Volume 2008, Article ID 472146,20pages doi:10.1155/2008/472146
Research Article
Upper Bounds for the Euclidean Operator Radius
and Applications
S. S. Dragomir
Research Group in Mathematical Inequalities & Applications, School of Engineering & Science, Victoria University, P.O. Box 14428, Melbourne, VIC 8001, Australia
Correspondence should be addressed to S. S. Dragomir,[email protected]
Received 5 September 2008; Accepted 3 December 2008
Recommended by Andr ´os Ront´a
The main aim of the present paper is to establish various sharp upper bounds for the Euclidean operator radius of ann-tuple of bounded linear operators on a Hilbert space. The tools used are provided by several generalizations of Bessel inequality due to Boas-Bellman, Bombieri, and the author. Natural applications for the norm and the numerical radius of bounded linear operators on Hilbert spaces are also given.
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
Following Popescu’s work1, we present here some basic properties of theEuclidean operator radiusof ann-tuple of operatorsT1, . . . , Tnthat are defined on a Hilbert spaceH;·,·.This
radius is defined by
weT1, . . . , Tn: sup
h1
n
i1
Tih, h2 1/2
. 1.1
We can also consider the following norm and spectral radius onBHn:BH×· · ·×BH, by setting1
T1, . . . , Tn
e: sup
λ1,...,λn∈Bn
λ1T1· · ·λnTn,
reT1, . . . , Tn sup
λ1,...,λn∈Bn
rλ1T1· · ·λnTn,
whererTdenotes the usual spectral radius of an operatorT ∈ BHandBnis the closed
unit ball inCn.
Notice that·eis a norm onBHn:
T1, . . . , Tn
eT1∗, . . . , Tn∗e, re
T1, . . . , TnreT1∗, . . . , Tn∗
. 1.3
Now, if we denote byT1, . . . , Tnthe square root of the normni1TiTi∗,that is,
T1, . . . , Tn:
n
i1
TiTi∗
1/2
, 1.4
then we can present the following result due to Popescu 1 concerning some sharp inequalities between the normsT1, . . . , TnandT1, . . . , Tne.
Theorem 1.1see1. IfT1, . . . , Tn∈BHn,then
1
√
nT1, . . . , Tn≤T1, . . . , Tne≤T1, . . . , Tn, 1.5
where the constants1/√nand1are best possible in1.5.
Following 1, we list here some of the basic properties of the Euclidean operator radius of ann-tuple of operatorsT1, . . . , Tn∈BHn.
iweT1, . . . , Tn 0 if and only ifT1· · ·Tn0; iiweλT1, . . . , λTn |λ|weT1, . . . , Tnfor anyλ∈C; iiiweT1T1, . . . , TnTn≤weT1, . . . , Tn weT1, . . . , Tn;
ivweU∗T1U, . . . , U∗TnU weT1, . . . , Tnfor any unitary operatorU:K → H; vweX∗T1X, . . . , X∗TnX≤ X2weT1, . . . , Tnfor any operatorX:K → H; vi 1/2T1, . . . , Tne≤weT1, . . . , Tn≤ T1, . . . , Tne;
viireT1, . . . , Tn≤weT1, . . . , Tn;
viiiweIε⊗T1, . . . , Iε⊗Tn weT1, . . . , Tnfor any separable Hilbert spaceε; ixweis a continuous map in the norm topology;
xweT1, . . . , Tn supλ1,...,λn∈Bnwλ1T1· · ·λnTn;
xi 1/2√nT1, . . . , Tn ≤ weT1, . . . , Tn ≤ T1, . . . , Tn and the inequalities are
sharp.
ForA∈BH,letwAandAdenote the numerical radius and the usual operator norm ofA,respectively. It is well known thatw·defines a norm onBH,and for every
A∈BH,
1
2A ≤wA≤ A. 1.6
For other results concerning the numerical range and radius of bounded linear operators on a Hilbert space, see2,3.
