Almost

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International Journal of Mathematics and Mathematical Sciences Volume 2011, Article ID 801313,5pages

doi:10.1155/2011/801313

Research Article

Almost

α

-Hyponormal Operators with

Weyl Spectrum of Area Zero

Vasile Lauric

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

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

Received 27 December 2010; Accepted 20 March 2011

Academic Editor: Ra ¨ul Curto

Copyrightq2011 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.

We define the class of almostα-hyponormal operators and prove that for an operatorT in this class,T∗Tα−TT∗αis trace-class and its trace is zero whenα∈0,1and the area of the Weyl spectrum is zero.

This note is dedicated to Professor Carl M. Pearcy with the occasion of his 75th birthday.

LetHbe a complex, separable, infinite-dimensional Hilbert space, and letLHdenote the

algebra of all linear bounded operators onH, and for 1 ≤ p < ∞, let CpH denote the

p-Schatten class on H. For K ∈ CpH, the expression ||K||p : ∞n1μnKp1/p, where

μ1K≥μ2K≥ · · · are the singular values ofK, is a norm forp≥1, and is only a quasinorm

for 0< p <1it does not satisfy the triangle inequality. Nevertheless, the latter case will be

used in what follows.

For T ∈ LH,σT and σwT will denote the spectrum and the Weyl spectrum,

respectively. Recall that Weyl spectrum is the union of the essential spectrum, σeT, and

all bounded components of \σeTassociated with nonzero Fredholm index. An operator

T ∈ LHis calledCp, α-normalnotation:T ∈ Nα

pHifCαT : T∗Tα−TT∗αbelongs

to CpH, andT is calledCp, α-hyponormalnotation: T ∈ Hα

pHif CαT is the sum of a

positive definite operator and an operator inCpH, or equivalently,Cα_T₋the negative part

of Cα

Tbelongs to CpH, whereαis a positive number. This note will be concerned with

the particular classHα

1H, which by some parallelism with some terminology used in1,

would be appropriate to be referred as almostα-hyponormal operators.

Voiculescu’s 1generalization of Berger-Shaw inequality gives an estimate for the

trace ofC1

T. The result was extended in2. The combination of these results will be stated

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T inLH, denoted bymT, is the smallest cardinal numbermwith the property that there

aremvectorsx1, . . . , xminHsuch that

∨fTxj|1≤j≤m, f ∈RatσTH, 1

where RatσTis the algebra of complex-valued rational functions with poles oﬀσT.

For a Borel subsetE⊆ andα >0, denoteμαE α/2_Eρα−1 dρdθ. In particular,

μ2is the planar Lebesgue measure.

Theorem Asee1,2. SupposeT ∈H₁1H. If there existsK ∈ C2Hsuch that eithermT

K<∞orμ2σTK 0, thenT ∈N1

1H. Moreover, whenmTK<∞,

trC1T

≤ mTK

π · μ2σTK, 2

and whenμ2σTK 0, trC1

T≤0, and consequently, trC1T 0.

In fact, it was observed in2that the inequality can be improved by replacingmTK

withτTK, where

τS:lim infrankI−PSP, 3

and the lim inf is taken over all sequences of finite-rank orthogonal projections such that

P → Iin the strong operator topology.

Corollary Bsee2. LetT ∈H1

1Hsuch thatμ2σwT 0. ThenT ∈N11Hand trCT1 0.

On the other hand, Berger-Shaw inequality was extended to operators inH₁αHusing

similar circle of ideas used in1. This was done in3for the caseα∈1/2,1and later on

in4for the caseα∈0,1/2.

Theorem Csee3,4. Let 0< α≤1, and letT ∈H₁αHandK∈ C2αHwithmTK<∞.

ThenT ∈Nα

1Hand

trCα

T ≤

mTK

π ·μ2ασTK. 4

The case in whichmTK ∞andμ2ασTK 0 was not discussed in4or3.

It is the goal of this note to make some progress towards this case. We have the following.

Theorem 1. Letα ∈ 0,1and letT ∈ Hα

1HandK ∈ CαHwithμ2ασT K 0. Then

T ∈Nα₁Hand trC_Tα 0.

Remark. It would have been desirable that Theorem1 be proved with the hypothesis that

K∈ C2αH.

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Corollary 2. Letα∈0,1and letT ∈H₁αHsuch thatμ2σwT 0. ThenT ∈N₁αHand

trCα_T 0.

Proof. Ifα1, then conclusion holds according to Corollary B. Letα∈0,1. First, a careful

inspection of the proof of a result of Stampfli5leads to the following. ForT ∈ LHand

α >0, there existsKα∈ CαHsuch thatσTKα\σwTconsists of a countable set which

clusters only onσwT. Thereforeμ2σTKα 0 and thus Theorem1applies.

The proof of Theorem1makes use of the following three inequalities.

Proposition DHansen’s inequality6. IfA, B∈LH,A≥0,||B|| ≤1, andα∈0,1, then

B∗Aα_B_≤_B∗_ABα_.

Proposition E Lowner’s inequality 7. IfA, B ∈ LH,A ≥ B ≥ 0, and α ∈ 0,1, then

Aα_≥_Bα_.

The following is a consequence of Theorem 3.4 of8.

Proposition FJocic’s inequality8. LetA, B∈LH,A, B≥0,α∈0,1, and 1≤p <∞. If

A−B∈ CαpH, thenAα₋_Bα_{∈ CpH}_and_||Bα₋_Aα_||p_{≤ || |B}₋_A|α _||p.

Proof of Theorem1. Letα ∈0,1,T ∈Hα

1H, andK ∈ CαHwithμ2ασT K 0, and

assumemTK ∞, otherwise Theorem C impliesT∈Nα

1H.

