Báo cáo hóa học: " Research Article Upper Bounds for the Euclidean Operator Radius and Applications" ppt

Dragomir,sever.dragomir@vu.edu.au 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

Volume 2008, Article ID 472146, 20 pages

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,sever.dragomir@vu.edu.au

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

Copyrightq 2008 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 the Euclidean operator radius of an n-tuple of operators T1, , T n  that are defined on a Hilbert space H; ·, · This

radius is defined by

w e

T1, , T n

: sup

h1

n



i1

T i h, h2

1/2

We can also consider the following norm and spectral radius onBH n: BH×· · ·×BH,

by setting1

1, , T n e: sup

λ1, ,λn∈Bn

1T1 · · ·  λ n T n

r eT1, , T n sup

whererT denotes the usual spectral radius of an operator T ∈ BH and B nis the closed unit ball inCn

Notice that·eis a norm onBH n:

1, , T n e ∗

1, , T∗

n e , r eT1, , T n r eT∗

1, , T∗

n

Now, if we denote byT1, , T n the square root of the norm n

i1 T i T∗

i , that is,

1, , T n : n

i1

T i T∗

i

1/2

then we can present the following result due to Popescu 1 concerning some sharp inequalities between the normsT1, , T n  and T1, , T ne

Theorem 1.1 see 1 If T1, , T n  ∈ BH n , then

1

√

n 1, , T n 1, , T n e≤ 1, , T n 1.5

where the constants 1/√n and 1 are best possible in 1.5.

Following 1, we list here some of the basic properties of the Euclidean operator radius of ann-tuple of operators T1, , T n  ∈ BH n

i w e T1, , T n   0 if and only if T1 · · ·  T n 0;

ii w e λT1, , λT n   |λ|w e T1, , T n  for any λ ∈ C;

iii w e T1 T

1, , T n  T

n  ≤ w e T1, , T n   w e T

1, , T

n;

iv w e U∗T1U, , U∗T n U  w e T1, , T n  for any unitary operator U : K → H;

v w e X∗T1X, , X∗T n X ≤ X2w e T1, , T n  for any operator X : K → H;

vi 1/2T1, , T ne ≤ w e T1, , T n  ≤ T1, , T ne;

vii r e T1, , T n  ≤ w e T1, , T n;

viii w e I ε ⊗ T1, , I ε ⊗ T n   w e T1, , T n  for any separable Hilbert space ε;

ix w eis a continuous map in the norm topology;

x w e T1, , T n  supλ1, ,λn∈Bn wλ1T1 · · ·  λ n T n;

xi 1/2√nT1, , T n  ≤ w e T1, , T n  ≤ T1, , T n and the inequalities are sharp

Due to the fact that the particular casesn  2 and n  1 are related to some classical

and new results of interest which naturally motivate the research, we recall here some facts

of significance for our further considerations

ForA ∈ BH, let wA and A denote the numerical radius and the usual operator

norm ofA, respectively It is well known that w· defines a norm on BH, and for every

A ∈ BH,

1

For other results concerning the numerical range and radius of bounded linear operators on

a Hilbert space, see2,3

In4, Kittaneh has improved 1.6 in the following manner:

1 4

∗A  AA∗ 2A ≤ 1

2

with the constants 1/4 and 1/2 as best possible.

LetC, D be a pair of bounded linear operators on H, the Euclidean operator radius

is

w e C, D : sup

x1 Cx,x2Dx,x21/2

1.8

and, as pointed out in1, w e :B2H → 0, ∞ is a norm and the following inequality holds:

√ 2 4

∗C  D∗D 1/2 ≤ w e C, D ≤ ∗C  D∗D 1/2 , 1.9

where the constants√

2/4 and 1 are best possible in 1.9

We observe that, ifC and D are self-adjoint operators, then 1.9 becomes

√ 2 4

We observe also that ifA ∈ BH and A  B  iC is the Cartesian decomposition of A, then

w2

e B, C  sup

x1 Bx,x2Cx,x2

 sup

x1 Ax,x2

 w2A.

