Volume 2008, Article ID 763102,9pages doi:10.1155/2008/763102
Research Article
Some Inequalities of the Gr ¨uss Type
for the Numerical Radius of Bounded Linear Operators in Hilbert Spaces
S. S. Dragomir
School of Computer Science and Mathematics, Victoria University, P.O. Box 14428, Melbourne VIC 8001, Australia
Correspondence should be addressed to S. S. Dragomir,[email protected] Received 27 May 2008; Accepted 4 August 2008
Recommended by Yeol Je Cho
Some inequalities of the Gr ¨uss type for the numerical radius of bounded linear operators in Hilbert spaces are established.
Copyrightq2008 S. S. Dragomir. 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
LetH;·,·be a complex Hilbert space. The numerical range of an operatorTis the subset of the complex numbersCgiven by1, page 1:
WT {Tx, x, x∈H,x1}. 1.1
The numerical radiuswTof an operatorT onHis given by1, page 8:
wT sup{|λ|, λ∈WT}sup{|Tx, x|,x1}. 1.2
It is well known thatw·is a norm on the Banach algebraBHof all bounded linear operatorsT : H → H.This norm is equivalent to the operator norm. In fact, the following more precise result holds1, page 9.
Theorem 1.1equivalent norm. For anyT ∈BH,one has
wT≤ T ≤2wT. 1.3
For other results on numerical radiussee2, Chapter 11.
We recall some classical results involving the numerical radius of two linear operators A, B.
The following general result for the product of two operators holds1, page 37.
Theorem 1.2. IfA, Bare two bounded linear operators on the Hilbert spaceH,·,·,then
wAB≤4wAwB. 1.4
In the case thatABBA,then
wAB≤2wAwB. 1.5
The following results are also well known1, page 38.
Theorem 1.3. IfAis a unitary operator that commutes with another operatorB,then
wAB≤wB. 1.6
IfAis an isometry andABBA,then1.6also holds true.
We say thatAandBdouble commute, ifABBAandAB∗B∗A.
The following result holds1, page 38.
Theorem 1.4double commute. If the operatorsAandBdouble commute, then
wAB≤wBA. 1.7
As a consequence of the above, one has1, page 39the following.
Corollary 1.5. LetAbe a normal operator commuting withB.Then
wAB≤wAwB. 1.8
For other results and historical comments on the abovesee 1, pages 39–41. For more results on the numerical radius, see2.
In the recent survey paper3, we provided other inequalities for the numerical radius of the product of two operators. We list here some of the results.
Theorem 1.6. LetA, B :H → Hbe two bounded linear operators on the Hilbert spaceH,·,·, then
A∗AB∗B 2
≤wB∗A 1
2A−B2, AB
2
2 ≤ 1 2
A∗AB∗B 2
wB∗A
,
1.9
respectively.
If more information regarding one of the operators is available, then the following results may be stated as well.
Theorem 1.7. LetA, B :H →H be two bounded linear operators onH, andBis invertible such that, for a givenr >0,
A−B ≤r. 1.10
Then
A ≤ B−1
wB∗A 1 2r2
,
0≤AB −wB∗A≤ 1
2r2 B2B−12−1 2B−12 ,
1.11
respectively.
Motivated by the natural questions that arise, in order to compare the quantity wABwith other expressions comprising the norm or the numerical radius of the involved operatorsAandBor certain expressions constructed with these operators, we establish in this paper some natural inequalities of the form
wBA≤wAwB K1, additive Gr ¨uss’type inequality, 1.12 or
wBA
wAwB ≤K2, multiplicative Gr ¨uss’type inequality, 1.13 where K1 and K2 are specified and desirably simple constants depending on the given operatorsAandB.
Applications in providing upper bounds for the non-negative quantities
A2−w2A, w2A−wA2, 1.14
and the superunitary quantities
A2
w2A, w2A
wA2 1.15
are also given.
