A Note on the Range of the Operator X↦TX−XT Defined on

International Journal of Mathematics and Mathematical Sciences Volume 2009, Article ID 603041,6pages

doi:10.1155/2009/603041

Research Article

A Note on the Range of the Operator

X

→

TX

−

XT

Defined on

C

2

H

Vasile Lauric

Department of Mathematics, Florida A&M University, Tallahassee, FL 32307, USA

Correspondence should be addressed to Vasile Lauric,[email protected]

Received 10 November 2008; Accepted 11 May 2009

Recommended by Manfred H. Moller

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 ifσTis finite.

Copyrightq2009 Vasile Lauric. 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

LetHbe a complex, separable, infinite dimensional Hilbert space, and letLHdenote the algebra of all linear bounded operators onH.The Hilbert-Schmidt class, denoted byC2H,is a Hilbert space with the||·||2-norm that arises from the inner productX, YtrXY∗,where tr is the scalar-valued trace. ForT ∈ LH,defineΔT :LH → LHbyΔTX TX−XT, and letσTdenote the spectrum ofT. Let the range of a linear operatorSbe denoted by

RS.For a normal operatorN∈ LH,Anderson and Foias¸1proved thatRΔNis norm closed if and only ifσNis a finite set. In2, Stampfli extended this result to the class of hyponormal operators.

Theorem A2. LetT ∈ LHbe a hyponormal operator. ThenRΔTis norm closed if and only

ifσTis 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 seeProposition 2.2. For an operatorT ∈ LH,letσnapTdenote its normal approximate point spectrum, that is, the set

of thoseλ∈Cfor which there exists an orthonormal sequence{φn}ninHsuch that

_{T−λφnT−λ}∗_φ

Define the classGHas follows:

GH:T ∈ LH|σnapTis an infinite set

. 1.2

Some classes of hyponormal related operators, such asM-hyponormal operators, that is,

m·T−λ∗φ≤T−λφ, ∀φ∈ H, ∀λ∈C, for some m >0, 1.3

p-hyponormal operators, that is, T∗Tp ≥ TT∗p for some p > 0, or log-hyponormal operators, that is, invertible operators such that logT∗_{T ≥ logTT}∗_{, have spectrum that}

is finite or they belong toGH.Particularly, the hyponormal operatorsi.e., 1-hyponormal have this property.

In3Stampfli proved the following lemma which will be used inSection 2.

Lemma B. Let T ∈ GH and let {λn}∞n1 be a sequence of distinct points of σnapT.Then for

any sequence {εn}∞n1 of positive numbers converging to zero, there exists an orthonormal sequence {φn}∞n1 of vectors in Hsuch that

T−λnφnT−λn∗φn< εn for n1,2, . . . , 1.4

φn, Tφk0 for k1, . . . , n−1. 1.5

2. The Closedness of the Range of

Δ

The operatorΔTdefined on the Hilbert-Schmidt class will be denoted in the remainder of this note byΔ_T2,that is, Δ_T2 :C2H → C2H,Δ_T2X TX−XT.LetHMHdenote the set ofM-hyponormal operators.

Proposition 2.1. Let T ∈HM_{H. If σT is finite, then RΔ}2

T is closed.

Proof. It is well known that an operatorT ∈HMHwith finite spectrum is normal. Indeed, for such an operator, the restriction to an invariant subspaceMbelongs toHMM.On the other hand, if T ∈ HMH with σT {λ}, then T λI,cf. 4. Thus, we can write

T n0

i1λiEi,whereEi’s are the spectral projections.

LetXnandCbe inC2Hsuch thatΔ_T2Xn−C₂ → 0.ThereforeΔTXn−C → 0 in theLHnorm, and according to Theorem A, there existsX0_{∈ LHsuch thatCTX}0_−X0_T.

For an arbitraryX ∈ LH,letXijbe the block-matrix representation ofX relative to the decompositionHn0

i1⊕EiH.Thus

Cijλi−λj X0ij, 2.1

for alli, j1, . . . , n0.This implies that eachXij0 1/λi−λjCijis a Hilbert-Schmidt operator. MoreoverX0

Proposition 2.2. LetT ∈ GH.Then RΔ_T2 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 ofσnapTso thatλn → λ0.Let

ηn maxλj1−λj−1/2|j1, . . . , n

, 2.2

and choose a nonincreasing sequence {εn}n≥1 so that 0 < εn ≤ |λn1−λn|2, n ≥ 1, and

n≥1ε2nηn2 < ∞.According to Lemma B, there exists an orthonormal sequence{φn}n≥1 that

satisfies1.4and1.5. LetH1∨{φn|n≥1},H2H⊥₁,and letδnsuch that

Tφnμnφnδn, δn⊥φn, n≥1. 2.3

It results that

_{μn−λn< εn, δn<2εn, n≥1. 2.4}

DefineV :H → HbyV φn|λj1−λj|−1/2 φn1, n≥1,andV g 0, g ∈ H2.LetMn∨{φj|

j1, . . . , n}and letPnbe the orthogonal projection ontoMn,and defineVnV Pn.A tedious calculation shows that

ΔTVnφj

⎧ ⎨ ⎩

vjμj1−μj φj1vjδj1−Vnδj,

−Vnδj,

j≤n,

j > n, 2.5

wherevj |λj1−λj|−1/2.DenotingΔTVn−ΔTVmbyΔn,mT ,then forn < m,

Δn,m_T φj

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

0,

−vjμj1−μj φj1vjδj1 Vm−Vnδj, Vm−Vnδj,

j≤n,

n < j ≤m,

j > m.