Let{en}_n∈be an orthonormal basis ofHand let

Hn∨rTKej| j1, . . . , n, r∈RatσTK. 5

Assume that with respect to the decompositionHHn⊕ H⊥_n, operatorsTandKare written

as

T

T1n T2n

T3n T4n

, K

K1n K2n

K3n K4n

. 6

SinceHnis a rationally invariant subspace forTK, we haveT3nK3n 0, and thusT3n

−K3n∈ CαHn⊆ C2αHn, andσT1nK1n⊆σTK, which impliesμ2ασT1nK1n 0.

LetPnbe the orthogonal projection ontoHn, and thusPn ↑ Istrongly. We will prove

next thatT1n∈Hα

1Hnby first establishing that

PnC_TαPn−Cα_T₁_n −QnKn, 7a

whereQn∈LHnis positive semidefinite andKn ∈ C1Hn.

Assuming that equality7awas already proved and writingCα_T QKwithQ≥0

andK∈ C1H, then we have

Cα_T₁_n PnQPnPnKPnQn −Kn, 7b

that is,Cα

T1nis the sum ofPnQPnQ

n, which is a positive semidefinite operator, and ofPnKPn−

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Indeed, the expressionPnCα_TPn−C_Tα₁_ncan be written asD1−D2, where

D1PnT∗TαPn−T1n∗ T1n

α ,

D2PnTT∗αPn−T1nT_1n∗ α.

8

We can writeD1−QnKn, where

Qn

PnT∗_TPnα₋_PnT∗_Tα

Pn, 9

which according to Hansen’s inequality is a positive semidefinite operator, and

Kn

PnT∗_TPnα₋_PnT∗_PnTPnα

, 10

which according to Jocic’s inequality is a trace-class operator that satisfies

_K

n1≤|PnT∗TPn−PnT∗PnTPn|

α

1T3n∗ T3nα1

T3n∗ T3n

α

α≤T3n∗

α_{· T3n}α

α≤ Tα· T3nαα.

11

Concerning operatorD2, we can writeD2Qn Kn, where

Qn PnTT∗αPn−PnTPnT∗αPn, 12

which according to Lowner’s inequality is a positive semidefinite operator, and

KnPnTPnT∗αPn−PnTPnT∗PnαPn

TPnT∗α₋_PnTPnT∗_Pnα

Pn, 13

which is also a trace-class operator since

TPnT∗−PnTPnT∗Pn TPnT∗−TPnT∗Pn TPnT∗Pn−PnTPnT∗Pn

TPnT∗I−Pn I−PnTPnT∗Pn

TT3n∗ T3nT∗Pn∈ CαH,

14

and according to Jocic’s inequality

_K

n1≤TPnT∗

α₋_PnTPn

T∗Pnα₁ ≤TT3n∗ T3nT∗Pn

α

1

TT3n∗ T3nT∗Pn

α

α≤CTT3n∗

α

αT3nT∗Pn

α α

≤CTαT_3n∗α_αT3nαα 2CTαT3nαα.

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Therefore,

D2Qn Kn, withQn ≥0, Kn∈C1H, 16

and consequently, D1−D2 −QnKn−Qn Kn −Qn Qn Kn−Kn, where

QnQn : Qnis positive semidefinite andKn−Kn : Kn is trace-class, which establishes

equality7a.

According to7b,T1n ∈ H₁αHn, and sincemT1n K1n ≤ nand σT1nK1n ⊆

σT K, Theorem C implies that trCα

T1n≤0, and furthermore, by replacingT1nwithT

∗

1n,

trCα

T1n 0. Furthermore, equality7aimplies

PnCα_TPn≤Cα_T₁_nKn, 17

which further implies

trPnCαTPn ≤tr

Kn . 18

Similar utilization of Lowner’s and Hansen’s inequalities implies that Kn and −Kn are

positive semidefinite, and thus so isKn Kn−Kn. Therefore

trKn ≤Kn 1Kn 1≤12CT

α_T3nα

α. 19

SinceT3n−K3n∈ CpHnandK3n → 0 weakly and both|T3n|and |T_3n∗ | ≤ ||T||I, we have

||T3n||α → 0, and thus trCα

T≤0. ReplacingTwithT∗we conclude that trCαT 0.

References

1 D. Voiculescu, “A note on quasitriangularity and trace-class self-commutators,” Acta Scientiarum

Mathematicarum, vol. 42, no. 1-2, pp. 1303–1320, 1980.

2 D. Hadwin and E. Nordgren, “Extensions of the Berger-Shaw theorem,” Proceedings of the American

Mathematical Society, vol. 102, no. 3, pp. 517–525, 1988.

3 R. E. Curto, P. S. Muhly, and D. Xia, “A trace estimate forp-hyponormal operators,” Integral Equations

and Operator Theory, vol. 6, no. 4, pp. 507–514, 1983.

4 A. Aluthge and D. Xia, “A trace estimate ofT∗Tp₋_TT∗p_{,” Integral Equations and Operator Theory,}

vol. 12, no. 2, pp. 300–303, 1989.

5 J. G. Stampfli, “Compact perturbations, normal eigenvalues and a problem of Salinas,” Journal of the

London Mathematical Society, vol. 9, no. 2, pp. 165–175, 1974-1975.

6 F. Hansen, “An operator inequality,” Mathematische Annalen, vol. 246, no. 3, pp. 249–250, 1980. 7 E. Heinz, “Beitr¨age zur st ¨orungstheorie der spektralzerlegung,” Mathematische Annalen, vol. 123, pp.

415–438, 1951.

8 D. R. Joci´c, “Integral representation formula for generalized normal derivations,” Proceedings of the

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Research Article

Almost

α

-Hyponormal Operators with

Weyl Spectrum of Area Zero

Vasile Lauric

References

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