Spectrum of Class wF( p,r, q) Operators

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Journal of Inequalities and Applications Volume 2007, Article ID 27195,10pages doi:10.1155/2007/27195

Research Article

Spectrum of Class wF( p,r, q) Operators

Jiangtao Yuan and Zongsheng Gao

Received 23 November 2006; Accepted 16 May 2007

Dedicated to Professor Daoxing Xia on his 77th birthday with respect and aﬀection Recommended by Jozsef Szabados

This paper discusses some spectral properties of class wF(p,r,q) operators for p >0, r >0, p+r≤1, and q≥1. It is shown that if T is a class wF(p,r,q) operator, then the Riesz idempotentEλ ofT with respect to each nonzero isolated point spectrum λ is selfadjoint andEλᏴ=ker (T−λ)=ker (T−λ)^∗. Afterwards, we prove that every class wF(p,r,q) operator has SVEP and property (β), and Weyl’s theorem holds forf(T) when

f ∈H(σ(T)).

Copyright © 2007 J. Yuan and Z. Gao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

A capital letter (such asT) means a bounded linear operator on a complex Hilbert space Ᏼ. Forp >0, an operatorTis said to bep-hyponormal if (T^∗T)^p≥(TT^∗)^p, whereT^∗ is the adjoint operator ofT. An invertible operatorT is said to be log-hyponormal if log(T^∗T)≥log(TT^∗). Ifp=1,Tis called hyponormal, and if p=1/2Tis called semi- hyponormal. Log-hyponormality is sometimes regarded as 0-hyponormal since (X^p− 1)/ p→logXasp→0 forX >0.

See Martin and Putinar [1] and Xia [2] for basic properties of hyponormal and semi- hyponormal operators. Log-hyponormal operators were introduced by Tanahashi [3], Aluthge and Wang [4], and Fujii et al. [5] independently. Aluthge [6] introduced p- hyponormal operators.

As generalizations ofp-hyponormal and log-hyponormal operators, many authors introduced many classes of operators. Aluthge and Wang [4] introducedw-hyponormal operators defined by|T| ≥ |T| ≥ |(T) ^∗|, where the polar decomposition ofTisT=U|T| andT= |T|^1/2U|T|^1/2is called Aluthge transformation ofT. Forp >0 andr >0, Ito [7]

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introduced classwA(p,r) defined by

T^∗^r|T|^2pT^∗^r^r/(p+r)≥T^∗^2r, |T|^pT^∗^2r|T|^ps/(p+r)

≤ |T|^2p. (1.1) Note that the two exponentsr/(p+r) and p/(p+r) in the formula above satisfyr/

(p+r) +p/(p+r)=1, Yang and Yuan [8] introduced classwF(p,r,q).

Definition 1.1 (see [8,9]). Forp >0,r >0, andq≥1, an operatorTbelongs to classwF(p, r,q) if

|T^∗^r|T|^2pT^∗^r^1/q≥T^∗^2(p+r)/q, |T|^2(p+r)(1⁻^1/q)≥

|T|^pT^∗^2r|T|^p(1−1/q)

. (1.2) Denote (1−q⁻¹)⁻¹byq^∗whenq >1 becauseqand (1−q⁻¹)⁻¹are a couple of conju- gate exponents. It is clear that classwA(p,r) equals classwF(p,r, (p+r)/r).

w-hyponormality equalswA(1/2, 1/2) [7]. Ito and Yamazaki [10] showed that class wA(p,r) coincides with classA(p,r) (introduced by Fujii et al. [11]) for each p >0 and r >0. Consequently, classwA(1, 1) equals classA(i.e., |T²| ≥ |T|², introduced by Fu- ruta et al. [12]). Reference [9] showed that classwF(p,r,q) coincides with classF(p,r,q) (introduced by Fujii and Nakamoto [13]) whenrq≤p+r.

Recently, there are great developments in the spectral theory of the classes of operators above. We cite [8,14–22]. In this paper, we will discuss several spectral properties of class wF(p,r,q) forp >0,r >0,p+r≤1, andq≥1.

InSection 2, we prove that Riesz idempotentEλofTwith respect to each nonzero isolated point spectrumλis selfadjoint andEλᏴ=ker(T−λ)=ker(T−λ)^∗. InSection 3, we will show that each classwF(p,r,q) operator has SVEP (single-valued extension property) and Bishop’s property (β). InSection 4, we show that Weyl’s theorem holds for class wF(p,r,q).