1.11

By the inequality1.10 and since see 4

A∗A  AA∗ 2B2 C2

then we have

1 16

∗A  AA∗ 2A ≤ 1

2

We remark that the lower bound forw2A in 1.13 provided by Popescu’s inequality 1.9

is not as good as the first inequality of Kittaneh from1.7 However, the upper bounds for

w2A are 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 in5 the following result

Theorem 1.2 Let B, C : H → H be two bounded linear operators on the Hilbert space H; ·, ·.

Then

√ 2 2

wB2 C2 1/2 ≤ w e B, C≤ ∗B  C∗C 1/2

The constant√

2/2 is best possible in the sense that it cannot be replaced by a larger constant.

Corollary 1.3 For any two self-adjoint bounded linear operators B, C on H,one has

√ 2 2

2 C2 1/2

≤ w e B, C≤ 2 C2 1/2

The constant√

2/2 is sharp in 1.15.

Remark 1.4 The inequality 1.15 is better than the first inequality in 1.10 which follows from Popescu’s first inequality in1.9 It also provides, for the case that B, C are the

self-adjoint operators in the Cartesian decomposition ofA, exactly the lower bound obtained by

Kittaneh in1.7 for the numerical radius wA.

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

n  2, which is of special interest since it generates for the Cartesian decomposition of a

bounded linear operator various interesting results for the norm and the usual numerical radius, is carefully analyzed

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, , y n in an inner product space is known in the literature as

the Boas-Bellman inequalitysee 6,7, or 8, chapter 4:

n



i1

x,y i2≤ x2

⎡

⎣max

1≤i≤n i

2





1≤i / j≤n

y i , y j2

1/2⎤

for anyx ∈ H.

Obviously, if{y1, , y n} is an orthonormal family, then 2.1 becomes the classical

Bessel’s inequality

n



i1

The following result provides a natural upper bound for the Euclidean operator radius

ofn bounded linear operators.

Theorem 2.1 If T1, , T n  ∈ BH n , then

w eT1, , T n≤

⎡

⎣max

1≤i≤n i

2

1≤i / j≤n

w2

T∗

j T i

1/2⎤

⎦

1/2

Proof Utilizing the Boas-Bellman inequality for x  h, h  1 and y i  T i h, i  1, , n, we

have

n



i1

T i h, h2 ≤ max

1≤i≤n i h 2





1≤i / j≤n

T∗

j T i h, h2

1/2

Taking the supremum overh  1 and observing that

sup

h1

 max

1≤i≤n i h 2

 max

1≤i≤n i

2,

sup

h1

1≤i / j≤n T∗

j T i h, h2

1/2

≤

1≤i / j≤n

sup

h1 T∗

j T i h, h2

1/2







1≤i / j≤n

w2

T∗

j T i

1/2

,

2.5

then by2.4 we deduce the desired inequality 2.3

Remark 2.2 If T1, , T n  ∈ BH nis such thatT∗

j T i  0 for i, j ∈ {1, , n}, then from 2.3,

we have the inequality:

w eT1, , T n≤ max

We observe that a sufficient condition for T∗

j T i  0, with i / j, i, j ∈ {1, , n} to hold, is that

RangeTi  ⊥ RangeT j  for i, j ∈ {1, , n}, with i / j.

Remark 2.3 If we apply the above result for two bounded linear operators on H, B, C : H →

H, then we get the simple inequality

w2

e B, C ≤ maxB2, C2

Remark 2.4 If A : H → H is a bounded linear operator on the Hilbert space H and if we

denote by

B : A  A∗

2 , C : A − A∗

its Cartesian decomposition, then

w2

e B, C  w2A,

wB∗C wC∗B 1

4wA∗− AA  A∗ ,

2.9

and from2.7, we get the inequality

w2A ≤ 1

4



√2wA∗− AA  A∗ . 2.10

In9, the author has established the following Boas-Bellman type inequality for the vectorsx, y1, , y nin the real or complex inner product spaceH, ·, ·:

n



i1 x,y i2≤ x2

max

1≤i≤n i

2 n − 1 max

1≤i / j≤n y i , y j. 2.11

For orthonormal vectors,2.11 reduces 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 of2.1 is smaller than that of2.11 and vice versa

