ForT ∈B(H),ifσ(T)=σ1(T),thenT∈(WE) and satisfies a-Weyl’s theorem according to Theorem 2.1.In this case,if there is an injective quasi-nilpotent operatorQcommuting withT,thenE(T+Q)=?.

In fact,ifλ0∈E(T+Q),thenT-λIis a Weyl operator if 0＜|λ-λ0|is sufficiently small.Sinceσ(T)=σ1(T)andσ1(T)?σw(T),we knowλ ∈ρ(T)if 0＜|λ-λ0|is sufficiently small.This meansλ0∈isoσ(T)∪ρ(T).Note that isoσ(T)?ρ1(T)andσ(T)=σ1(T)imply that isoσ(T)=?.Thenλ0∈ρ(T),and henceT+Q-λ0Iis a Weyl operator.From Lemma 3.2,we known(T+Q-λ0I)=0,which meansλ0∈ρ(T+Q).It is in contradiction to the factλ0∈E(T+Q).

Corollary 3.6LetT ∈B(H) withσ(T)=σ1(T).IfQis an injective quasi-nilpotent operator withQT=TQ,thenT+Qsatisfies a-Weyl’s theorem and has the property(WE).

The converse of Corollary 3.6 is not true.For example,letT ∈B(?2)be defined by

Corollary 3.7LetT ∈B(H).IfQis an injective quasi-nilpotent operator withQT=TQ.Thenσ(T)=σ1(T)if and only ifT+Qis isoloid,σa(T+Q)=σ(T+Q)andT+Q∈(WE)satisfying a-Weyl’s theorem.

LetT ∈B(?2)be defined by

Then we haveTis a-isoloid,σ(T)=σ1(T),and

(1)T+Fis a-isoloid such that both a-Weyl’s theorem and the property(WE)hold forT+Ffor every commuting finite rank operatorF?

(2)T+Nsatisfies a-Weyl’s theorem and has the property (WE) for every commuting nilpotent operatorN?

(3)T+Qsatisfies a-Weyl’s theorem and has the property (WE) for every commuting injective quasi-nilpotent operatorQ.