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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^∗T^α−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 : _∞

n1μnK^p^1/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 ∈ N_p^αHifC^α_T : T^∗T^α−TT^∗^αbelongs to CpH, andT is calledCp, α-hyponormalnotation: T ∈ H_p^αHif C^α_T is the sum of a positive definite operator and an operator inCpH, or equivalently,C^α_T₋the negative part of C^α_Tbelongs to CpH, whereαis a positive number. This note will be concerned with the particular classH₁^αH, 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 ofC_T¹. The result was extended in2. The combination of these results will be stated after recalling some terminology and notation. The rational cyclic multiplicity of an operator

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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 ∈RatσT

H, 1 where RatσTis the algebra of complex-valued rational functions with poles oﬀσT.

For a Borel subsetE⊆ andα >0, denoteμαE α/2

Eρ^α−1 dρdθ. In particular, μ2is the planar Lebesgue measure.

Theorem Asee1,2. SupposeT ∈H₁¹H. If there existsK ∈ C2Hsuch that eithermT K<∞orμ2σTK 0, thenT ∈N₁¹H. Moreover, whenmTK<∞,

tr C¹_T

≤ mTK

π · μ2σTK, 2

and whenμ2σTK 0, trC¹_T≤0, and consequently, trC¹_T 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 ∈H₁¹Hsuch thatμ2σwT 0. ThenT ∈N₁¹Hand trC_T¹ 0.

On the other hand, Berger-Shaw inequality was extended to operators inH₁^αHusing 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 ∈H₁^αHandK∈ C2αHwithmTK<∞.

ThenT ∈N₁^αHand

tr

C_T^α ≤ 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 ∈ H₁^αHandK ∈ CαHwithμ2ασT K 0. Then T ∈N^α₁Hand trC_T^α 0.

Remark. It would have been desirable that Theorem1 be proved with the hypothesis that K∈ C2αH.

Before we prove Theorem1, we extract a similar consequence to Corollary B.

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Corollary 2. Letα∈0,1and letT ∈H₁^αHsuch thatμ2σwT 0. ThenT ∈N₁^αHand trC^α_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^∗A^αB≤B^∗AB^α.

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

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, thenA^α−B^α∈ CpHand||B^α−A^α||p≤ || |B−A|^α ||p.

Proof of Theorem1. Letα ∈0,1,T ∈H₁^αH, andK ∈ CαHwithμ2ασT K 0, and assumemTK ∞, otherwise Theorem C impliesT∈N₁^αH.

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∈H₁^αHnby first establishing that

PnC_T^αPn−C^α_T_1n −Q_nK_n, 7a

whereQ_n∈LHnis positive semidefinite andK_n ∈ C1Hn.

Assuming that equality7awas already proved and writingC^α_T QKwithQ≥0 andK∈ C1H, then we have

C^α_T_1n PnQPnPnKPnQ_n −K_n, 7b

that is,C^α_T_1nis the sum ofPnQPnQ_n, which is a positive semidefinite operator, and ofPnKPn− K_n, which is a trace-class operator.

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Indeed, the expressionPnC^α_TPn−C_T^α_1ncan be written asD1−D2, where D1PnT^∗T^αPn−

T_1n^∗ T1n α

, D2PnTT^∗^αPn−

T1nT_1n^∗ ^α.

8

We can writeD1−Q_nK_n, where Q_n

PnT^∗TPn^α−PnT^∗T^αPn

, 9

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

PnT^∗TPn^α−PnT^∗PnTPn^α

, 10

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

1≤|PnT^∗TPn−PnT^∗PnTPn|^α

1T_3n^∗ T3n^α

1

T_3n^∗ T3n^α

α≤T_3n^∗^α· T3n^α_α≤ T^α· T3n^α_α.

11

Concerning operatorD2, we can writeD2Q_n K_n, where

Q_n PnTT^∗^αPn−PnTPnT^∗^αPn, 12

which according to Lowner’s inequality is a positive semidefinite operator, and K_nPnTPnT^∗^αPn−PnTPnT^∗Pn^αPn

TPnT^∗^α−PnTPnT^∗Pn^α

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

TT_3n^∗ T3nT^∗Pn∈ CαH,

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and according to Jocic’s inequality K_n

1≤TPnT^∗^α−PnTPnT^∗Pn^α

1 ≤TT_3n^∗ T3nT^∗Pn^α

1

TT_3n^∗ T3nT^∗Pn^α

α≤CTT_3n^∗ ^α

αT3nT^∗Pn^α_α

≤CT^αT_3n^∗^α

αT3n^α_α 2CT^αT3n^α_α.

15

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Therefore,

D2Q_n K_n, withQ_n ≥0, K_n∈C1H, 16 and consequently, D1−D2 −Q_nK_n−Q_n K_n −Q_n Q_n K_n−K_n, where Q_nQ_n : Q_nis positive semidefinite andK_n−K_n : K_n is trace-class, which establishes equality7a.

According to7b,T1n ∈ H₁^αHn, and sincemT1n K1n ≤ nand σT1nK1n ⊆ σT K, Theorem C implies that trC^α_T

1n≤0, and furthermore, by replacingT1nwithT_1n^∗, trC^α_T_1n 0. Furthermore, equality7aimplies

PnC^α_TPn≤C^α_T_1nK_n, 17

which further implies

tr

PnC^α_TPn ≤tr

Kn . 18

Similar utilization of Lowner’s and Hansen’s inequalities implies that K_n and −K_n are positive semidefinite, and thus so isK_n K_n−K_n. Therefore

tr

K_n ≤K_n ₁K_n ₁≤12CT^αT3n^α_α. 19 SinceT3n−K3n∈ CpHnandK3n → 0 weakly and both|T3n|and |T_3n^∗ | ≤ ||T||I, we have

||T3n||α → 0, and thus trC_T^α≤0. ReplacingTwithT^∗we conclude that trC^α_T 0.

References

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