Journal of Inequalities and Applications Volume 2010, Article ID 584642,10pages doi:10.1155/2010/584642
Research Article
A Note on
C
p
, α
-Hyponormal Operators
Xiaohuan Wang
1and Zongsheng Gao
21LMIB and School of Mathematics and Systems Science, Beihang University, Beijing 100191, China 2LMIB and Department of Mathematics, Beihang University, Beijing 100191, China
Correspondence should be addressed to Xiaohuan Wang,xiaohuan@smss.buaa.edu.cn
Received 20 January 2010; Accepted 22 April 2010
Academic Editor: Sin Takahasi
Copyrightq2010 X. Wang 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.
We studyCp, α-normal operators and Cp, α-hyponormal operators. We show the inclusion
relation between these classes under various hypotheses forpandα. We also obtain some sufficient conditions for Aluthge transformTs,t|T|sU|T|tandT2ofCp, α-hyponormal operators still to be Cp, α-hyponormal.
1. Introduction
LetHbe a separable, infinite dimensional, complex Hilbert space, and denote byLHthe algebra of all bounded linear operators on H. Recently, Lauric in 1 introduced Cp, α
-hyponormal operators. For α > 0 and T ∈ LH, denote byDα
T T∗T
α −TT∗α. We
denote thatCpH, 1≤p <∞, the ideal of operators in the Schattenp-class2. Although, for
0 < p < 1, the usual definition of · p does not satisfy the triangle inequality, nevertheless Cp, · pis closed andTKp ≤ T · Kp, whenT ∈ LHandK ∈ CpH. An operator T inLHisCp, α-normal ifDαT ∈ CpH, and denote the class ofCp, α-normal operators
byNα
pH. An operatorTinLHwill be calledCp, α-hyponormal ifDTαPK, whereP
is a positive semidefinite operatorP ≥0andK ∈ CpH. The class ofCp, α-hyponormal
operators will be denoted byHα
pH. In particular, an operatorT inH11Hwill be called almost hyponormal. Furthermore, an operatorT ∈ LHwhoseDTαis positive semidefinite is calledα-hyponormalnotation:T ∈ Hα
0H.
In this paper, we first study the inclusion relation between these classes under various hypotheses forpandαinSection 2. Then we study the Aluthge transformTs,t|T|sU|T|tand T2ofC
p, α-hyponormal operators inSection 3.
Theorem FSee Furuta inequality in3. IfA≥B≥0, then, for eachr ≥0,
Br/2ApBr/21/q≥Br/2BpBr/21/q,
Ar/2ApAr/21/q≥Ar/2BpAr/21/q,
1.1
as long as real numbersp, r, qsatisfy
p≥0, q≥1with 1rq≥pr. 1.2
Lemma 1.1see 1. LetA ∈ LH, A ≥ 0, α ∈ 0,1, p ≥ α, andK ∈ CpH, such that AK≥0. ThenAKαAαK1, whereK1∈ Cp/αH. If in additionK≥0, thenK1≥0.
Lemma 1.2see1. LetA∈ LH,A≥0,p≥1, andK∈ CpH, such thatAK≥0, and let α∈1,∞. ThenAKαAαK
1, whereK1∈ CpH.
2. Some Inclusions
According to L ¨owner-HeinzL-Hinequality4,5thatA ≥B≥0 ensures thatAα ≥ Bαfor
eachα ∈ 0,1, we obtainHα0H ⊇ Hβ0Hwhenα ≤ β. However, the similar inclusions for the classesNαpHandHαpHare less obvious. In this section, we will examine various inclusions between these classes of operators.1ofTheorem 2.1has been already shown in
1. But we will give a proof for the readers’ convenience.
Theorem 2.1. Letα >0,p≥1, and letTbe inNα pH.
1Ifβ≥α, thenT belongs toNpβH, and thereforeNαpH⊆ N β pH. 2If0< β≤α, thenTbelongs toNβαp/βH, and thereforeNα
pH⊆ N β αp/βH.
