Volume 2009, Article ID 492154,11pages doi:10.1155/2009/492154
Research Article
Numerical Radius and Operator Norm Inequalities
Khalid Shebrawi
1and Hussien Albadawi
21Department of Applied Sciences, Al-Balqa’ Applied University, 19117 Al-Salt, Jordan
2Department of Basic Sciences and Mathematics, Philadelphia University, 19392 Amman, Jordan
Correspondence should be addressed to Hussien Albadawi,[email protected] Received 4 November 2008; Accepted 2 March 2009
Recommended by Sever Dragomir
A general inequality involving powers of the numerical radius for sums and products of Hilbert space operators is given. This inequality generalizes several recent inequalities for the numerical radius, and includes that ifAandBare operators on a complex Hilbert spaceH, thenwrA∗B≤ 1/2|A|2r|B|2rforr≥1. It is also shown that ifXiis normali1,2, . . . , n, thenn
i1Xir≤ nr−1n
i1|Xi|r. Related numerical radius and usual operator norm inequalities for sums and products of operators are also presented.
Copyrightq2009 K. Shebrawi and H. Albadawi. 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
Let H be a complex Hilbert space with inner product ·,·, and let BH denote the C∗- algebra of all bounded linear operators onH. For A ∈ BH, the usual operator norm of an operatorAis defined by
A sup
x1Ax, 1.1
wherexx, x1/2.
The numerical range ofA, known also as the field of values ofA, is defined as the set of complex numbers given by
WA
Ax, x:x∈H,x1
. 1.2
The most important properties of the numerical range are that it is convex and its closure contains the spectrum of the operator.
A unitarily invariant norm||| · |||onHis a norm on the idealC|||·|||ofBH, satisfying
|||UAV||| |||A|||for allA ∈ BHand all unitary operatorsUandV inBH. It is called weakly unitarily invariant normor invariant under similaritiesif|||UAU∗||||||A|||for all A∈BHand all unitary operatorsU∈BH.
The most familiar example of weakly unitarily invariant norm is the numerical radius wA,defined by
wA sup
|λ|:λ∈WA
. 1.3
It is well known thatwAdefines a norm onBHand for everyA∈BH, we have
1
2A ≤wA≤ A. 1.4
Thus, the usual operator norm and the numerical radius norm are equivalent. The inequalities in1.4are sharp: ifA2 0, then the first inequality becomes an equality, while the second inequality becomes an equality ifAis normal. In fact, for a nilpotant operatorAwithAn0, Haagerup and Harpe1show thatwA≤ Acosπ/n1. In particular, whenn2, we get the reverse inequality of the first inequality in1.4. For a comprehensive account on the theory of the numerical range and numerical radius, the reader is referred to2,3. A detailed study for the field of values of a matrix is given in4.
The inequalities in1.4have been improved considerably by Kittaneh in5,6. It has been shown that ifA∈BH, then
wA≤ 1
2|A|A∗≤ 1 2
AA21/2
, 1.5
1
4A∗AAA∗≤w2A≤ 1
2A∗AAA∗, 1.6
where|A| A∗A1/2 is the absolute value ofA. The second inequality in1.5refines the second inequality in1.4. For diverse applications of these inequalities we refer to5,7.
Considerable generalizations of the first inequality in1.5and the second inequality in1.6have been established in8for the numerical radius of one operator and for the sum of two operators. It has been shown that ifA,B∈BH, then
wrA≤ 1 2
A2rαA∗2r1−α, 1.7
wrAB≤2r−2A2rαA∗2r1−αB2rαB∗2r1−α 1.8
for 0 < α < 1 andr ≥ 1. Other recent inequalities have been obtained in9,10, which are related to the Euclidean radius of two Hilbert space operators andα, β-normal operators in Hilbert spaces, respectively.
A general numerical radius inequality has been proved by Kittaneh, it has been shown in6that ifA, B, C, D, S, T ∈BH, then
wATBCSD≤ 1 2
AT∗21−αA∗B∗|T|2αBCS∗21−αC∗D∗S2αD 1.9
for allα∈0,1. In particular,
wAB±BA≤ 1
2A∗AAA∗B∗BBB∗. 1.10 Usual operator norm inequalities for sums of operators have attracted the attention of several mathematicians. Some of these inequalities have been introduced in3,11. It has been shown in6that ifAandBare normal andr≥1, then
ABr ≤2r−1|A|r|B|r. 1.11 Another important norm inequalities for unitarily invariant norms, which are related to1.11 assert that ifA1, A2, . . . , An∈BHare positive andr≥1, then
n i1
Ari ≤
n
i1
Ai r
, 1.12
n
i1
Ai
r ≤nr−1
n i1
Ari
1.13
see, e.g.,12.
