# A Note on the Range of the OperatorX↦TX−XT Defined on 𝒞2(ℋ)

Authors: Vasile Lauric
Journal: International Journal of Mathematics and Mathematical Sciences
 (2009)

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## Abstract

We show how a proof of J. Stampfli can be extended to prove that the operatorX↦TX−XT defined on the Hilbert-Schmidt class, when T is an M-hyponormal, p-hyponormal, or log-hyponormal operator, has a closed range if and only if σ(T) is finite.


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## Body

## 1. Introduction

Letℋ be a complex, separable, infinite dimensional Hilbert space, and let ℒ(ℋ) denote the algebra of all linear bounded operators on ℋ. The Hilbert-Schmidt class, denoted by 𝒞2(ℋ), is a Hilbert space with the ||·||2-norm that arises from the inner product 〈X,Y〉=tr(XY*), where tr is the scalar-valued trace. For T∈ℒ(ℋ), define ΔT:ℒ(ℋ)→ℒ(ℋ) by ΔT(X)=TX-XT, and let σ(T) denote the spectrum of T. Let the range of a linear operator S be denoted by ℛ(S). For a normal operator N∈ℒ(ℋ), Anderson and Foiaş [1] proved that ℛ(ΔN) is norm closed if and only if σ(N) is a finite set. In [2], Stampfli extended this result to the class of hyponormal operators.Theorem 1 A ([2]).
LetT∈ℒ(ℋ) be a hyponormal operator. Then ℛ(ΔT) is norm closed if and only ifσ(T) is finite.In fact, Stampfli provided a proof of the “only if" implication which extends to a larger class of operators than the class of hyponormal operators (see Proposition2.2). For an operator T∈ℒ(ℋ), let σnap(T) denote its normal approximate point spectrum, that is, the set of those λ∈ℂ for which there exists an orthonormal sequence {ϕn}n in ℋ such that
(1.1)∥(T-λ)ϕn∥+∥(T-λ)*ϕn∥→0.
Define the class 𝒢(ℋ) as follows:
(1.2)𝒢(ℋ):={T∈ℒ(ℋ)∣σnap(T)is  an  infinite  set}.
Some classes of hyponormal related operators, such as M-hyponormal operators, that is,
(1.3)m·∥(T-λ)*ϕ∥≤∥(T-λ)ϕ∥,(∀)ϕ∈ℋ,(∀)λ∈ℂ,for  somem>0,p-hyponormal operators, that is, (T*T)p≥(TT*)p for some p>0, or log-hyponormal operators, that is, invertible operators such that log(T*T)≥log(TT*), have spectrum that is finite or they belong to 𝒢(ℋ). Particularly, the hyponormal operators (i.e., 1-hyponormal) have this property.In [3] Stampfli proved the following lemma which will be used in Section 2.Lemma 1 B.
LetT∈𝒢(ℋ) and let  {λn}n=1∞ be a sequence of distinct points of  σnap(T).Then for any sequence {εn}n=1∞ of positive numbers converging to zero, there exists an orthonormal sequence  {ϕn}n=1∞ of vectors in   ℋ such that
(1.4)∥(T-λn)ϕn∥+∥(T-λn)*ϕn∥<εnforn=1,2,…,(1.5)〈ϕn,Tϕk〉=0fork=1,…,n-1.