2. Numerical radius inequalities of Gr ¨uss type
For the complex numbersα, βand the bounded linear operatorT, we define the following transform:
Cα,βT: T∗−αIβI−T, 2.1
where byT∗we denote the adjoint ofT.
We list some properties of the transformCα,β·that are useful in the following.
iFor anyα, β∈CandT ∈BH,we have
Cα,βI 1−αβ−1I, Cα,αT −αI−T∗αI−T, Cα,βγT |γ|2Cα/γ,β/γT, for eachγ∈C\ {0},
Cα,βT∗Cβ,αT, Cβ,αT∗−Cα,βT T∗T−TT∗.
2.2
iiThe operatorT ∈BHis normal, if and only ifCβ,αT∗ Cα,βTfor eachα, β∈C.
We recall that a bounded linear operatorT on the complex Hilbert spaceH,·,·is called accretive, if ReTy, y ≥0, for anyy∈H.
Utilizing the following identity
ReCα,βTx, xReCβ,αTx, x 1
4|β−α|2−
T−αβ 2 I
x
2, 2.3 that holds for any scalarsα, β,and any vector x ∈ H withx 1,we can give a simple characterization result that is useful in the following.
Lemma 2.1. Forα, β∈CandT ∈BH,the following statements are equivalent.
iThe transformCα,βT or, equivalentlyCβ,αTis accretive.
iiThe transformCα,βT∗ or, equivalentlyCβ,αT∗is accretive.
iiiOne has the norm inequality
T−αβ 2 ·I
≤ 1
2|β−α|, 2.4
or, equivalently,
T∗− αβ 2 ·I
≤ 1
2|β−α|. 2.5
Remark 2.2. In order to give examples of operators T ∈ BHand numbers α, β ∈ Csuch that the transformCα,βTis accretive, it suffices to select a bounded linear operatorSand the complex numbersz, wwith the property thatS−zI ≤ |w|, and by choosingT S, α 1/2zw,andβ 1/2z−w,we observe thatTsatisfies2.4, that is,Cα,βTis accretive.
The following results compare the quantities wABand wAwB provided that some information about the transformsCα,βAandCγ,δBare available, whereα, β, γ, δ∈K.
Theorem 2.3. LetA, B∈BHandα, β, γ, δ∈Kbe such that the transformsCα,βAandCγ,δB are accretive, then
wBA≤wAwB 1
4|β−α| |γ−δ|. 2.6
Proof. SinceCα,βAand Cγ,δBare accretive, then, on making use ofLemma 2.1, we have that
Ax−αβ 2 x
≤ 1 2|β−α|, B∗x−γδ
2 x ≤ 1
2γ−δ,
2.7
for anyx∈H, x1.
Now, we make use of the following Gr ¨uss type inequality for vectors in inner product spaces obtained by the author in4 see also5or6, page 43.
Let H,·,· be an inner product space over the real or complex number field K, u, v, e∈H, e1,andα, β, γ, δ∈Ksuch that
Reβe−u, u−αe ≥0, Reδe−v, v−γe ≥0, 2.8 or equivalently,
u− αβ 2 e
≤ 1
2|β−α|,
v− γδ 2 e
≤ 1
2|δ−γ|, 2.9
then
|u, v − u, ee, v| ≤ 1
4|β−α| |δ−γ|. 2.10 Applying2.10foruAx, vB∗x, andexwe deduce
|BAx, x − Ax, xBx, x| ≤ 1
4|β−α| |δ−γ|, 2.11 for anyx∈H, x1,which is an inequality of interest in itself.
Observing that
|BAx, x| − |Ax, xBx, x| ≤ |BAx, x − Ax, xBx, x|, 2.12 then by2.10, we deduce the inequality
|BAx, x| ≤ |Ax, xBx, x|1
4|β−α| |δ−γ|, 2.13 for anyx ∈ H, x 1.On taking the supremum overx 1 in2.13, we deduce the desired result2.6.
The following particular case provides an upper bound for the nonnegative quantity A2−wA2when some information about the operatorAis available.
