Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2010, Article ID 584642, 10 pages
doi:10.1155/2010/584642
Research Article
A Note on Cp , α-Hyponormal Operators
Xiaohuan Wang1 and Zongsheng Gao2
1
2
LMIB and School of Mathematics and Systems Science, Beihang University, Beijing 100191, China
LMIB 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
Copyright q 2010 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 study Cp , α-normal operators and Cp , α-hyponormal operators. We show the inclusion
relation between these classes under various hypotheses for p and α. We also obtain some sufficient
conditions for Aluthge transform Ts,t |T |s U|T |t and T 2 of Cp , α-hyponormal operators still to be
Cp , α-hyponormal.
1. Introduction
Let H be a separable, infinite dimensional, complex Hilbert space, and denote by LH the
algebra of all bounded linear operators on H. Recently, Lauric in 1 introduced Cp , αhyponormal operators. For α > 0 and T ∈ LH, denote by DTα T ∗ T α − T T ∗ α . We
denote that Cp H, 1 ≤ p < ∞, the ideal of operators in the Schatten p-class 2. Although, for
0 < p < 1, the usual definition of · p does not satisfy the triangle inequality, nevertheless
Cp , · p is closed and T Kp ≤ T · Kp , when T ∈ LH and K ∈ Cp H. An operator
T in LH is Cp , α-normal if DTα ∈ Cp H, and denote the class of Cp , α-normal operators
by Nαp H. An operator T in LH will be called Cp , α-hyponormal if DTα P K, where P
is a positive semidefinite operator P ≥ 0 and K ∈ Cp H. The class of Cp , α-hyponormal
operators will be denoted by Hαp H. In particular, an operator T in H11 H will be called
almost hyponormal. Furthermore, an operator T ∈ LH whose DTα is positive semidefinite
is called α-hyponormal notation: T ∈ Hα0 H.
In this paper, we first study the inclusion relation between these classes under various
hypotheses for p and α in Section 2. Then we study the Aluthge transform Ts,t |T |s U|T |t and
T 2 of Cp , α-hyponormal operators in Section 3.
Before proceeding, we will make use of the following inequality.
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Journal of Inequalities and Applications
Theorem F See Furuta inequality in 3. If A ≥ B ≥ 0, then, for each r ≥ 0,
1/q
1/q
≥ Br/2 Bp Br/2
,
1/q
1/q
≥ Ar/2 Bp Ar/2
,
Br/2 Ap Br/2
Ar/2 Ap Ar/2
1.1
as long as real numbers p, r, q satisfy
p ≥ 0, q ≥ 1 with 1 rq ≥ p r.
1.2
Lemma 1.1 see 1. Let A ∈ LH, A ≥ 0, α ∈ 0, 1, p ≥ α, and K ∈ Cp H, such that
A K ≥ 0. Then A Kα Aα K1 , where K1 ∈ Cp/α H. If in addition K ≥ 0, then K1 ≥ 0.
Lemma 1.2 see 1. Let A ∈ LH, A ≥ 0, p ≥ 1, and K ∈ Cp H, such that A K ≥ 0, and let
α ∈ 1, ∞. Then A Kα Aα K1 , where K1 ∈ Cp H.
2. Some Inclusions
According to Löwner-Heinz L-H inequality 4, 5 that A ≥ B ≥ 0 ensures that Aα ≥ Bα for
β
each α ∈ 0, 1, we obtain Hα0 H ⊇ H0 H when α ≤ β. However, the similar inclusions
α
α
for the classes Np H and Hp H are less obvious. In this section, we will examine various
inclusions between these classes of operators. 1 of Theorem 2.1 has been already shown in
1. But we will give a proof for the readers’ convenience.
Theorem 2.1. Let α > 0, p ≥ 1, and let T be in Nαp H.
β
β
1 If β ≥ α, then T belongs to Np H, and therefore Nαp H ⊆ Np H.
β
β
2 If 0 < β ≤ α, then T belongs to Nαp/β H, and therefore Nαp H ⊆ Nαp/β H.
Proof. Let α, p, and T be as in the hypotheses and let T U|T | be the polar decomposition
of T .
For T ∈ Nαp H, we have
DTα T ∗ T α − T T ∗ α |T |2α − |T ∗ |2α K,
2.1
with K ∈ Cp H. Then we obtain
|T |2α |T ∗ |2α K ≥ 0.
2.2
1 First we consider the case β ≥ α. According to Lemma 1.2, we obtain
β/α
|T ∗ |2β K1 ,
|T |2β |T ∗ |2α K
β
with K1 ∈ Cp H. Then T ∈ Np H.