In4, Kittaneh has improved1.6in the following manner:
1 4A
∗AAA∗≤w2A≤ 1
2A
∗AAA∗, 1.7
with the constants 1/4 and 1/2 as best possible.
LetC, Dbe a pair of bounded linear operators onH,the Euclidean operator radius is
weC, D: sup
x1
Cx, x2Dx, x21/2
1.8
and, as pointed out in1,we :B2H → 0,∞is a norm and the following inequality holds:
√
2 4 C
∗CD∗D1/2≤w
eC, D≤C∗CD∗D1/2, 1.9
where the constants√2/4 and 1 are best possible in1.9.
We observe that, ifCandDare self-adjoint operators, then1.9becomes
√
2 4 C
2D21/2≤w
eC, D≤C2D21/2. 1.10
We observe also that ifA∈BHandABiCis theCartesian decompositionofA,then
w2
eB, C sup
x1
Bx, x2Cx, x2
sup
x1
Ax, x2
w2A.
1.11
By the inequality1.10and sincesee4
then we have
1 16A
∗AAA∗≤w2A≤ 1
2A
∗AAA∗. 1.13
We remark that the lower bound forw2Ain1.13provided by Popescu’s inequality1.9
is not as good as the first inequality of Kittaneh from1.7. However, the upper bounds for
w2Aare the same and have been proved using different arguments.
In order to get a natural generalization of Kittaneh’s result for the Euclidean operator radius of two operators, we have obtained in5the following result.
Theorem 1.2. LetB, C:H → Hbe two bounded linear operators on the Hilbert spaceH;·,·. Then
√
2 2
wB2C21/2≤w
eB, C≤B∗BC∗C1/2. 1.14
The constant√2/2is best possible in the sense that it cannot be replaced by a larger constant.
Corollary 1.3. For any two self-adjoint bounded linear operatorsB, ConH,one has
√
2 2 B
2C21/2≤w
eB, C≤B2C21/2. 1.15
The constant√2/2is sharp in1.15.
Remark 1.4. The inequality 1.15is better than the first inequality in 1.10 which follows from Popescu’s first inequality in1.9. It also provides, for the case thatB, Care the self-adjoint operators in the Cartesian decomposition ofA,exactly the lower bound obtained by Kittaneh in1.7for the numerical radiuswA.
For other inequalities involving the Euclidean operator radius of two operators and their applications for one operator, see the recent paper 5, where further references are given.
Motivated by the useful applications of the Euclidean operator radius concept in multivariable operator theory outlined in 1, we establish in this paper various new sharp upper bounds for the general case n ≥ 2. The tools used are provided by several generalizations of Bessel inequality due to Boas-Bellman, Bombieri, and the author. Also several reverses of the Cauchy-Bunyakovsky-Schwarz inequalities are employed. The case
2. Upper bounds via the Boas-Bellman-type inequalities
The following inequality that naturally generalizes Bessel’s inequality for the case of nonorthonormal vectors y1, . . . , yn in an inner product space is known in the literature as
theBoas-Bellman inequalitysee6,7, or8, chapter 4:
n
i1
x, yi2≤ x2
⎡
⎣max
1≤i≤nyi 2
1≤i /j≤n
yi, yj2 1/2⎤
⎦, 2.1
for anyx∈H.
Obviously, if{y1, . . . , yn} is an orthonormal family, then 2.1 becomes the classical Bessel’s inequality
n
i1
x, yi2≤ x2, x∈H. 2.2
The following result provides a natural upper bound for the Euclidean operator radius ofnbounded linear operators.