Proof. Letα,p, andT be as in the hypotheses and letT U|T|be the polar decomposition ofT.
ForT ∈ Nα
pH, we have
DαT T∗Tα−TT∗α|T|2α− |T∗|2αK, 2.1
withK∈ CpH. Then we obtain
|T|2α|T∗|2αK≥0. 2.2
1First we consider the caseβ≥α. According toLemma 1.2, we obtain
|T|2β|T∗|2αKβ/α|T∗|2βK1, 2.3
2Next we consider the case 0< β≤α. According toLemma 1.1, we obtain
|T|2β|T∗|2αKβ/α|T∗|2βK1, 2.4
withK1 ∈ Cαp/βH. ThenT ∈ Nβαp/βH.
The following corollary is a consequence ofTheorem 2.1.
Corollary 2.2. Letα >0,p≥1, then, for0< β≤α,
Nβ
pH⊆ NαpH⊆ N β
αp/βH⊆ N α
αp/βH. 2.5
Theorem 2.3. Letα >0,p≥1, and letT be inHα
pH. If0< β≤α, thenT belongs toH β αp/βH, and thereforeHα
pH⊆ H β αp/βH.
Proof. Letα,p, andTbe as in the hypotheses and letT U|T|be the polar decomposition of T.
ForT ∈ Hα
pH, we have
DαT T∗Tα−TT∗α|T|2α− |T∗|2αPK, 2.6
withP ≥0,K∈ CpH. Then we obtain
|T|2α|T∗|2αPK≥0. 2.7
For 0< β≤α, according toLemma 1.1and L-H inequality, we obtain
|T|2β|T∗|2αPKβ/α
|T∗|2αPβ/αK1
≥ |T∗|2βK1,
2.8
withK1 ∈ Cαp/βH. Then we obtainT ∈ Hβαp/βH.
3. Some Properties of
C
p, α
-Hyponormal Operators
LetT U|T|be the polar decomposition of an operator T on a Hilbert space H, whereU is a partial isometry operator. Recently, Lauric 1 shows some theorems on the Aluthge transform T |T|1/2U|T|1/2 ofCp, α-hyponormal operators. In this section, we will show
T2 ofC
p, α-hyponormal operators to beCp, α-hyponormal. Aluthge transform Ts,t arose
in the study ofp-hyponormal operators6,7and has since been studied in many different contexts8–15.
LetT belong to Hα
pH, for someα > 0, p > 0, such thatDTα P K with P ≥ 0, K ∈ CpH. SinceK K∗ K−K− andK,K− ≥0 areCp-class operators, in what follows
we will assume thatDα
TP1−K1withP1≥0 andK1≥0, K1 ∈ CpH.
Theorem 3.1. Letp≥ 1,α≥max{s, t}, andT ∈ Hα
pHsuch thatDαT P−KwithP,K ≥0, K∈ CpH, and letε∈0,1/2such thatαε≤1. ThenTs,t∈ H2ααp/εαεsH.
Proof. We may assume thatT U|T|withUbeing unitary. The equalityDα
T P−KwithP, K ≥0 implies that|T|2αK≥U|T|2αU∗. Multiplying this inequality byU∗to the left and by Uto the right, we obtain
AU∗|T|2αUU∗KU≥ |T|2αB. 3.1
According toLemma 1.1,
As/αU∗|T|2αKUs/α U∗|T|2αKs/αUU∗|T|2sK1
U, 3.2
withK1 ∈ Cαp/sH. SettingK2|T|tU∗K1U|T|t, by Theorem F we have
Ts,t∗ Ts,tK2
αε
|T|t U∗|T|2sK1
U|T|tαε
|T|t U∗|T|2αKUs/α|T|t αε
Bt/2αAs/αBt/2ααε
≥Bstαε/α
|T|2stαε.