Now, utilizing the inequality2.11 and making use of the same argument from the proof ofTheorem 2.1, we can state the following result as well

Theorem 2.5 If T1, , T n  ∈ BH n , then

w e

T1, , T n

≤

 max

1≤i≤n i

2 n − 1 max

1≤i / j≤n wT∗

j T i1/2

If in2.12 one assumes that T∗

j T i  0 for each i, j ∈ {1, , n} with i / j, then one gets

the result from2.6

Remark 2.6 We observe that, for n  2, we get from 2.12 a better result than 2.7, namely,

w2

e B, C ≤ maxB2, C2

whereB, C are arbitrary linear bounded operators on H The inequality 2.13 is sharp This follows from the fact that forB  C  A ∈ BH, A a normal operator, we have

w2

e A, A  2w2A  2A2,

and we obtain in 2.13 the same quantity in both sides The inequality 2.13 has been obtained in5,12.23 on utilizing a different argument

Also, for the operatorA : H → H, we can obtain from 2.13 the following inequality:

w2A ≤ 1

4



 wA∗− AA  A∗ , 2.15

which is better than2.10 The constant 1/4 in 2.15 is sharp The case of equality in 2.15 follows, for instance, ifA is assumed to be self-adjoint.

Remark 2.7 If in2.13 we choose C  A, B  A∗, A ∈ BH, and take into account that

w2

e

then we get the inequality

w2A ≤ 1

2

A2 wA2 ≤ A2

for anyA ∈ BH The constant 1/2 is sharp.

Note that this inequality has been obtained in10 by the use of a different argument based on the Buzano inequality11

A different approach is incorporated in the following result

Theorem 2.8 If T1, , T n  ∈ BH n , then

w2

e

T1, , T n

≤ max

1≤i≤nwT i

i1

T∗

i T i



1≤i / j≤n

wT∗

j T i1/2

Proof We use the following Boas-Bellman-type inequality obtained in9 see also 8, page 132:

n



i1

x,y i2≤ x max

1≤i≤nx,y in

2 

1≤i / j≤n

y i , y j1/2 , 2.19

wherex, y1, , y nare arbitrary vectors in the inner product spaceH; ·, ·.

Now, forx  h, h  1, y i  T i h, i  1, , n, we get from 2.19 that

n



i1 T i h, h2≤ max

1≤i≤nT i h, hn

h 2 

1≤i / j≤nT i h, T j h1/2 . 2.20 Observe that

n



h 2n

i1



T i h, T i hn

i1



T∗

i T i

h, h

n

i1



T∗

i T i

h, h

forh ∈ H, h  1.

Therefore, on taking the supremum in 2.20 and noticing that wn i1 T∗

i T i 

n

i1 T∗

i T i , we get the desired result 2.18

Remark 2.9 If T1, , T n  ∈ BH n satisfies the condition that T∗

i T j  0 for each i, j ∈ {1, , n} with i / j, then from 2.18 we get

w2

e

T1, , T n≤ max

1≤i≤nwT i· n

i1

T∗

i T i

1/2

Remark 2.10 If we apply Theorem 2.8 to n  2, then we can state the following simple

inequality:

w2

for any bounded linear operatorsB, C ∈ BH.

Moreover, ifB and C are chosen as the Cartesian decomposition of the bounded linear

operatorA ∈ BH, then we can state that

w2A ≤ 1

2max

2

1

2wA  A∗

A − A∗ 1/2

2.24

The constant 1/2 is best possible in 2.24 The equality case is obtained if A is a self-adjoint

operator onH.

If we choose in2.23, C  A, B  A∗, A ∈ BH, then we get

w2A ≤ 1

≤ A2

The constant 1/2 is best possible in 2.25

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 the Bombieri inequalitysee 12, 13, page 394, or 8, page 134

n



i1 x,y i2 ≤ x2max

1≤i≤n

n

wherex, y1, , y nare 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 to 3.1 which, 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 If T1, , T n  ∈ BH n , then

w2

e

T1, , T n≤ max

1≤i≤n

 n



j1

wT∗

j T i



Proof Follows by Bombieri’s inequality applied for x  h, h  1 and y i  T i h, i  1, , n.