Theorem 2.1. IfT1, . . . , Tn∈BHn,then
weT1, . . . , Tn≤
⎡
⎣max
1≤i≤n
Ti2
1≤i /j≤n
w2T∗
jTi
1/2⎤
⎦
1/2
. 2.3
Proof. Utilizing the Boas-Bellman inequality forxh, h1 andyi Tih, i1, . . . , n,we
have
n
i1
Tih, h2 ≤
max
1≤i≤nTih 2
1≤i /j≤n
T∗
jTih, h2 1/2
. 2.4
Taking the supremum overh1 and observing that
sup
h1
max
1≤i≤nTih 2
max
1≤i≤nTi 2,
sup
h1
1≤i /j≤n
T∗
jTih, h2 1/2
≤
1≤i /j≤n
sup
h1
T∗
jTih, h2
1/2
1≤i /j≤n
w2T∗ jTi
1/2
,
2.5
Remark 2.2. IfT1, . . . , Tn∈BHnis such thatTj∗Ti0 fori, j ∈ {1, . . . , n},then from2.3,
we have the inequality:
weT1, . . . , Tn≤max 1≤i≤n
Ti. 2.6
We observe that a sufficient condition forTj∗Ti 0,withi /j, i, j ∈ {1, . . . , n}to hold, is that RangeTi⊥RangeTjfori, j∈ {1, . . . , n},withi /j.
Remark 2.3. If we apply the above result for two bounded linear operators onH, B, C:H →
H,then we get the simple inequality
w2
eB, C≤maxB2,C2
√
2wB∗C. 2.7
Remark 2.4. IfA :H → His a bounded linear operator on the Hilbert spaceH and if we denote by
B: AA∗
2 , C:
A−A∗
2i 2.8
its Cartesian decomposition, then
w2
eB, C w2A,
wB∗CwC∗B 1
4w
A∗−AAA∗,
2.9
and from2.7, we get the inequality
w2A≤ 1
4
maxAA∗2,A−A∗2√2wA∗−AAA∗. 2.10
In9, the author has established the following Boas-Bellman type inequality for the vectorsx, y1, . . . , ynin the real or complex inner product spaceH,·,·:
n
i1
x, yi2≤ x2max
1≤i≤nyi 2 n−
1 max
1≤i /j≤nyi, yj
. 2.11
For orthonormal vectors,2.11reduces to Bessel’s inequality as well. It has also been shown in 9 that the Boas-Bellman inequality 2.1 and the inequality 2.11 cannot be compared in general, meaning that in some instances the right-hand side of2.1is smaller than that of2.11and vice versa.
Theorem 2.5. IfT1, . . . , Tn∈BHn,then
weT1, . . . , Tn
≤
max
1≤i≤nTi 2 n−
1 max
1≤i /j≤nw
T∗
jTi 1/2
. 2.12
If in2.12one assumes thatTj∗Ti0 for eachi, j ∈ {1, . . . , n}withi /j,then one gets
the result from2.6.
Remark 2.6. We observe that, forn2,we get from2.12a better result than2.7, namely,
w2
eB, C≤maxB2,C2wB∗C, 2.13
whereB, Care arbitrary linear bounded operators onH.The inequality2.13is sharp. This follows from the fact that forBCA∈BH, Aa normal operator, we have
w2
eA, A 2w2A 2A2,
wA∗AA2, 2.14
and we obtain in 2.13 the same quantity in both sides. The inequality 2.13 has been obtained in5,12.23on utilizing a different argument.
Also, for the operatorA:H → H,we can obtain from2.13the following inequality:
w2A≤ 1
4
maxAA∗2,A−A∗2wA∗−AAA∗, 2.15
which is better than2.10. The constant 1/4 in2.15is sharp. The case of equality in2.15 follows, for instance, ifAis assumed to be self-adjoint.
Remark 2.7. If in2.13we chooseCA, BA∗, A∈BH,and take into account that
w2
eA∗, A2w2A, 2.16
then we get the inequality
w2A≤ 1
2
A2wA2≤ A2, 2.17
for anyA∈BH.The constant 1/2 is sharp.
Note that this inequality has been obtained in10by the use of a different argument based on the Buzano inequality11.