2.3
Journal of Inequalities and Applications
3
2 Next we consider the case 0 < β ≤ α. According to Lemma 1.1, we obtain
β/α
|T ∗ |2β K1 ,
|T |2β |T ∗ |2α K
2.4
β
with K1 ∈ Cαp/β H. Then T ∈ Nαp/β H.
The following corollary is a consequence of Theorem 2.1.
Corollary 2.2. Let α > 0, p ≥ 1, then, for 0 < β ≤ α,
β
β
Np H ⊆ Nαp H ⊆ Nαp/β H ⊆ Nααp/β H.
2.5
β
Theorem 2.3. Let α > 0, p ≥ 1, and let T be in Hαp H. If 0 < β ≤ α, then T belongs to Hαp/β H,
β
and therefore Hαp H ⊆ Hαp/β H.
Proof. Let α, p, and T be as in the hypotheses and let T U|T | be the polar decomposition of
T.
For T ∈ Hαp H, we have
DTα T ∗ T α − T T ∗ α |T |2α − |T ∗ |2α P K,
2.6
with P ≥ 0, K ∈ Cp H. Then we obtain
|T |2α |T ∗ |2α P K ≥ 0.
2.7
For 0 < β ≤ α, according to Lemma 1.1 and L-H inequality, we obtain
β/α
|T |2β |T ∗ |2α P K
β/α
|T ∗ |2α P
K1
2.8
≥ |T ∗ |2β K1 ,
β
with K1 ∈ Cαp/β H. Then we obtain T ∈ Hαp/β H.
3. Some Properties of Cp , α-Hyponormal Operators
Let T U|T | be the polar decomposition of an operator T on a Hilbert space H, where U
is a partial isometry operator. Recently, Lauric 1 shows some theorems on the Aluthge
transform T |T |1/2 U|T |1/2 of Cp , α-hyponormal operators. In this section, we will show
some sufficient conditions for the generalized Aluthge transform Ts,t |T |s U|T |t s, t > 0 and
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Journal of Inequalities and Applications
T 2 of Cp , α-hyponormal operators to be Cp , α-hyponormal. Aluthge transform Ts,t arose
in the study of p-hyponormal operators 6, 7 and has since been studied in many different
contexts 8–15.
Let T belong to Hαp H, for some α > 0, p > 0, such that DTα P K with P ≥ 0,
K ∈ Cp H. Since K K ∗ K − K− and K , K− ≥ 0 are Cp -class operators, in what follows
we will assume that DTα P1 − K1 with P1 ≥ 0 and K1 ≥ 0, K1 ∈ Cp H.
Theorem 3.1. Let p ≥ 1, α ≥ max{s, t}, and T ∈ Hαp H such that DTα P − K with P , K ≥ 0,
αε
K ∈ Cp H, and let ε ∈ 0, 1/2 such that α ε ≤ 1. Then Ts,t ∈ H2αp/αεs H.
Proof. We may assume that T U|T | with U being unitary. The equality DTα P − K with P ,
K ≥ 0 implies that |T |2α K ≥ U|T |2α U∗ . Multiplying this inequality by U∗ to the left and by
U to the right, we obtain
A U∗ |T |2α U U∗ KU ≥ |T |2α B.
3.1
s/α
s/α
U∗ |T |2α K
U U∗ |T |2s K1 U,
As/α U∗ |T |2α K U
3.2
According to Lemma 1.1,
with K1 ∈ Cαp/s H. Setting K2 |T |t U∗ K1 U|T |t , by Theorem F we have
∗ Ts,t K2
Ts,t
αε
αε
|T |t U∗ |T |2s K1 U |T |t
s/α αε
t
2α
∗
|T | U |T | K U
|T |t
αε
Bt/2α As/α Bt/2α
3.3
≥ Bstαε/α
|T |2stαε .
On the other hand, according to Lemma 1.1,
∗ Ts,t K2
Ts,t
αε
αε
∗ Ts,t
K3 ,
Ts,t
3.4
with K3 ∈ Cαp/αεs H. Then we have
∗ Ts,t
Ts,t
αε
K3 ≥ |T |2stαε .
3.5
C |T |2α K ≥ U|T |2α U∗ D,
3.6
According to the following inequality
Journal of Inequalities and Applications
5
p
1 rq p r
q1
pq
1, 1
1, 0
q
0, −r
Figure 1: Domain of Furuta inequality.
by Theorem F, we have
αε
≤ Cstαε/α .
3.7
s/2α
|T |s K4 ,
Cs/2α |T |2α K
3.8
t/α
U|T |2t U∗ .