In Section 2 of this paper, we establish a general numerical radius inequality that generalizes1.6,1.7,1.8, and1.9, from which numerical radius inequalities for sums, products, and commutators of operators are obtained. Usual operator norm inequalities that generalize1.11and related to1.13are presented inSection 3.
2. A General Numerical Radius Inequality
In this section, we establish a general numerical radius inequality for Hilbert space operators which yields well known and new numerical radius inequalities as special cases. To prove our generalized inequality, we need the following basic lemmas. The first lemma is a generalized form of the mixed Schwarz inequality, which has been proved by Kittaneh13.
Lemma 2.1. Let A be an operator inBH, and letfandgbe nonnegative functions on0,∞which are continuous and satisfy the relationftgt tfor allt∈0,∞.Then
Ax, y≤f
|A|
xgA∗y 2.1
for allxandyinH.
The second lemma, which is called H ¨older-McCarthy inequality, is a well-known result that follows from the spectral theorem for positive operators and Jensen’s inequality see 13.
Lemma 2.2. LetAbe a positive operator inBHand letx∈Hbe any unit vector. Then
Ax, xr ≤ Arx, x ∀r≥1. 2.2
The third lemma concerned with positive real numbers, and it is a consequence of the convexity of the functionft tr,r ≥1.
Lemma 2.3. Letaibe a positive real numberi1,2, . . . , n.Then n
i1
ai r
≤nr−1 n
i1
ari ∀r ≥1. 2.3
The fourth lemma is a norm inequality for the sum of two operators, which can be found in14.
Lemma 2.4. IfAandBare positive operators inBH, then AB ≤max
A,B
A1/2B1/2. 2.4
Another important usual operator norm inequality which will be used in this section says that for any positive operatorsA,B∈BHwe havesee11
ArBr ≤ ABr ∀0≤r≤1. 2.5
Our main result of this paper, which leads to a generalization of1.6,1.7,1.8, and 1.9, can be stated as follows.
Theorem 2.5. Let Ai,Bi,Xi ∈BH i 1,2, . . . , n, and letfandg be nonnegative functions on 0,∞which are continuous and satisfy the relationftgt tfor allt∈0,∞. Then
wr n
i1
A∗iXiBi
≤ nr−1 2
n i1
A∗ig2Xi∗Air
Bi∗f2XiBir
2.6 for allr≥1.
Proof. For every unit vectorx∈H, we have
n
i1
A∗iXiBi
x, x
r
n i1
A∗iXiBix, x
r
≤ n
i1
A∗iXiBix, xr
n
i1
XiBix, Aixr
≤ n
i1
f2XiBix, Bix1/2
g2Xi∗Aix, Aix1/2r
by2.1
≤nr−1 n
i1
Bi∗f2XiBix, xr/2
A∗ig2Xi∗Aix, xr/2
by2.3
≤nr−1 n
i1
Bi∗f2XiBir
x, x1/2
A∗ig2Xi∗Air
x, x1/2
by2.2
≤ nr−1 2
n i1
B∗if2XiBir
x, x
A∗ig2Xi∗Air
x, x
by the arithmetic-geometric mean inequality nr−1
2 n
i1
B∗if2XiBir
A∗ig2Xi∗Air x, x
.
2.7
Now the result follows by taking the supremum over all unit vectors inH.
Inequality 2.6 includes several numerical radius inequalities as special cases.
Samples of inequalities are demonstrated in what follows.
For ft tα and gt t1−α, α ∈ 0,1, in inequality2.6, we get the following inequality that generalizes1.9.
Corollary 2.6. LetAi,Bi,Xi∈BH i1,2, . . . , n,r≥1, and 0< α <1. Then
wr n
i1
A∗iXiBi
≤ nr−1 2
n i1
A∗iXi∗21−αAir
B∗iXi2αBir
. 2.8 In particular,
w n
i1
A∗iXiBi
≤ 1 2
n i1
A∗iX∗iAiB∗iXiBi
. 2.9
ForAi Bi I i 1,2, . . . , n in inequality2.6, we get the following numerical radius inequalities for sums of operators that generalizes1.8.