Theorem 2.8. IfT1, . . . , Tn∈BHn,then
w2
eT1, . . . , Tn≤max 1≤i≤nw
Ti·
n
i1
T∗
iTi
1≤i /j≤n
wT∗
jTi
1/2
. 2.18
Proof. We use the following Boas-Bellman-type inequality obtained in9 see also8, page 132:
n
i1
x, yi2≤ x
max
1≤i≤nx, yi
n
i1
yi2
1≤i /j≤n
yi, yj1/2, 2.19
wherex, y1, . . . , ynare arbitrary vectors in the inner product spaceH;·,·.
Now, forxh, h1, yiTih, i1, . . . , n,we get from2.19that
n
i1
Tih, h2≤
max
1≤i≤n
Tih, hn
i1
Tih2
1≤i /j≤n
Tih, Tjh1/2. 2.20
Observe that
n
i1
Tih2n i1
Tih, Tih n
i1
T∗
iTih, h
n
i1
T∗
iTih, h
, 2.21
forh∈H,h1.
Therefore, on taking the supremum in 2.20 and noticing that wni1Ti∗Ti
n
i1Ti∗Ti,we get the desired result2.18.
Remark 2.9. If T1, . . . , Tn ∈ BHn satisfies the condition that Ti∗Tj 0 for each i, j ∈
{1, . . . , n}withi /j,then from2.18we get
w2
eT1, . . . , Tn≤max 1≤i≤nw
Ti·
n
i1
T∗
iTi
1/2
. 2.22
Remark 2.10. If we apply Theorem 2.8 to n 2, then we can state the following simple inequality:
w2
eB, C≤maxB,CB∗BC∗C2wB∗C1/2, 2.23
Moreover, ifBandCare chosen as the Cartesian decomposition of the bounded linear operatorA∈BH,then we can state that
w2A≤ 1
2maxAA
∗,A−A∗A∗AAA∗
2
1
2w
AA∗A−A∗1/2.
2.24
The constant 1/2 is best possible in2.24. The equality case is obtained ifAis a self-adjoint operator onH.
If we choose in2.23,CA, BA∗, A∈BH,then we get
w2A≤ 1
2AAA
∗A∗A2wA21/2≤ A2. 2.25
The constant 1/2 is best possible in2.25.
3. Upper bounds via the Bombieri-type inequalities
A different generalization of Bessel’s inequality for nonorthogonal vectors than the one mentioned above and due to Boas and Bellman is theBombieri inequalitysee12,13, page 394, or8, page 134
n
i1
x, yi2 ≤ x2max
1≤i≤n
n
j1
yi, yj, 3.1
wherex, y1, . . . , ynare vectors in the real or complex inner product spaceH;·,·.
Note that the Bombieri inequality was not stated in the general case of inner product spaces in12. However, the inequality presented there easily leads to3.1which, apparently, was firstly mentioned as is in13, page 394.
The following upper bound for the Euclidean operator radius may be obtained as follows.
Theorem 3.1. IfT1, . . . , Tn∈BHn,then
w2
eT1, . . . , Tn≤max 1≤i≤n
n
j1
wT∗
jTi
. 3.2
Proof. Follows by Bombieri’s inequality applied forxh,h 1 andyi Tih, i 1, . . . , n.
Remark 3.2. If we apply the above theorem for two operatorsBandC,then we get
w2
eB, C≤maxwB∗BwC∗B, wB∗CwC∗C maxB2wB∗C, wB∗CC2
maxB2,C2wB∗C,
3.3
which is exactly the inequality2.13that has been obtained in a different manner above.
In order to get other bounds for the Euclidean operator radius, we may state the following result as well.