Corollary 2.4. LetA∈BHandα, β∈Kbe such that the transformCα,βAis accretive, then
0≤A2−w2A≤ 1
4|β−α|2. 2.14
Proof. Follows on applyingTheorem 2.3above for the choiceBA∗,taking into account that Cα,βAis accretive implies thatCα,βA∗is the same andwA∗A A2.
Remark 2.5. Let A ∈ BHand M > m > 0be such that the transform Cm,MA A∗ − mIMI−Ais accretive. Then
0≤A2−w2A≤ 1
4M−m2. 2.15
A sufficient simple condition forCm,MAto be accretive is thatAis a self-adjoint operator onHand such thatMI≥A≥mIin the partial operator order ofBH.
The following result may be stated as well.
Theorem 2.6. LetA, B ∈BHandα, β, γ, δ ∈Kbe such that Reβ α>0, Reδ γ >0 and the transformsCα,βA, Cγ,δBare accretive, then
wBA
wAwB ≤11
4· |β−α| |δ−γ|
Reβ αReδ γ1/2, wBA≤wAwB |αβ|−2Reβ α1/2
× |δγ| −2Reδ γ1/21/2
wAwB1/2, 2.16
respectively.
Proof. With the assumptions2.8 or, equivalently,2.9in the proof ofTheorem 2.3and if Reβ α>0, Reδ γ>0 then
|u, v − u, ee, v| ≤
⎧⎪
⎪⎪
⎪⎪
⎨
⎪⎪
⎪⎪
⎪⎩ 1 4
|β−α| |δ−γ|
Reβ αReδ γ1/2|u, ee, v|,
|αβ| −2Reβ α1/2 |δγ| −2Reδ γ1/21/2
×|u, ee, v|1/2.
2.17
The first inequality has been established in7 see6, page 62while the second one can be obtained in a canonical manner from the reverse of the Schwarz inequality given in8. The details are omitted.
Applying2.10foruAx, vB∗x, andexwe deduce
|BAx, x − Ax, xBx, x| ≤
⎧⎪
⎪⎪
⎪⎪
⎨
⎪⎪
⎪⎪
⎪⎩ 1 4
|β−α| |δ−γ|
Reβ αReδ γ1/2|A, xBx, x|,
|αβ| −2Reβ α1/2 |δγ| −2Reδ γ1/21/2
×|A, xBx, x|1/2,
2.18 for anyx∈H, x1,which are of interest in themselves.
A similar argument to that in the proof ofTheorem 2.3yields the desired inequalities 2.16. The details are omitted.
Corollary 2.7. LetA∈BHandα, β∈Kbe such that Reβ α>0 and the transformCα,βAis accretive, then
1≤ A2
w2A ≤11
4·|β−α|2 Reβ α,
0≤A2−w2A≤|αβ| −2Reβ α1/2wA,
2.19
respectively.
The proof is obvious fromTheorem 2.6on choosingBA∗and the details are omitted.
Remark 2.8. Let A ∈ BHand M > m > 0be such that the transform Cm,MA A∗ − mIMI−Ais accretive. Then, on making use ofCorollary 2.7, we may state the following simpler results:
1≤ A wA ≤ 1
2·M√ m Mm, 0≤A2−w2A≤√
M−√
m2wA,
2.20
respectively. These two inequalities were obtained earlier by the author using a different approachsee9.
Problem 1. Find general examples of bounded linear operators realizing the equality case in each of inequalities2.6,2.16, respectively.
3. Some particular cases of interest
The following result is well known in the literaturesee, e.g.,10:
wAn≤wnA, 3.1
for each positive integernand any operatorA∈BH.
The following reverse inequalities forn2 can be stated.