Dt/α U|T |2α U∗
3.9
Cs/2α Dt/α Cs/2α
Again, according to Lemma 1.1,
with K4 ∈ C2αp/s H.
Next, obviously,
Then we have
Cs/2α Dt/α Cs/2α
αε
|T |s K4
U|T |2t U∗
|T |s K4
αε
|T |s U|T |2t U∗ |T |s K5
∗
Ts,t Ts,t
K5
αε
αε
∗
Ts,t Ts,t
K6 ,
with K5 ∈ C2αp/s H, K6 ∈ C2αp/αεs H.
αε
3.10
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Journal of Inequalities and Applications
1 First we consider the case 0 ≤ s t/α ≤ 1. According to Lemma 1.1, we have
C
st/α
αε
2α
|T |
K
st/α αε
αε
|T |2st K7
3.11
|T |2stαε K8 ,
with K7 ∈ Cαp/st H and K8 ∈ Cαp/αεst H.
Then by 3.7 and 3.10, set K9 K6 − K8 ∈ C2αp/αεs H, and
αε
∗
K9 .
|T |2stαε ≥ Ts,t Ts,t
3.12
Combining 3.5 and 3.12, we obtain
∗ Ts,t
Ts,t
αε
αε
∗
− Ts,t Ts,t
≥ K10 ,
3.13
where K10 K9 − K3 ∈ C2αp/αεs H.
2 Next we consider the case s t/α > 1. According to Lemmas 1.1 and 1.2,
Cst/α
αε
|T |2α K
st/α αε
αε
|T |2st K7
3.14
|T |2stαε K8
,
with K7
∈ Cp H and K8
∈ Cp/αε H. and
Then by 3.7 and 3.10, set K9
K6 − K8
∈ C2αp/αεs H,
αε
∗
K9
.
|T |2stαε ≥ Ts,t Ts,t
3.15
Combining 3.5 and 3.15, we obtain
∗ Ts,t
Ts,t
αε
αε
∗
− Ts,t Ts,t
≥ K10
,
K9
− K3 ∈ C2αp/αεs H.
where K10
αε
By 3.13 and 3.16, we obtain Ts,t ∈ H2αp/αεs H.
3.16
Journal of Inequalities and Applications
7
Remark 3.2. The main theorem of 1 was considered in the case α ∈ 1/2, 1. Apparently,
Theorem 3.1 implies Theorems 13 in 1 when s t 1/2. And we also obtain the following
theorem.
Theorem 3.3. Let p ≥ 1, 0 < α ≤ min{s, t}, and T ∈ Hαp H such that DTα P − K with P , K ≥ 0,
K ∈ Cp H, and let ε ≥ 0 such that α ε ≤ 2α/s t.
αε
1 If s ≥ 2α, then Ts,t ∈ Hp/αε H.
αε
2 If 0 < s < 2α, then Ts,t ∈ H2αp/αεs H.
Proof. The proof of Theorem 3.3 is similar to the proof of Theorem 3.1.
Corollary 3.4. Let p ≥ 1, T ∈ Hαp H such that DTα P − K with P , K ≥ 0, K ∈ Cp H, and let
ε ∈ 0, 1/2.
αε
1 If α ∈ 0, 1/4, then T ∈ Hp/αε H.
αε
2 If α ∈ 1/4, 1/2, then T ∈ H4αp/αε H.
Proof. Put s t 1/2 in Theorem 3.3.
1 When α ∈ 0, 1/4, we have s ≥ 2α. According to 1 of Theorem 3.3, then T ∈
αε
Hp/αε H.
2 When α ∈ 1/4, 1/2, we have 0 < s < 2α. According to 2 of Theorem 3.3, then
αε
T ∈ H4αp/αε H.
Next, we will study T 2 of Cp , α-hyponormal operators. And first we will prove the
following lemma.
Lemma 3.5. Let p ≥ 1, α ∈ 0, 1, and T ∈ Hαp H such that DTα P K with P ≥ 0, K ∈ Cp H,
and DTα P1 − K1 with P1 ≥ 0, K1 ≥ 0, K1 ∈ Cp H. Then if |T |2α − P ≥ 0, one has the following
inequalities
α/2
1 There exists K ∈ C2p/α H such that |T ||T ∗ |2 |T |
K ≤ |T |2α .
α/2
2 There exists K ∈ C2p/α H such that |T ∗ ||T |2 |T ∗ |
K ≥ |T ∗ |2α .