Corollary 2.7. LetXi∈BH i1,2, . . . , n, and letfandgbe as inLemma 2.1. Then
wr n
i1
Xi
≤ nr−1 2
n i1
f2rXig2rX∗i
∀r≥1. 2.10 In particular,
wr n
i1
Xi
≤ nr−1 2
n i1
Xi2rαXi∗2r1−α
∀α∈0,1. 2.11 It should be mentioned here that the inequality in2.11generalizes1.7in the case X1 X2· · ·Xn.
Remark 2.8. The caseα1/2 in2.11gives
wr n
i1
Xi
≤ nr−1 2
n i1
|Xi|rXi∗r
∀r ≥1, 2.12 which generalizes the second inequality in 1.6, while the choice n 1 will give a generalization of the first inequality in1.5and can be stated as
wrX≤ 1 2
|X|rX∗r ∀r ≥1. 2.13
Note that using2.4and2.5, a related inequality can be derived from2.13. Indeed,
wrX≤ 1 2
|X|rX∗r
≤ 1 2
max|X|r,X∗r
|X|r/2X∗r/2 1
2
Xr |X|rX∗r1/2 .
2.14
The above inequality generalizes the second inequality in1.5. In fact, for 1≤r ≤2, we have
wrX≤ 1 2
Xr|X|r/2X∗r/2
≤ 1 2
Xr|X|X∗r/2
1 2
XrX2r/2 .
2.15
The last equality can be proved using the polar decomposition. In fact, ifXU|X|andX∗ V|X∗|are the polar decompositions ofXandX∗, respectively, then|X||X∗|U∗XXV X2.
It is known that wAB ≤ wA wB. However, the numerical radius is not submultiplicative, even for commuting operators. On the other hand, we have the power inequality, which asserts that ifA∈BH, then
wAn≤wnA forn1,2, . . .. 2.16
It is evident from the first inequality in1.4that ifA, B∈BH, then
wAB≤4wAwB. 2.17
Moreover, ifABBA, then
wAB≤2wAwB. 2.18
These inequalities, among other related ones, can be found in2.
ForXi I i 1,2, . . . , nin inequality2.6, we get the following numerical radius inequalities for products of operators that are related to the above inequalities.
Corollary 2.9. LetAi,Bi∈BH i1,2, . . . , nandr ≥1. Then
wr n
i1
A∗iBi
≤ nr−1 2
n i1
Ai2rBi2r
. 2.19 In particular,
w n
i1
A∗iBi
≤ 1 2
n i1
A∗iAiBi∗Bi
. 2.20
Remark 2.10. The casen1 in2.19, provides the following inequality
wrA∗B≤ 1
2A∗Ar
B∗Br, 2.21
which is a numerical radius inequality for the product of operators and is related to the arithmetic-geometric mean inequality for operators. Note that a more general inequality can be obtained by lettingα1/2 andn1 in2.8. In fact, we have
wr A∗XB
≤ 1
2A∗|X∗|Ar
B∗|X|Br. 2.22
Forr1 in2.22, we obtain the inequality
w A∗XB
≤ 1
2A∗X∗AB∗|X|B 2.23
as well as
w A∗XB
≤ 1
2AA∗XXBB∗ 2.24
which follows from the the arithmetic-geometric mean inequality for operatorssee15.
Inequalities2.23and2.24are not equivalent. This can be seen from the exampleA 1 00 0, XI,B 0 10 0.
The inequality in2.22can be used to give an upper bound for the numerical radius ofA2andA3. In fact, we have
wrA2≤ 1
2AA∗r A∗Ar, 2.25
wrA3≤ 1
2A|A∗|A∗A∗|A|A. 2.26
The commutator ofAandBis the operatorAB−BA. Commutators play an important role in operator theory. It follows by the triangle inequality that ifA, B∈BH, thenAB− BA ≤2AB.
Forn 2 in inequality2.6, we get the following numerical radius inequalities that generalize1.9, and give an estimate for the numerical radius of commutators.
Corollary 2.11. LetA, B, C, D, S, T ∈BH, and letfandgbe as inTheorem 2.5. Then
wrATBCSD
≤2r−2
Ag2T∗A∗r
B∗f2
|T|
Br
Cg2S∗C∗r
D∗f2
|S|
Dr
2.27
forr ≥1. In particular,
wrATBCSD≤2r−2
AT∗A∗r
B∗|T|Br
CS∗C∗r
D∗|S|Dr. 2.28
We end this section by the following remark.