2.6

Furthermore, from2.3it results that

δj⊥φj, φj1, φj2, . . . 2.7

and from2.4

Vnδj≤2ηjεj, ∀j, n≥1. 2.8

We will show next thatΔn,m_{T 2} → 0 whenm, n → ∞,thus there existsC ∈ C2H such thatΔTVn−C2 → 0,that is,C∈ RΔ

2

First, we will show thatΔn,m_T |_H12₂ → 0,whenm, n → ∞.Indeed,

_Δn,m T |H1

2 2

∞

j1

_Δn,m T φj2

2.6

m jn1

_−vj

μj1−μj φj1vjδj1 Vm−Vnδj2

∞ jm1

Vm−Vnδj2.

2.9

The first sum of the right-hand side of the above can be majorized by

2· m

jn1

_−vj

μj1−μj φj1vjδj122·

m

jn1

_Vm−Vnδj2_{. 2.10}

Sinceφj1⊥δj1,we have

_Δn,m T |H1

2 2≤2

⎡ ⎣m

jn1

v2

j μj1−μj2vj2 δj12

∞ jn1

Vm−Vnδj2

⎤

⎦. 2.11

According to2.8,

_Vm−Vnδj2_≤

16η2_jε2_j, 2.12

and according to2.4,

v2

jδj12 ≤4η2jε2j1≤4ηj2ε2j,

μj1−μj2≤2εjλj1−λj 2≤8ε2j 2λj1−λj2,

2.13

which implies

v2

jμj1−μj2≤8ηj2εj22λj1−λj. 2.14

Therefore

_Δn,m T |H1

2 2≤c1·

∞

jn1

η2

jεj2c2·

m

jn1

λj1−λj, 2.15

wherec1andc2are some constants. After a careful review of the proof, one can see that the

sequence{λn}can be assumed to converge fast enoughotherwise choose a subsequence of it, more preciselym

We show next thatΔn,m_T |_H22₂ → 0,whenm, n → ∞.Indeed, we can write

T∗_φnμ

nφnγn with

γn, φn0, γn≤2εn, n≥1. 2.16

Obviously, we can writeT∗φnθnφnγnwithγn, φn0,which implies

θnθnφnγn, φnT∗φn, φnφn, Tφnφn, μnφnδnμn,

γnT∗−μ_n φn≤T−λn∗φnλn−μ_n

1.4,2.4

≤ 2εn.

2.17

For an orthonormal basis{ψi}i≥1ofH2,we will show that

∞

i1

_Δn,m

T ψi2−→0 whenn, m−→ ∞. 2.18

For eachi,writeTψi∞k1a

i

k φkwiwithwi∈ H2.Thus

VmTψi

m

k1

a_kiVmφkVmwi m

k1

a_kivkφk1. 2.19

SinceVmψi0,we haveΔTVmψi−VmTψi,and consequently, forn < m,

Δn,m_T ψi m

kn1

a_kivkφk1. 2.20

Since the sequence{φk}is orthonormal, we have

_Δn,m T ψi2

m

kn1

a_ki2·v2

k. 2.21

Therefore

∞

i1

_Δn,m T ψi2

∞

i1

m

kn1

a_ki2·v2

k m

kn1

v2

k

_∞

i1

a_ki2

For a fixedk,

∞

i1

a_ki2∞ i1

_{Tψi, φk}2∞

i1

_{ψi, T}∗_φk2

2.16∞

i1

ψi, μkφkγk2

∞

i1

ψi, γk2≤γk2

2.16

≤ 4ε2_k.

2.23

Consequently,∞_i1Δn,m_T ψi2≤4mkn1vk2·ε2k → 0 forn, m → ∞.

The operatorC is not inRΔ_T2since, according to the proof of Theorem A in 2,

C /∈ RΔT.

Theorem 2.3. LetT ∈HM_H.ThenRΔ2

T is closed if and only ifσTis finite.

Proof. If T ∈ HMH and σT are finite, then according to Proposition 2.1, RΔ_T2is closed. Conversely, ifT ∈ HMHhas 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 inσapT.

Since T ∈ HMH, we have σpT, σapT ⊆ σnapT. Thus T ∈ GH and according to

Proposition 2.2,RΔ_T2is not closed.

References

1 J. Anderson and C. Foias¸, “Properties which normal operators share with normal derivations and related operators,” Pacific Journal of Mathematics, vol. 61, no. 2, pp. 313–325, 1975.

2 J. G. Stampfli, “On the range of a hyponormal derivation,” Proceedings of the American Mathematical Society, vol. 52, pp. 117–120, 1975.

3 J. G. Stampfli, “Compact perturbations, normal eigenvalues and a problem of Salinas,” Journal of the London Mathematical Society, vol. 9, no. 1, pp. 165–175, 1974.