Proof. Let α, p, and T be as in the hypotheses and let T U|T | be the polar decomposition of
T . Then we have
DTα T ∗ T α − T T ∗ α |T |2α − |T ∗ |2α P K,
3.17
DTα |T |2α − |T ∗ |2α P1 − K1 ,
3.18
with P ≥ 0, K ∈ Cp H.
with and P1 ≥ 0, K1 ≥ 0, and K1 ∈ Cp H.
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Journal of Inequalities and Applications
By 3.17, we have
A1 |T |2α ≥ |T ∗ |2α K B1 ≥ 0.
3.19
1/α
|T ∗ |2 K2 ,
B11/α |T ∗ |2α K
3.20
And according to Lemma 1.2,
with K2 ∈ Cp H. Setting K3 |T |K2 |T |, by Theorem F we have
|T ||T ∗ |2 |T | K3
α/2
α/2
|T | |T ∗ |2 K2 |T |
α/2
1/α 1/2α
A1/2α
B
A
1
1
1
3.21
≤ A1
|T |2α .
By 3.18, we have
A2 |T |2α K1 ≥ |T ∗ |2α B2 .
3.22
1/α
A1/α
|T |2α K1
|T |2 K4 ,
2
3.23
And according to Lemma 1.2,
with K4 ∈ Cp H. Setting K5 |T ∗ |K4 |T ∗ |, by Theorem F we have
|T ∗ ||T |2 |T ∗ | K5
α/2
α/2
|T ∗ | |T |2 K4 |T ∗ |
α/2
1/2α
B21/2α A1/α
B
2
2
3.24
≥ B2
|T ∗ |2α .
On the other hand, by Lemma 1.1,
|T ||T ∗ |2 |T | K3
∗
2
∗
|T ||T | |T | K5
with K , K ∈ C2p/α H.
α/2
α/2
α/2
|T ||T ∗ |2 |T |
K
,
∗
2
∗
|T ||T | |T |
α/2
K ,
3.25
Journal of Inequalities and Applications
9
Then by 3.21 and 3.24, we obtain
α/2
K ≤ |T |2α ,
|T ||T ∗ |2 |T |
α/2
K ≥ |T ∗ |2α ,
|T ∗ ||T |2 |T ∗ |
3.26
with K , K ∈ C2p/α H.
Theorem 3.6. Let p ≥ 1, α ∈ 0, 1, andT ∈ Hαp H such that DTα P K with P ≥ 0, K ∈ Cp H,
and DTα P1 − K1 with P1 ≥ 0, K1 ≥ 0, and K1 ∈ Cp H. Then if |T |2α − P ≥ 0, one has T 2 ∈
H.
Hα/2
2p/α
Proof. Let α, p, and T be as in the hypotheses. We may assume that T U|T | with U being
unitary. Then obviously,
T2
∗
T2
α/2
∗ α/2
T2 T2
U |T ||T ∗ |2 |T |
U∗ ,
α/2
α/2
α/2
|T |U∗ |T |2 U|T |
U∗ |T ∗ ||T |2 |T ∗ |
U.
3.27
3.28
By Lemma 3.5, there exits K , K ∈ C2p/α H such that
α/2
K ≤ |T |2α ,
|T ||T ∗ |2 |T |
|T ∗ ||T |2 |T ∗ |
α/2
K ≥ |T ∗ |2α .
3.29
3.30
Multiplying 3.29 by U to the left and by U∗ to the right, we obtain
α/2
U |T ||T ∗ |2 |T |
U∗ UK U∗ ≤ U|T |2α U∗ |T ∗ |2α .
3.31
Multiplying 3.30 by U∗ to the left and by U to the right, we obtain
α/2
U∗ |T ∗ ||T |2 |T ∗ |
U U∗ K U ≥ U∗ |T ∗ |2α U |T |2α .
3.32
By 3.27 and 3.31, we have
∗ α/2
T2 T2
UK U∗ ≤ |T ∗ |2α .
3.33
By 3.28 and 3.32, we have
T2
∗
T2
α/2
U∗ K U ≥ |T |2α .
3.34
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Journal of Inequalities and Applications
Setting K2 UK U∗ − U∗ K U, K2 ∈ C2p/α H, we have
T2
∗
T2
α/2
∗ α/2
− T2 T2
≥ |T |2α − |T ∗ |2α K2 .
3.35
Therefore, for K3 K K2 , K3 ∈ C2p/α H, we have
T2
∗
T2
α/2
∗ α/2
− T2 T2
≥ P K3 .
3.36
Then the proof of Theorem 3.6 is finished.
Acknowledgment
The authors would like to express their cordial gratitude to the referee for his kind comments.
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