Proposition 3.1. LetA∈BHandα, β∈Kbe such that the transformCα,βAis accretive, then 0≤w2A−wA2≤ 1
4|β−α|2. 3.2
Proof. On applying inequality 2.11 from Theorem 2.3 for the choice B A, we get the following inequality of interest in itself:
|Ax, x2− A2x, x| ≤ 1
4|β−α|2, 3.3
for anyx∈H, x1.Since obviously,
|Ax, x|2− |A2x, x| ≤ |Ax, x2− A2x, x|, 3.4 then by3.3, we get
|Ax, x|2≤ |A2x, x| 1
4|β−α|2, 3.5
for anyx∈H, x 1.Taking the supremum overx1 in3.5, we deduce the desired result3.2.
Remark 3.2. Let A ∈ BH andM > m > 0 be such that the transformCm,MA A∗ − mIMI−Ais accretive. Then
0≤w2A−wA2≤ 1
4M−m2. 3.6
IfMI≥A≥mIin the partial operator order ofBH,then3.6is valid.
Finally, we also have the following proposition.
Proposition 3.3. LetA∈BHandα, β∈Kbe such that Reβ α>0 and the transformCα,βA is accretive, then
1≤w2A
wA2 ≤1 1
4·|β−α|2 Reβ α,
0≤w2A−wA2≤|αβ| −2Reβ α1/2wA,
3.7
respectively.
Proof. On applying inequality 2.18 from Theorem 2.6 for the choice B A, we get the following inequality of interest in itself:
|Ax, x2− A2x, x| ≤
⎧⎪
⎪⎨
⎪⎪
⎩ 1
4·|β−α|2
Reβ α|A, x|2,
|αβ| −2Reβ α1/2|A, x|,
3.8
for anyx∈H, x1.
Now, on making use of a similar argument to the one in the proof ofProposition 3.1, we deduce the desired results3.7. The details are omitted.
Remark 3.4. Let A ∈ BH andM > m > 0 be such that the transformCm,MA A∗ − mIMI−Ais accretive. Then, on making use ofProposition 3.3, we may state the following simpler results:
1≤w2A wA2 ≤ 1
4·Mm2 Mm , 0≤w2A−wA2≤√
M−√
m2wA,
3.9
respectively.
References
1 K. E. Gustafson and D. K. M. Rao, Numerical Range: The Field of Values of Linear Operators and Matrices, Universitext, Springer, New York, NY, USA, 1997.
2 P. R. Halmos, A Hilbert Space Problem Book, vol. 19 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 2nd edition, 1982.
3 S. S. Dragomir, “A survey of some recent inequalities for the norm and numerical radius of operators in Hilbert spaces,” Banach Journal of Mathematical Analysis, vol. 1, no. 2, pp. 154–175, 2007.
4 S. S. Dragomir, “A generalization of Gr ¨uss’s inequality in inner product spaces and applications,”
Journal of Mathematical Analysis and Applications, vol. 237, no. 1, pp. 74–82, 1999.
5 S. S. Dragomir, “Some Gr ¨uss type inequalities in inner product spaces,” Journal of Inequalities in Pure and Applied Mathematics, vol. 4, no. 2, article 42, pp. 1–10, 2003.
6 S. S. Dragomir, Advances in Inequalities of the Schwarz, Gr ¨uss and Bessel Type in Inner Product Spaces, Nova Science, Hauppauge, NY, USA, 2005.
7 S. S. Dragomir, “Reverses of Schwarz, triangle and Bessel inequalities in inner product spaces,” Journal of Inequalities in Pure and Applied Mathematics, vol. 5, no. 3, article 76, pp. 1–18, 2004.
8 S. S. Dragomir, “Reverses of the Schwarz inequality generalising a Klamkin-McLenaghan result,”
Bulletin of the Australian Mathematical Society, vol. 73, no. 1, pp. 69–78, 2006.
9 S. S. Dragomir, “Reverse inequalities for the numerical radius of linear operators in Hilbert spaces,”
Bulletin of the Australian Mathematical Society, vol. 73, no. 2, pp. 255–262, 2006.
10 C. Pearcy, “An elementary proof of the power inequality for the numerical radius,” The Michigan Mathematical Journal, vol. 13, no. 3, pp. 289–291, 1966.