2. Riesz idempotent

Letσ(T),σ_p(T),σ_{j p}(T),σ_a(T),σ_ja(T), andσ_r(T) mean the spectrum, point spectrum, joint point spectrum, approximate point spectrum, joint approximate point spectrum, and residual spectrum of an operatorT, respectively (cf. [8,23]).σ_r^Xia(T) andσiso(T) mean the setσ(T)−σa(T) and the set of isolated points ofσ(T), see [23,2].

Ifλ∈σiso(T), the Riesz idempotentE_λofTwith respectλis defined by E_λ=

∂Ᏸ(z−T)⁻¹dz, (2.1)

whereᏰis an open disk which is far from the rest ofσ(T) and∂Ᏸmeans its boundary.

Stampfli [24] showed that ifTis hyponormal, thenE_λis selfadjoint andE_λᏴ=ker(T− λ)=ker(T−λ)^∗. The recent developments of this result are shown in [16,17,20,22], and so on.

In this section, it is shown that whenλ=0, this result holds for classwF(p,r,q) with p+r≤1 and q≥1. It is always assumed that λ∈σiso(T) when the idempotent Eλ is considered.

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Theorem 2.1. LetT belong to classwF(p,r,q) with p+r≤1,λ= |λ|e^iθ∈Ꮿ, andλp+r=

|λ|^p+re^iθ, then the following assertions hold.

(1) Ifλ=0, thenEλ=Eλ(p,r) andEλᏴ=ker(T−λ)=ker(T−λ)^∗, whereEλ(p,r) is the Riesz idempotent ofT(p,r)= |T|^pU|T|^r (the generalized Aluthge transfor- mation ofT) with respect toλp+r.

(2) Ifλ=0, then kerT=E0Ᏼ=E0(p,r)Ᏼ=ker(T(p,r)).

Reference [21] gave an example that the operatorTisw-hyponormal,E0is not selfadjoint, and kerT=kerT^∗.

An operatorTis said to be isoloid ifσiso(T)⊆σp(T), is said to be reguloid if (T−λ)Ᏼ, is closed for eachλ∈σiso(T).

Theorem 2.2. IfTbelongs to classwF(p,r,q) withp+r≤1, thenTis isoloid and reguloid.

To give proofs, we prepare the following results.

Theorem 2.3 (see [14]). Letλ=0, and let{xn}be a sequence of vectors. Then the following assertions are equivalent.

(1) (T−λ)xn→0 and (T^∗−λ)xn→0.

(2) (|T| − |λ|)xn→0 and (U−e^iθ)xn→0.

(3) (|T|^∗− |λ|)x_n→0 and (U^∗−e⁻^iθ)x_n→0.

Theorem 2.4 (see [8]). IfTis a classwF(p,r,q) operator forp+r≤1 andq≥1, then the following assertions hold.

(1) IfTx=λx,λ=0, thenT^∗x=λx.

(2)σ_a(T)− {0} =σ_ja(T)− {0}.

(3) IfTx=λx,T y=μyandλ=μ, then (x,y)=0.

Theorem 2.5 (see [9]). IfT is a classwF(p,r,q) operator, then there existsα0>0, which satisfies

T(p,r)^2α⁰≥ |T|^2α⁰^(p+r)≥T(p,r)^∗^2α⁰. (2.2) Lemma 2.6. IfTbelongs to classwF(p,r,q) forp+r≤1,λ=|λ|e^iθ∈Ꮿ, andλ_p+r=|λ|^p+re^iθ, then ker(T−λ)=ker(T(p,r)−λp+r).

Proof. We only prove ker(T−λ)⊇ker(T(p,r)−λp+r) because ker(T−λ)⊆ker(T(p,r)− λ_p+r) is obvious by Theorems2.3-2.4.