Remark 2.12. Inequality 2.28 gives a numerical radius inequality for commutators of operators that generalizes1.10. IfDA,CB, andS±T X, then
wrAXB±BXA≤2r−2
A∗X∗Ar
A|X|A∗r
B∗|X|Br
BX∗B∗r. 2.29
In particular,
wrAB±BA≤2r−2|A|2rA∗2r|B|2rB∗2r. 2.30 In fact, by lettingB A∗in2.29and2.30, respectively, we get the following inequalities for the generalized commutator and the self commutator
wr
A∗XA±AXA∗
≤2r−1
A∗|X|Ar
A|X|A∗r, 2.31 wr
A∗A±AA∗
≤2r−1|A|2rA∗2r. 2.32
3. A General Norm Inequality
In this section, we introduce a general norm inequality for Hilbert space operators, from which new inequalities for operators and generalizations of earlier results can be derived. The proof of this general inequality is similar to that ofTheorem 2.5under slight modification.
Theorem 3.1. LetAi,Bi,Xi∈BH i1,2, . . . , n, and letfandgbe as inTheorem 2.5. Then
n i1
A∗iXiBi
r
≤ nr−1 2
n i1
A∗ig2Xi∗Ar
n i1
B∗if2XiBir
3.1
forr ≥1.
Inequality 3.1 yields several norm inequalities as special cases. Samples of these inequalities are demonstrated below.
Corollary 3.2. LetAi,Bi,Xi∈BH i1,2, . . . , n,r≥1, andα∈0,1. Then
n i1
A∗iXiBi
r
≤ nr−1 2
n i1
A∗iXi∗21−αAr
n i1
Bi∗Xi2αBi
r
. 3.2
In particular,
A∗XBr ≤ 1 2
A∗X∗ArB∗|X|Br. 3.3
For Ai Bi I i 1,2, . . . , nin inequality 3.2, we get the following operator inequalities for sums of operators.
Corollary 3.3. LetXi∈BH i1,2, . . . , n,r≥1, andα∈0,1. Then
n i1
Xi
r
≤ nr−1 2
n i1
Xi2αr
n i1
X∗i21−αr
. 3.4
In particular, ifXiis normali1,2, . . . , n, then
n i1
Xi
r
≤nr−1
n i1
Xir
. 3.5
Remark 3.4. The inequality 3.5 is a generalized form of 1.11. The normality of Xi is necessary, this inequality is not true for arbitrary operatorsXi, as may be seen forn2, X1 0 10 0andX2 1 00 0.
ForXi I i1,2, . . . , nin inequality3.2, we get norm inequalities for products of operators.
Corollary 3.5. LetAi, Bi∈BH i1,2, . . . , nandr≥1. Then
n i1
A∗iBi
r
≤ nr−1 2
n i1
Ai2r
n i1
Bi2r
. 3.6
In particular,
n i1
A∗iBi ≤ 1
2
n i1
A∗iAi
n i1
Bi∗Bi
. 3.7
For n 2 in inequality 3.2, we get the following norm inequalities that give an estimate for the usual norm of commutators.
Corollary 3.6. LetA, B, C, D, S, T ∈BH, and let r≥1. Then ATBCSDr ≤2r−2AT∗A∗r
CS∗C∗rB∗|T|Br
D∗|S|Dr. 3.8
Finally, we end this paper by the following remark.
Remark 3.7. Inequality3.8gives a norm inequality for commutators of operators. IfDA, CB, andT ±SX, then we get
AXB±BXAr ≤2r−2A|X∗|A∗r
B|X∗|B∗rB∗|X|Br
A∗|X|Ar. 3.9
In particular,
AB±BAr ≤2r−2|A|2r A∗2r|B|2rB∗2r. 3.10 In fact, by lettingBA∗in3.10, we get the following inequality for self commutator
A∗A±AA∗r ≤2r−1A2rA∗2r. 3.11
Moreover a related inequality to3.11can be derived from1.12and1.13. Indeed,
A∗Ar
AA∗r≤
A∗AAA∗r≤2r−1
A∗Ar
AA∗r. 3.12
Acknowledgment
The authors thank the anonymous referee for his valuable comments and suggestions for improving this paper.
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