Then taking the supremum overh  1 and utilizing its properties, we easily deduce the

desired inequality3.2

Remark 3.2 If we apply the above theorem for two operators B and C, then we get

w2

e B, C ≤ maxwB∗B wC∗B, wB∗C wC∗C

 maxB2 wB∗C, wB∗C C2

 maxB2, C2

 wB∗C,

3.3

which is exactly the inequality2.13 that 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 If T1, , T n  ∈ BH n , then

w2

e

T1, , T n≤

⎧

⎪

⎪

D w,

E w,

F w,

3.4

meaning that the left side is less than each of the quantities in the right side, where

D w:

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎩

max

1≤k≤n



wT k

n

i,j1

wT∗

j T i

1/2

,

max

1≤k≤n



wT k1/2n

i1

wT i r

1/2r⎡

⎣n

i1

 n



j1

wT∗

j T i s⎤⎦1/2s

,

wherer, s > 1, 1r 1s  1,

max

1≤k≤n



wT k1/2

n

i1

wT i

1/2 max

1≤i≤n

n

j1

wT∗

j T i

1/2

,

E w:

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎩

n

k1

wT k p

1/2p max

1≤k≤n



wT k1/2⎡⎣n

i1

 n



j1

wT∗

j T i q⎤⎦1/2q

,

wherep > 1, 1p 1q  1,

n

k1

wT k p

1/2pn

i1

wT i t

1/2t⎡

⎣n

i1

n

j1

wT∗

j T i q

u/q⎤

⎦

1/2u

,

wherep > 1, 1p 1q  1, t > 1, 1t u1  1,

n



k1

wT k p

1/2p n



i1

wT i

1/2 max

1≤i≤n

⎧

⎨

⎩

 n



j1

wT∗

j T i q

1/2p⎫⎬

⎭ ,

wherep > 1, 1p 1q  1,

F w:

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

n



k1

wT k

1/2 max

1≤i≤n



wT i1/2n

i1

 max

1≤j≤n



wT∗

j T i1/2

,

n

k1

wT k

1/2n

i1

wT i m

1/2m n

i1

 max

1≤j≤n

wT∗

j T i l

1/2l

,

wherem > 1, m1  1l  1,

n



k1

wT k

· max

1≤i,j≤n



wT∗

j T i1/2

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, , y nin the inner product spaceH, ·, · and

the scalarsc1, , c n∈ K:







n



i1

c ix, y i





2

≤ x2×

⎧

⎪

⎪

D, E, F,

3.6

where

D :

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

max

1≤k≤nc k2



i,j1

y i , y j ,

max

1≤k≤nc kn

i1

c ir

1/r⎡

⎣n

i1

 n



j1

y i , y j s⎤⎦

1/s

, where r, s > 1, 1r 1s  1,

max

1≤k≤nc kn

i1 c i max

1≤i≤n

n

j1 y i , y j,

E :

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎩

 n



k1

c kp

1/p max

1≤i≤n|c i|

⎡

⎣n

i1

n



j1

y i , y j q⎤⎦

1/q

, wherep > 1, 1pq1  1,

n

k1

c kp

1/pn

i1 c it

1/t⎡

⎣n

i1

n

j1 y i , y jq

u/q⎤

⎦

1/u

, where p > 1, p1 1q  1,

t > 1,1t u1  1,

 n



k1

c kp

1/p n



i1

c i max

1≤i≤n

⎧

⎨

⎩

 n



j1

y i , y jq

1/q⎫⎬

⎭ , wherep > 1,

1

p

1

q  1,

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...

Remark 3.2 If we apply the above theorem for two operators B and C, then we get

w2... norm and the usual numerical radius, is carefully analyzed

2 Upper bounds via the Boas-Bellman-type