Ifλ=0, let 0=x∈ker(T(p,r)−λp+r). ByTheorem 2.5,T(p,r) isα0-hyponormal and we have

T(p,r)x= |λ|^p+rx=T(p,r)^∗x,

T(p,r)^2α⁰−T(p,r)^∗^2α⁰≥T(p,r)^2α⁰− |T|^2α⁰^(p+r)≥0. (2.3) Hence (|T(p,r)|^2α⁰− |T|^2α⁰^(p+r))x=0,

|T|^2α⁰^(p+r)x− |λ|^2α⁰^(p+r)x

≤ |T|^2α⁰^(p+r)x−T(p,r)^2α⁰x+T(p,r)^2α⁰x− |λ|^2α⁰^(p+r)x=0. (2.4)

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On the other hand, (T(p,r))^∗x= |λ|^p+re⁻^iθx implies that |T|^rU^∗x= |λ|^re⁻^iθx,T^∗=

|λ|e⁻^iθx. Therefore,

(T−λ)x²= Tx²−λ(x,Tx)−λ(Tx,x) +|λ|²x²

= |T|x²−λT^∗x,x−λx,T^∗x+|λ|²x²=0. (2.5) Ifλ=0, let 0=x∈kerT(p,r), thenx∈ker|T| =kerTbyTheorem 2.5so that ker(T

−λ)⊇ker(T(p,r)−λp+r).

Lemma 2.7 (see [18,25]). IfAis normal, then for every operatorB,σ(AB)=σ(BA).

LetᏲbe the set of all strictly monotone increasing continuous nonnegative functions on᏾⁺=[0,∞). LetᏲ0= {Ψ∈Ᏺ:Ψ(0)=0}. ForΨ∈Ᏺ0, the mappingΨis defined by Ψ(ρe^iθ)=e^iθΨ(ρ) andΨ(T)=UΨ(|T|).

Theorem 2.8 (see [26]). IfΨ∈Ᏺ0, then for every operatorT,σja(Ψ(T)) =Ψ(σ ja(T)).

Lemma 2.9. Let T belong to class wF(p,r,q) with p+r≤1, λ= |λ|eîθ∈Ꮿ, T(t)= U|T|¹⁻^t+t(p+r), andτt(ρeîθ)=eîθρ^1+t(p+r⁻¹⁾, wheret∈[0, 1]. Then

σa

T(t)=τt

σa(T), σ_r^XiaT(t)=τt

σ_r^Xia(T), σT(t)=τt

σ(T). (2.6) Proof. We only need to show that σ_a(T(t))=τ_t(σ_a(T)) by homotopy property of the spectrum [2, page 19].

SinceT belongs to classwF(p,r,q) with p+r≤1, T(t) belongs to classwF(p/(1 + t(p+r−1)),r/(1 +t(p+r−1),q)) withp/(1 +t(p+r−1)) +r/(1 +t(p+r−1))≤1. By Theorems2.4(2) and2.8,

σa

T(t)− {0} =σja

T(t)− {0} =τt

σja(T)− {0}

=τt

σa(T)− {0}. (2.7) On the other hand, if 0∈σ_a(T), then there exists a sequence{x_n}of unit vectors such thatU|T|xn→0. Hence|T|xn=U^∗U|T|xn→0, so that|T|^1/(2^m⁾xn→0 for each positive integermby induction. Take a positive integerm(t) such that 1/(2^m(t))≤1 +t(p+r−1), then

|T|^1+t(p+r⁻¹⁾x_n= |T|^1+t(p+r⁻¹⁾⁻^1/(2^m(t)⁾|T|^1/(2^m(t)⁾x_n−→0 (2.8) and 0∈σa(T(t)). It is obvious that if 0∈σa(T(t)), then 0∈σa(T) because of p+r≤1.

Thereforeσ_a(T(t))=τ_t(σ_a(T)).

Theorem 2.10 (see [15]). IfT isp-hyponormal or log-hyponormal, thenE_λ is selfadjoint andEλᏴ=ker(T−λ)=ker(T−λ)^∗.

Riesz and Sz.-Nagy [27] gave the the formulaEλᏴ= {x∈Ᏼ:(T−λ)ⁿx^1/n→0}. Lemma 2.11. For any operatorT,|T|^pker(T−λ)⊆ |T|^pEλᏴ⊆Eλ(p,r)Ᏼforp+r=1.

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Proof. Letx∈Eλ, by the formula above we have

T(p,r)−λⁿ|T|^px^1/n= |T|^p(T−λ)ⁿx^1/n−→0. (2.9)

Hence|T|^px∈Eλ(p,r)Ᏼ.

Lemma 2.12. IfTbelongs to classwF(p,r,q) withp+r≤1, then

kerT=E0Ᏼ=E0(p,r)Ᏼ=kerT(p,r). (2.10) Note thatλp+r∈σiso(T(t)) ifλ∈σiso(T) byLemma 2.9, so the notionE0(p,r) inLemma 2.11is reasonable.