Theorem 3.3. IfT1, . . . , Tn∈BHn,then
w2
eT1, . . . , Tn≤
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ Dw, Ew, Fw, 3.4
meaning that the left side is less than each of the quantities in the right side, where
Dw:
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ max
1≤k≤n
wTk
n
i,j1
wT∗
jTi
1/2
,
max
1≤k≤n
wTk1/2
n
i1
wTir 1/2r⎡
⎣n
i1
n
j1
wT∗
jTi
s⎤⎦1/2s
,
wherer, s >1, 1
r
1
s 1,
max
1≤k≤n
wTk1/2
n
i1
wTi 1/2
max
1≤i≤n
n
j1
wT∗
jTi
1/2
,
Ew:
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ n
k1
wTkp 1/2p
max
1≤k≤n
wTk1/2
⎡
⎣n
i1
n
j1
wT∗
jTi
q⎤⎦1/2q
,
wherep >1, 1
p
1
q1,
n
k1
wTkp
1/2pn
i1
wTit 1/2t⎡
⎣n
i1
n
j1
wT∗
jTi
q u/q⎤⎦1/2u
,
wherep >1, 1
p
1
q1, t >1,
1
t
1
u 1,
n
k1
wTkp
1/2pn
i1
wTi 1/2
max
1≤i≤n
⎧ ⎨ ⎩
n
j1
wT∗
jTiq
1/2p⎫⎬
⎭,
wherep >1, 1
p
1
Fw: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ n
k1
wTk 1/2
max
1≤i≤n
wTi1/2 n
i1
max
1≤j≤n
wT∗
jTi1/2
,
n
k1
wTk
1/2n
i1
wTim
1/2m n
i1
max
1≤j≤n
wT∗
jTil
1/2l
,
wherem >1, 1
m
1
l 1,
n
k1
wTk·max 1≤i,j≤n
wT∗
jTi
1/2.
3.5
Proof. In our paper 14 see also 8, page 141-142, we have established the following sequence of inequalities for the vectorsx, y1, . . . , ynin the inner product spaceH,·,·and
the scalarsc1, . . . , cn∈K:
n
i1
cix, yi
2
≤ x2×
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ D, E, F, 3.6 where D: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ max
1≤k≤nck 2
n
i,j1
yi, yj ,
max
1≤k≤nck
n
i1
cir 1/r
⎡
⎣n
i1
n
j1
yi, yj s⎤⎦
1/s
, wherer, s >1, 1
r
1
s 1,
max
1≤k≤nck
n
i1
ci max
1≤i≤n
n
j1
yi, yj,
E: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ n
k1
ckp 1/pmax 1≤i≤n|ci|
⎡
⎣n
i1
n
j1
yi, yj q⎤⎦
1/q
, wherep >1, 1
p
1
q 1,
n
k1
ckp 1/p
n
i1
cit 1/t
⎡
⎣n
i1
n
j1
yi, yjq u/q
⎤ ⎦
1/u
, wherep >1, 1
p
1
q 1,
t >1,1
t
1
u 1,
n
k1
ckp 1/p
n
i1
ci max
1≤i≤n
⎧ ⎨ ⎩
n
j1
yi, yjq 1/q
⎫ ⎬
⎭, wherep >1,
1
p
1
F : ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ n
k1
ck max
1≤i≤nci n
i1
max
1≤j≤nyi, yj
,
n
k1
ckn
i1
cim 1/m n i1
max
1≤j≤nyi, yj
l1/l, wherem >1, 1
m
1
l 1,
n
k1
ck 2
max
1≤i,j≤nyi, yj.
3.7
If in this inequality we chooseci x, yi, i1, . . . , nand take the square root, then we get the inequalities
n
i1
x, yi2≤ x ×
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ % D, % E, % F, 3.8 where % D: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ max
1≤k≤nx, yk
n
i,j1
yi, yj 1/2,
max
1≤k≤nx, yk 1/2
n
i1
x, yir 1/2r
⎡
⎣n
i1
n
j1
yi, yj s⎤⎦
1/2s
,
wherer, s >1, 1
r
1
s 1,
max
1≤k≤nx, yk 1/2
n
i1
x, yi 1/2max
1≤i≤n
n
j1
yi, yj1/2,
% E: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ n
k1
x, ykp 1/2pmax 1≤i≤n
x, yi1/2
⎡
⎣n
i1
n
j1
yi, yj q⎤⎦
1/2q
,
wherep >1, 1
p
1
q 1,
n
k1
x, ykp 1/2p
n
i1
x, yit 1/2t
⎡
⎣n
i1
n
j1
yi, yjq u/q
⎤ ⎦
1/2u
,
wherep >1, 1
p
1
q 1, t >1,
1
t
1
u 1,
n
k1
x, ykp 1/2p
n
i1
x, yi 1/2max
1≤i≤n
⎧ ⎨ ⎩
n
j1
yi, yjq 1/q
⎫ ⎬
⎭,
wherep >1, 1
p
1
%
F:
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
n
k1
x, yk 1/2max
1≤i≤nx, yi 1/2n
i1
max
1≤j≤nyi, yj
1/2
,
n
k1
x, yk 1/2n
i1
x, yim 1/2m
n
i1
max
1≤j≤nyi, yj l1/2l,
wherem >1, 1
m
1
l 1,
n
k1
x, ykmax
1≤i,j≤nyi, yj 1/2.