3.3
On the other hand, according toLemma 1.1,
Ts,t∗Ts,tK2
αε
Ts,t∗ Ts,t αε
K3, 3.4
withK3 ∈ Cαp/αεsH. Then we have
Ts,t∗Ts,t
αε
K3≥ |T|2stαε. 3.5
According to the following inequality
0,−r
1,0 1,1
q q1 pq
[image:5.600.201.398.97.283.2]1rqpr p
Figure 1:Domain of Furuta inequality.
by Theorem F, we have
Cs/2αDt/αCs/2ααε≤Cstαε/α. 3.7
Again, according toLemma 1.1,
Cs/2α|T|2αKs/2α|T|sK4, 3.8
withK4 ∈ C2αp/sH.
Next, obviously,
Dt/αU|T|2αU∗t/αU|T|2tU∗. 3.9
Then we have
Cs/2αDt/αCs/2ααε|T|sK4
U|T|2tU∗|T|sK4
αε
|T|sU|T|2tU∗|T|sK5
αε
Ts,tTs,t∗ K5
αε
Ts,tTs,t∗ αε
K6,
3.10
1First we consider the case 0≤st/α≤1. According toLemma 1.1, we have
Cst/ααε|T|2αKst/α αε
|T|2stK7
αε
|T|2stαεK8,
3.11
withK7∈ Cαp/stHandK8∈ Cαp/αεstH.
Then by3.7and3.10, setK9 K6−K8∈ C2αp/αεsH, and
|T|2stαε≥Ts,tTs,t∗ αε
K9. 3.12
Combining3.5and3.12, we obtain
Ts,t∗ Ts,t
αε
−Ts,tTs,t∗ αε
≥K10, 3.13
whereK10 K9−K3 ∈ C2αp/αεsH.
2Next we consider the casest/α>1. According to Lemmas1.1and1.2,
Cst/ααε|T|2αKst/α αε
|T|2stK7αε
|T|2stαεK8,
3.14
withK7∈ CpHandK8 ∈ Cp/αεH. and
Then by3.7and3.10, setK9 K6−K8 ∈ C2αp/αεsH,
|T|2stαε≥Ts,tTs,t∗ αε
K9. 3.15
Combining3.5and3.15, we obtain
Ts,t∗ Ts,t
αε
−Ts,tTs,t∗ αε
≥K10, 3.16
whereK10 K9 −K3 ∈ C2αp/αεsH.
Remark 3.2. The main theorem of1was considered in the caseα ∈ 1/2,1. Apparently,
Theorem 3.1impliesTheorems 13 in1 whenst1/2. And we also obtain the following theorem.
Theorem 3.3. Letp≥1,0< α≤min{s, t}, andT ∈ Hα
pHsuch thatDαTP−KwithP,K≥0, K∈ CpH, and letε≥0such thatαε≤2α/st.
1Ifs≥2α, thenTs,t∈ Hp/αεαεH.
2If0< s <2α, thenTs,t∈ H2ααp/εαεsH.
Proof. The proof ofTheorem 3.3is similar to the proof ofTheorem 3.1.
Corollary 3.4. Letp ≥1,T ∈ HαpHsuch thatDαT P−KwithP,K ≥0,K ∈ CpH, and let ε∈0,1/2.
1Ifα∈0,1/4, thenT∈ Hp/ααεεH.
2Ifα∈1/4,1/2, thenT∈ H4ααp/εαεH.
Proof. Putst1/2 inTheorem 3.3.
1When α ∈ 0,1/4, we have s ≥ 2α. According to 1 of Theorem 3.3, then T ∈
Hαε
p/αεH.
2When α∈ 1/4,1/2, we have 0 < s <2α. According to2ofTheorem 3.3, then
T ∈ H4ααp/εαεH.
Next, we will studyT2ofC
p, α-hyponormal operators. And first we will prove the
following lemma.
Lemma 3.5. Letp≥1,α∈0,1, andT ∈ HαpHsuch thatDαTPKwithP≥0,K∈ CpH, andDαT P1−K1 withP1 ≥ 0, K1 ≥ 0, K1 ∈ CpH. Then if|T|2α−P ≥ 0, one has the following inequalities
1There existsK∈ C2p/αHsuch that|T||T∗|2|T| α/2
K≤ |T|2α.