Proof. SinceT(p,r) isα0-hyponormal byTheorem 2.5, we only need to prove thatE0Ᏼ⊆ E0(p,r)ᏴforE0Ᏼ⊇E0(p,r)Ᏼholds byLemma 2.6andTheorem 2.10. We may also assume thatp+r=1 byLemma 2.6.

It follows fromLemma 2.11that

|T|^pE0(p,r)Ᏼ⊆ |T|^pE0Ᏼ⊆E0(p,r)Ᏼ, (2.11) thusE0(p,r)Ᏼis reduced by|T|^p.

Letx∈E0Ᏼandx=x1+x2∈E0(p,r)Ᏼ⊕(E0(p,r)Ᏼ)^⊥. Then |T|^px∈ |T|^pE0Ᏼ⊆ E0(p,r)Ᏼ,|T|^px1∈E0(p,r)Ᏼ,|T|^px2∈(E0(p,r)Ᏼ)^⊥by (2.11), andE0(p,r)Ᏼis reduced by|T|^p.

Thus|T|^px2= |T|^px− |T|^px1∈E0(p,r)Ᏼ,|T|^px2∈E0(p,r)Ᏼ∩(E0(p,r)Ᏼ)^⊥ so that x2∈ker|T|^p⊆ker(T(p,r))=E0(p,r)Ᏼ,x∈E0(p,r)Ᏼ.

Proof ofTheorem 2.1. We only need to prove (1) for (2) holds byLemma 2.12.

Sinceσ(T(p,r))=σ(U|T|^p+r)= {e^iθρ^p+r:e^iθρ∈σ(T)}by Lemmas2.7and2.9,λ_p+r∈ σiso(T(p,r)). Hence

Eλ(p,r)Ᏼ^⊥=kerEλ(p,r)=

I−Eλ(p,r)Ᏼ (2.12) byTheorem 2.10, soλp+r∈σ(T(p,r)|(Eλ(p,r)Ᏼ)^⊥). ByTheorem 2.4(1) andLemma 2.6, we haveT=λ⊕T22onᏴ=E_λ(p,r)Ᏼ⊕(E_λ(p,r)Ᏼ)^⊥, whereT22=T|(ker(T−λ))^⊥.

Since ker(T−λ) is reduced byT,T22also belongs to classwF(p,r,q) andT22(p,r)= T(p,r)|(Eλ(p,r)Ᏼ)^⊥so thatλ∈σ(T22) becauseλp+r∈σ(T22(p,r)). HenceT−λ=0⊕(T22− λ) and ker(T₋λ)^∗₌ker(T₋λ)_⊕ker(T22−λ)^∗₌ker(T₋λ).

Meanwhile,E_λ=

∂Ᏸ(z−λ)⁻¹⊕(z−T22)⁻¹dz=1⊕0=E_λ(p,r).

Proof ofTheorem 2.2. We only need to prove thatTis reguloid forTbeing isoloid follows byTheorem 2.1easily.

Ifλ∈σiso(T), thenᏴ=EλᏴ+ (I−Eλ)Ᏼ, whereEλᏴ, and (I−Eλ)Ᏼare topologically complemented [28, page 94]. ByT =T|EλᏴ+T|(I−Eλ)Ᏼ on Ᏼ=E_λᏴ+ (I−E_λ)Ᏼ and Theorem 2.1, we have

(T−λ)Ᏼ=

T_(I₋_E_λ_)Ᏼ−λI−E_λᏴ. (2.13) Therefore (T−λ)Ᏼis closed becauseσ(T|(I−Eλ)Ᏼ)=σ(T)− {λ}.

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3. SVEP and Bishop’s property (β)

Definition 3.1. An operatorTis said to have SVEP atλ∈Ꮿif for every open neighbor- hoodGofλ, the only function f ∈H(G) such that (T−λ)f(μ)=0 onGis 0∈H(G), whereH(G) means the space of all analytic functions onG.

WhenThave SVEP at eachλ∈Ꮿ, say thatThas SVEP.

This is a good property for operators. IfT has SVEP, then for eachλ∈Ꮿ,λ−T is invertible if and only if it is surjective (cf. [29,18]).