3.9
By making use of the inequality3.8for the choicesxh,h1, yiTih, i1, . . . , nand
taking the supremum, we get the following result3.4.
Remark 3.4. For n 2, the above inequalities 3.4 provide various upper bounds for the Euclidean operator radiusweB, C,for anyB, C ∈ BH.Out of these results and for the
sake of brevity, we only mention the following ones:
w2
eB, C≤max
B,C1/2wB wC1/2
maxB2,C2wB∗C1/2, 3.10
w2
eB, C≤
wB wCmaxB,C,wB∗C1/2, 3.11
for anyB, C∈BH.
Both inequalities are sharp. This follows by the fact that forB CA ∈BH, Aa normal operator, we get in both sides of3.10and3.11the same quantity 2A2.
Remark 3.5. If we choose in 3.10, the Cartesian decomposition of the operatorA, then we get
w2A≤ 1
4maxAA
∗,A−A∗1/2AA∗A−A∗1/2
×maxAA∗2,A−A∗2wA∗−AA∗A1/2.
3.12
The constant 1/4 is sharp. The equality case holds ifAA∗.The same choice in3.11will give
w2A≤ 1
4AA
∗A−A∗
×maxAA∗,A−A∗, w1/2A∗−AA∗A.
3.13
In15 see also8, page 233, we obtained the following inequality of Bombieri type:
n
i1
x, yi2≤ x
n
k1
x, ykp 1/p
n
i,j1
yi, yjq
1/2q
, 3.14
for any x, y1, . . . , yn vectors in the inner product space H,·,·, where p > 1 and
1/p1/q1.Out of this inequality, one can get forp q 2 the following inequality of Bessel-type firstly obtained in16:
n
i1
x, yi2≤ x2
n
i,j1
yi, yj2 1/2
. 3.15
The following upper bound for the Euclidean operator radius may be stated.
Theorem 3.6. IfT1, . . . , Tn∈BHn,then
w2 e
T1, . . . , Tn≤
w2 e
T∗
1T1, . . . , Tn∗Tn
1≤i /j≤n
w2T∗
jTi
1/2
≤n
i1
T∗
iTi2
1≤i /j≤n
w2T∗
jTi
1/2
⎛
⎝≤
n
i1
Ti4
1≤i /j≤n
w2T∗
jTi
1/2⎞
⎠.
3.16
Proof. Utilizing3.15, forxh,h1, yiTih, i1, . . . , n,we get
n
i1
Tih, h2≤
n
i1
Tih4
1≤i /j≤n
Tih, Tjh2
1/2
n
i1
T∗
iTih, h2
1≤i /j≤n
T∗
jTih, h2
1/2
.
3.17
Now, taking the supremum overh1,we deduce the first inequality in3.16. The second inequality follows by the propertyxifrom Introduction applied for the self-adjoint operatorsV1T1∗T1, . . . , Vn Tn∗Tn.