## 2. The Closedness of the Range ofΔT(2)

The operatorΔT defined on  the Hilbert-Schmidt class will be denoted in the remainder of this note by ΔT(2), that is, ΔT(2):𝒞2(ℋ)→𝒞2(ℋ),  ΔT(2)(X)=TX-XT. Let HM(ℋ) denote the set of M-hyponormal operators.Proposition 2.1.
LetT∈HM(ℋ). If   σ(T) is finite, then  ℛ(ΔT(2)) is closed.Proof.
It is well known that an operatorT∈HM(ℋ) with finite spectrum is normal. Indeed, for such an operator, the restriction to an invariant subspace ℳ belongs to HM(ℳ). On the other hand, if T∈HM(ℋ) with σ(T)={λ}, then T=λI, (cf. [4]). Thus, we can write T=∑i=1n0λiEi, where Ei's are the spectral projections.
LetXn and C be in 𝒞2(ℋ) such that∥ΔT(2)(Xn)-C∥2→0. Therefore ΔT(Xn)-C→0 in the ℒ(ℋ) norm, and according to Theorem A, there exists X0∈ℒ(ℋ) such that C=TX0-X0T. For an arbitrary X∈ℒ(ℋ), let [Xij] be the block-matrix representation of X relative to the decomposition ℋ=∑i=1n0⊕Eiℋ. Thus
(2.1)Cij=(λi-λj)Xij0,
for alli,j=1,…,n0. This implies that each Xij0=(1/(λi-λj))Cij is a Hilbert-Schmidt operator. Moreover Xii0 can be chosen 0, and thus X0∈𝒞2(ℋ).Proposition 2.2.
LetT∈𝒢(ℋ).Then   ℛ(ΔT(2)) is not closed.Proof.
We will use same notation and circle of ideas as in [2]. Let {λn}n≥1 be sequence of distinct points of σnap(T) so that λn→λ0. Let
(2.2)ηn=max{|λj+1-λj|-1/2∣j=1,…,n},
and choose a nonincreasing sequence {εn}n≥1 so that 0<εn≤|λn+1-λn|2,n≥1, and ∑n≥1εn2ηn2<∞. According to Lemma B, there exists an orthonormal sequence {ϕn}n≥1 that satisfies (1.4) and(1.5). Let ℋ1=∨{ϕn∣n≥1},ℋ2=ℋ1⊥, and let δn such that
(2.3)Tϕn=μnϕn+δn,δn⊥ϕn,n≥1.
It results that
(2.4)|μn-λn|<εn,∥δn∥<2εn,n≥1.
Define V:ℋ→ℋ  by Vϕn=|λj+1-λj|-1/2ϕn+1,n≥1, and  Vg=0,g∈ℋ2. Let ℳn=∨{ϕj∣j=1,…,n} and let Pn be the orthogonal projection onto ℳn, and define Vn=VPn. A tedious calculation shows that
(2.5)ΔT(Vn)ϕj={vj(μj+1-μj)ϕj+1+vjδj+1-Vnδj,-Vnδj,j≤n,j>n,
where vj=|λj+1-λj|-1/2. Denoting ΔT(Vn)-ΔT(Vm) by ΔTn,m,  then for n<m,(2.6)ΔTn,mϕj={0,-vj(μj+1-μj)ϕj+1+vjδj+1+(Vm-Vn)δj,(Vm-Vn)δj,j≤n,n<j≤m,j>m.
Furthermore, from (2.3) it results that
(2.7)δj⊥ϕj,ϕj+1,ϕj+2,…
and from (2.4)
(2.8)∥Vnδj∥≤2ηjεj,∀j,n≥1.
We will show next that∥ΔTn,m∥2→0 when m,n→∞, thus there exists C∈𝒞2(ℋ) such that ∥ΔT(Vn)-C∥2→0, that is, C∈ℛ(ΔT(2))¯.
First, we will show that∥ΔTn,m∣ℋ1∥22→0, when m,n→∞. Indeed,
(2.9)∥ΔTn,m∣ℋ1∥22=∑j=1∞∥ΔTn,mϕj∥2=(2.6)=∑j=n+1m∥-vj(μj+1-μj)ϕj+1+vjδj+1+(Vm-Vn)δj∥2+∑j=m+1∞∥(Vm-Vn)δj∥2.