Definition 3.2. An operatorT is said to have Bishop’s property (β) atλ∈Ꮿif for every open neighborhoodGofλ, the function fn∈H(G) with (T−λ)fn(μ)→0 uniformly on every compact subset ofG implies that f_n(μ)→0 uniformly on every compact subset ofG.

WhenThas Bishop’s property (β) at eachλ∈Ꮿ, simply say thatThas property (β).

This is a generalization of SVEP and it is introduced by Bishop [30] in order to develop a general spectral theory for operators on Banach space.

Theorem 3.3. Letpandrbe positive numbers. Ifp+r=1, thenThas SVEP if and only ifT(p,r) has SVEP,Thas property (β) if and only ifT(p,r) has property (β). In particular, every classwF(p,r,q) operatorTwithp+r≤1 has SVEP and property (β).

This result is a generalization of [18].Lemma 3.4and the relations betweenTand its transformationT(p,r) are important:

T(p,r)|T|^p= |T|^pU|T|^r|T|^p= |T|^pT,

U|T|^rT(p,r)=U|T|^r|T|^pU|T|^r=TU|T|^r. (3.1) Lemma 3.4 (see [18]). LetG be open subset of complex plane Ꮿand let f_n∈H(G) be functions such thatμ fn(μ)→0 uniformly on every compact subset of G, then fn(μ)→0 uniformly on every compact subset ofG.

Proof ofTheorem 3.3. We only prove that T has property (β) if and only ifT(p,r) has property (β) because the assertion thatThas SVEP if and only ifT(p,r) has SVEP can be proved similarly.

Suppose thatT(p,r) has property (β). LetG be an open neighborhood ofλand let fn∈H(G) be functions such that (μ−T)fn(μ)→0 uniformly on every compact subset of G. By (3.1), (T(p,r)−μ)|T|^pfn(μ)= |T|^p(T−μ)fn(μ)→0 uniformly on every compact subset ofG. HenceT f_n(μ)=U|T|^r|T|^pf_n(μ)→0 uniformly on every compact subset of GforT(p,r) has property (β), so thatμ fn(μ)→0 uniformly on every compact subset of G, andThaving property (β) follows byLemma 3.4.

Suppose thatT has property (β). LetGbe an open neighborhood of λand let f_n∈ H(G) be functions such that (μ−T(p,r))fn(μ)→0 uniformly on every compact subset ofG. By (3.1), (μ−T)(U|T|^rfn(μ))=U|T|^r(μ−T(p,r))fn(μ)→0 uniformly on every compact subset ofG. HenceT(p,r)f_n(μ)→0 uniformly on every compact subset ofG forT has property (β) so thatμ fn(μ)→0 uniformly on every compact subset ofG, and T(p,r) having property (β) follows byLemma 3.4.

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4. Weyl spectrum

For a Fredholm operatorT, indTmeans its (Fredholm) index. A Fredholm operatorTis said to be Weyl if indT=0.

Letσe(T),σw(T), andπ00(T) mean the essential spectrum, Weyl spectrum, and the set of all isolated eigenvalues of finite multiplicity of an operatorT, respectively (cf. [28,17]).

According to Coburn [31], we say that Weyl’s theorem holds for an operatorT if σ(T)−σw(T)=π00(T). Very recently, the theorem was shown to hold for several classes of operators includingw-hyponormal operators and paranormal operators (cf. [17,32, 20]).

In this section, we will prove that Weyl’s theorem and Weyl spectrum mapping theorem hold for classwF(p,r,q) operatorT with p+r_≤1. We also assume that p+r₌1 because of the inclusion relations among classwF(p,r,q) [9].

Theorem 4.1. LetTbelong to classwF(p,r,q) withp+r=1 and letH(σ(T)) be the space of all functions f analytic on some open setGcontainingσ(T), then the following assertions hold.

(1) Weyl’s theorem holds forT.

(2)σ_w(f(T))=f(σ_w(T)) whenf ∈H(σ(T)).

(3) Weyl’s theorem holds for f(T) when f ∈H(σ(T)).

This is a generalization of the related assertions of [17].

Theorem 4.2. LetTbelong to classwF(p,r,q) withp+r=1, then the following assertions hold.

(1) Ifm2(σ(T))=0 wherem2means the planar Lebesgue measure, thenTis normal.

(2) Ifσw(T)=0, thenTis compact and normal.

Theorem 4.2(1) is a generalization of [26] and (2) is a generalization of [24].

To give proofs, the following results are needful.