Remark 3.7. If in3.16we assume that the operatorsT1, . . . , Tnsatisfy the conditionTj∗Ti0
fori, j∈ {1, . . . , n},withi /j,then we get the inequality
w2
eT1, . . . , Tn
≤w2
eT1∗T1, . . . , Tn∗Tn≤
n
i1
T∗
iTi
2
1/2⎛
⎝≤
n
i1
Ti4
1/2⎞
⎠. 3.18
Remark 3.8. Forn2, the above inequality3.16provides
w2
eB, C≤we2B∗B, C∗C2w2B∗C1/2
≤B∗B2C∗C2
2w2B∗C1/2
≤B4C42w2B∗C1/2
≤ B2C2,
3.19
for anyB, C∈BH.
If in3.19 we choose B and C to be the Cartesian decompositions of the operator
A∈BH,then we get
w2A≤ 1
4AA
∗4A−A∗4
2w2A∗−AA∗A1/2
*
≤ 1
4AA
∗4A−A∗4
2w2A∗−AA∗A1/2
+
.
3.20
Here, the constant 1/4 is best possible.
Remark 3.9. If in3.19we chooseBA∗andCA, whereA∈BH,then we get
w4A≤ 1
2
AA∗2A∗A2
2
w2
A2
. 3.21
The constant 1/2 infront of the square bracket is best possible in3.21.
4. Other upper bounds
For ann-tuple of operatorsT1, . . . , Tn∈BHn, we use the notationΔTk:Tk1−Tk, k
1, . . . , n−1.
Theorem 4.1. IfT1, . . . , Tn∈BHn, n≥2,then
0≤1
nw2e
T1, . . . , Tn
−w2
1
n
n
j1
Tj ≤ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
n2−1
12 1≤maxk≤n−1w 2ΔT
k
n2−1
6n
n−1
k1
wpΔTk
1/pn−1
k1
wqΔTk
1/q
ifp >1, 1
p
1
q1;
n−1 2n
n−1
k1
wΔTk
2
.
4.1
Proof. We use the following scalar inequality that provides reverses of the Cauchy-Bunyakovsky-Schwarz result forncomplex numbers:
1
n
n
j1
zj2−
1
n
n
j1
zj 2 ≤ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
n2−1
12 1≤maxk≤n−1
Δzk2,
n2−1
6n
n−1
k1
Δzkp
1/pn−1
k1
Δzkq
1/q
if p >1, 1
p
1
q 1,
n−1 2n
n−1
k1
Δzk2
,
4.2
and the constants 1/12, 1/6, and 1/2 above cannot be replaced by smaller quantities for generaln.For complete proofs in the general setting of real or complex inner product spaces, see8, page 196–200.
Now, writing the inequality 4.2 for zj Tjh, h, where h ∈ H, h 1, j ∈
{1, . . . , n},yields
0≤1
n
n
j1
Tjh, h2−
1
n
n
j1
Tj h, h
2 ≤ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
n2−1
12 1≤maxk≤n−1
ΔTk
h, h2,
n2−1
6n
n−1
k1
ΔTk
h, hp
1/pn−1
k1
ΔTk
h, hq
1/q
if p >1, 1
p
1
q 1,
n−1 2n
n−1
k1
ΔTk
h, h
2
,
4.3
for anyh∈H, h1.
Remark 4.2. We observe that ifpq2 in4.3, then we have the inequality
0≤1
n
n
j1
Tjh, h2−
1
n
n
j1
Tj h, h
2
≤ n2−1
6n
n−1
k1
ΔTk
h, h2,
4.4
which implies, by taking the supremum overh∈H, h1,that
0≤1
nw2e
T1, . . . , Tn−w2
1
n
n
j1
Tj ≤ n 2−1
6n w
2
eT2−T1, . . . , Tn−Tn−1
. 4.5
Remark 4.3. The casen2 in all the inequalities4.1and4.5produces the simple inequality
w2
eB, C≤ 12w2BC w2B−C, 4.6
for anyB, C∈BH,that has been obtained in a different manner in5 see2.11.
The following result providing other upper bounds for the Euclidean operator radius holds.