2There existsK ∈ C2p/αHsuch that|T∗||T|2|T∗| α/2
K≥ |T∗|2α.
Proof. Letα,p, andTbe as in the hypotheses and letT U|T|be the polar decomposition of T. Then we have
DαT T∗Tα−TT∗α|T|2α− |T∗|2αPK, 3.17
withP ≥0, K∈ CpH.
DαT|T|2α− |T∗|2αP1−K1, 3.18
By3.17, we have
A1 |T|2α≥ |T∗|2αKB1≥0. 3.19
And according toLemma 1.2,
B11/α |T∗|2αK1/α |T∗|2K2, 3.20
withK2 ∈ CpH. SettingK3|T|K2|T|, by Theorem F we have
|T||T∗|2|T|K3
α/2
|T||T∗|2K2
|T|α/2
A11/2αB11/αA11/2αα/2
≤A1
|T|2α.
3.21
By3.18, we have
A2|T|2αK1≥ |T∗|2αB2. 3.22
And according toLemma 1.2,
A12/α |T|2αK1
1/α
|T|2K4, 3.23
withK4 ∈ CpH. SettingK5|T∗|K4|T∗|, by Theorem F we have
|T∗||T|2|T∗|K5
α/2
|T∗||T|2K4
|T∗|α/2
B12/2αA21/αB12/2αα/2
≥B2
|T∗|2α.
3.24
On the other hand, byLemma 1.1,
|T||T∗|2|T|K3
α/2
|T||T∗|2|T|α/2K,
|T∗||T|2|T∗|K5
α/2
|T∗||T|2|T∗|α/2K,
3.25
Then by3.21and3.24, we obtain
|T||T∗|2|T|α/2K≤ |T|2α, |T∗||T|2|T∗|α/2K ≥ |T∗|2α, 3.26
withK, K∈ C2p/αH.
Theorem 3.6. Letp≥1, α∈0,1,andT ∈ Hα
pHsuch thatDαT PKwithP ≥0, K∈ CpH, andDα
T P1 −K1 with P1 ≥ 0, K1 ≥ 0,andK1 ∈ CpH. Then if|T|2α −P ≥ 0, one hasT2 ∈
Hα/2 2p/αH.
Proof. Letα, p, andT be as in the hypotheses. We may assume thatT U|T|withUbeing unitary. Then obviously,
T2T2∗α/2U|T||T∗|2|T|α/2U∗, 3.27
T2∗T2α/2|T|U∗|T|2U|T|α/2U∗|T∗||T|2|T∗|α/2U. 3.28
ByLemma 3.5, there exitsK, K∈ C2p/αHsuch that
|T||T∗|2|T|α/2K≤ |T|2α, 3.29
|T∗||T|2|T∗|α/2K≥ |T∗|2α. 3.30
Multiplying3.29byUto the left and byU∗to the right, we obtain
U|T||T∗|2|T|α/2U∗UKU∗≤U|T|2αU∗|T∗|2α. 3.31
Multiplying3.30byU∗to the left and byUto the right, we obtain
U∗|T∗||T|2|T∗|α/2UU∗KU≥U∗|T∗|2αU|T|2α. 3.32
By3.27and3.31, we have
T2T2∗α/2UKU∗≤ |T∗|2α. 3.33
By3.28and3.32, we have
SettingK2UKU∗−U∗KU, K2∈ C2p/αH, we have
T2∗T2α/2−T2T2∗α/2 ≥ |T|2α− |T∗|2αK2. 3.35
Therefore, forK3KK2, K3∈ C2p/αH, we have
T2∗T2α/2−T2T2∗α/2≥PK
3. 3.36
Then the proof ofTheorem 3.6is finished.
Acknowledgment
The authors would like to express their cordial gratitude to the referee for his kind comments.
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