Theorem 4.3 [9]. Letp >0,r >0, andq≥1,s≥p,t≥r. IfTis a classwF(p,r,q) operator andT(s,t) is normal, thenTis normal.

Lemma 4.4. IfTbelongs to classwF(p,r,q) withp+r=1 and is Fredholm, then indT≤0.

This result can be regarded as a good complement ofTheorem 2.1.

Proof. SinceTis Fredholm,|T|^pis also Fredholm and ind(|T|^p)=0. By (3.1), indT=ind|T|^pT=indT(p,r)|T|^p

=indT(p,r). (4.1)

Hence, indT≤0 for ind(T(p,r))≤0 byTheorem 2.5.

Proof ofTheorem 4.1. (1) Letλ∈σ(T)−σw(T), thenT−λis Fredholm, ind(T−λ)=0, and dim ker(T−λ)>0.

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Ifλis an interior point ofσ(T), there would be an open subsetG⊆σ(T) includingλ such that ind(T−μ)=ind(T−λ)=0 for allμ∈G[28, page 357]. So dim ker(T−μ)>0 for allμ∈G, this is impossible forThas SVEP byTheorem 3.3[29, Theorem 10]. Thus λ∈∂σ(T)−σw(T),λ∈σiso(T) by [28, Theorem 6.8, page 366], andλ∈π00(T) follows.

Let λ∈π00(T), then the Riesz idempotent Eλ has finite rank byTheorem 2.1, and λ∈σ(T)−σ_w(T) follows.

(2) We only need to prove thatσw(f(T))⊇f(σw(T)) sinceσw(f(T))⊆ f(σw(T)) is always true for any operators.

Assume that f ∈H(σ(T)) is not constant. Let λ∈σ_w(f(T)) and f(z)−λ=(z− λ1)···(z₋λ_k)g(z), where_{λ_i_}^k₁are the zeros of f(z)₋λinG(listed according to multiplicity) andg(z)=0 for eachz∈G. Thus

f(T)−λ= T−λ1

···

T−λk

g(T). (4.2)

Obviously,λ∈ f(σ_w(T)) if and only ifλ_i∈σ_w(T) for somei. Next we prove thatλ_i∈ σw(T) for everyi∈ {1,. . .,k}, thusλ∈ f(σw(T)) andσw(f(T))⊇f(σw(T)).

In fact, for eachi,T−λiis also Fredholm because f(T)−λis Fredholm. ByTheorem 2.1andLemma 4.4, ind(T−λ_i)≤0 for eachi. Since 0=ind(f(T)−λ)=ind(T−λ1) +

···+ ind(T−λ_k), ind(T−λ_i)=0 andλ_i∈σ_w(T) for eachi.

(3) ByTheorem 2.2,Tis isoloid and it follows from [33] that σf(T)−π00

f(T)= fσ(T)−π00(T). (4.3) On the other hand, f(σ(T)−π00(T))= f(σw(T))=σw(f(T)) by (1)-(2). The proof is

complete.

Proof ofTheorem 4.2. (1) By α0-hyponormality ofT(p,r) and Putnam’s inequality for α0-hyponormal operators [26],T(p,r) is normal. Hence, (1) follows byTheorem 4.3.

(2) Since σw(T)=0, σ(T)− {0} =π00(T)⊆σiso(T) by Theorem 4.1(1). Hence m2(σ(T))=0 andTis normal by (1).

Next to prove thatTis compact, we may assume thatσ(T)− {0}is a countable infinite set forσ(T)− {0} ⊆σiso(T). Letσ(T)− {0} = {λn}^∞1 with|λ1| ≥ |λ2| ≥ ··· ≥0 andλ0= limn→∞|λn|, thenλ0=0. Since every Eλn has finite rank by Theorems2.1and 4.1, for everyε >0,_|_λ_n_|_>εE_λ_nalso has finite rank. ThereforeTis compact [28, page 271].

Acknowledgments

The authors would like to express their cordial gratitude to the referee for valuable ad- vice and suggestions, and Professor Atsushi Uchiyama for sending them [22]. This work was supported in part by the National Key Basic Research Project of China Grant no.

2005CB321902.

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Jiangtao Yuan: LMIB and Department of Mathematics, Beihang University, Beijing 100083, China Email address:[email protected]

Zongsheng Gao: LMIB and Department of Mathematics, Beihang University, Beijing 100083, China Email address:[email protected]