The first sum of the right-hand side of the above can be majorized by(2.10)2·∑j=n+1m∥-vj(μj+1-μj)ϕj+1+vjδj+1∥2+2·∑j=n+1m∥(Vm-Vn)δj∥2.
Since ϕj+1⊥δj+1, we have
(2.11)∥ΔTn,m∣ℋ1∥22≤2[∑j=n+1m(vj2|μj+1-μj|2+vj2∥δj+1∥2)+∑j=n+1∞∥(Vm-Vn)δj∥2].
According to (2.8),
(2.12)∥(Vm-Vn)δj∥2≤16ηj2εj2,
and according to (2.4),
(2.13)vj2∥δj+1∥2≤4ηj2εj+12≤4ηj2εj2,|μj+1-μj|2≤(2εj+|λj+1-λj|)2≤8εj2+2|λj+1-λj|2,
which implies
(2.14)vj2|μj+1-μj|2≤8ηj2εj2+2|λj+1-λj|.
Therefore
(2.15)∥ΔTn,m∣ℋ1∥22≤c1·∑j=n+1∞ηj2εj2+c2·∑j=n+1m|λj+1-λj|,
where c1 and c2 are some constants. After a careful review of the proof, one can see that the sequence {λn} can be assumed to converge fast enough (otherwise choose a subsequence of it), more precisely  ∑j=n+1m|λj+1-λj|→0  when  n,m→∞.
We show next that∥ΔTn,m∣ℋ2∥22→0, when m,n→∞. Indeed, we can write
(2.16)T*ϕn=μ¯nϕn+γnwith〈γn,ϕn〉=0,∥γn∥≤2εn,n≥1.
Obviously, we can writeT*ϕn=θnϕn+γn with 〈γn,ϕn〉=0, which implies
(2.17)θn=〈θnϕn+γn,ϕn〉=〈T*ϕn,ϕn〉=〈ϕn,Tϕn〉=〈ϕn,μnϕn+δn〉=μ¯n,∥γn∥=∥(T*-μ¯n)ϕn∥≤∥(T-λn)*ϕn∥+|λ¯n-μ¯n|≤(1.4),(2.4)2εn.
For an orthonormal basis {ψi}i≥1 of ℋ2, we will show that
(2.18)∑i=1∞∥ΔTn,mψi∥2→0whenn,m→∞.
For each i,  write Tψi=∑k=1∞ak(i)ϕk+wi  with  wi∈ℋ2. Thus
(2.19)VmTψi=∑k=1mak(i)Vmϕk+Vmwi=∑k=1mak(i)vkϕk+1.
Since Vmψi=0, we have ΔT(Vm)ψi=-VmTψi, and consequently, for n<m,(2.20)ΔTn,mψi=∑k=n+1mak(i)vkϕk+1.
Since the sequence {ϕk} is orthonormal, we have
(2.21)∥ΔTn,mψi∥2=∑k=n+1m|ak(i)|2·vk2.
Therefore
(2.22)∑i=1∞∥ΔTn,mψi∥2=∑i=1∞∑k=n+1m|ak(i)|2·vk2=∑k=n+1mvk2(∑i=1∞|ak(i)|2).
For a fixed k,(2.23)∑i=1∞|ak(i)|2=∑i=1∞|〈Tψi,ϕk〉|2=∑i=1∞|〈ψi,T*ϕk〉|2=(2.16)∑i=1∞|〈ψi,μ¯kϕk+γk〉|2=∑i=1∞|〈ψi,γk〉|2≤∥γk∥2≤(2.16)4εk2.
Consequently, ∑i=1∞∥ΔTn,mψi∥2≤4∑k=n+1mvk2·εk2→0  for  n,m→∞.
The operatorC is not in ℛ(ΔT(2)) since, according to the proof of  Theorem  A in [2],  C∉ℛ(ΔT).Theorem 2.3.
LetT∈HM(ℋ). Then ℛ(ΔT(2)) is closed if and only if σ(T) is finite.Proof.
IfT∈HM(ℋ) and σ(T) are finite, then according to Proposition 2.1, ℛ(ΔT(2))is closed. Conversely, if T∈HM(ℋ) has an infinite spectrum, then there are infinitely many distinct points {λn}n that are either isolated points of the spectrum, in which case they are eigenvalues, or accumulation points of the spectrum, in which case they are in σap(T). Since T∈HM(ℋ), we have σp(T),σap(T)⊆σnap(T). Thus T∈𝒢(ℋ) and according to Proposition 2.2, ℛ(ΔT(2)) is not closed.



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*Source: 603041-2009-07-06.xml*