Theorem 4.4. For anyn-tuple of operatorT1, . . . , Tn∈BHn,one has
0≤1
nw2e
T1, . . . , Tn−w2
1
n
n
j1
Tj
≤
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
inf
T∈BH
max
1≤j≤n
wTj−T n
j1
w
Tj−1n n
k1
Tk ,
inf
T∈BH
⎡ ⎣
n
j1
wqT j−T
1/q⎤
⎦
n
j1
wp
Tj−n1 n
k1
Tk
1/p
with p >1, 1
p
1
q1,
inf
T∈BH
n
j1
wTj−T max 1≤j≤n
w
Tj−n1 n
k1
Tk .
4.7
Proof. Utilizing the elementary identity for complex numbers,
1
n
n
j1
zj2−
1
n
n
j1
zj
2 1nn
j1
zj−n1 n
k1
which holds for anyz1, . . . , znandz,we can write that
1
n
n
j1
Tjh, h2−
1
n
n
j1
Tj h, h
2 n1n
j1
Tj−n1 n
k1
Tk h, h
h,Tj−Th,
4.9
for anyT1, . . . , Tn∈BHn, T a bounded linear operator onHandh∈Hwithh1.
By the H ¨older inequality, we also have
n
j1
Tj−n1 n
k1
Tk h, h
h,Tj−Th
≤ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ max
1≤j≤nTj−T
h, hn
j1
Tj−1n n
k1
Tk h, h
,
n
j1
Tj−n1 n
k1
Tk h, h
p 1/pn
j1
Tj−T
h, hq
1/q
withp>1, 1
p
1
q 1,
max
1≤j≤n
Tj−n1 n
k1
Tk h, h
n
j1
Tj−T
h, h,
4.10
for anyh∈H,h1.
On utilizing4.9and4.10, we thus have
1
n
n
k1
Tjh, h2
≤1n
n
j1
Tj h, h
2 ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ max
1≤j≤n
Tj−T
h, hn
j1
Tj−n1 n
k1
Tk h, h
, ⎛
⎝n
j1
Tj−1n n
k1
Tk h, h
p⎞ ⎠ 1/p n
j1
Tj−T
h, hq
1/q
withp>1, 1
p
1
q1,
max
1≤j≤n
Tj−n1 n
k1
Tk h, h
n
j1
Tj−T
h, h,
4.11
for anyh∈H,h1.
Taking the supremum overh, h 1 in4.11, we easily deduce the desired result
Remark 4.5. We observe that forpq2 in4.11we can also get the inequality of interest
0≤1
nw2eT1, . . . , Tn−w2
1
n
n
j1
Tj
≤ inf
T∈BH
weT1−T, . . . , Tn−Twe
T1−
1
n
n
k1
Tk, . . . , Tn−n1 n
k1
Tk .
4.12
In particular, we have
0≤1
nw2e
T1, . . . , Tn−w2
1
n
n
j1
Tj ≤we2
T1−1n n
k1
Tk, . . . , Tn−n1 n
k1
Tk . 4.13
The following particular case ofTheorem 4.4may be of interest for applications.
Corollary 4.6. Assume thatT1, . . . , Tn∈BHnare such that there exists an operatorT ∈BH
and a constantM >0such thatwTj−T≤Mfor eachj ∈ {1, . . . , n}.Then
0≤1
nw2eT1, . . . , Tn−w2
1
n
n
j1
Tj
≤M×
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
n
j1
w
Tj−n1 n
k1
Tk ,
n1/q
n
j1
wp
Tj−1n n
k1
Tk 1/p
withp >1, 1
p
1
q 1,
nmax
1≤j≤n
w
Tj−n1 n
k1
Tk .
4.14
Remark 4.7. Notice that by the H ¨older inequality, the first branch in4.14provides a tighter bound for the nonnegative quantity1/nw2
eT1, . . . , Tn−w21/n
n
j1Tjthan the other
two.
Remark 4.8. Finally, we observe that the casen2 in4.14provides the simple inequality
w2
eB, C≤ 12w2BC
√
2
2 wB−C·T∈infBHweB−T, C−T
*
≤ 1
2
w2BC w2B−C+,
4.15
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