Journal of Inequalities in Pure and
Applied Mathematics
NORM INEQUALITIES FOR SEQUENCES OF OPERATORS RELATED
TO THE SCHWARZ INEQUALITY
volume 7, issue 3, article 97,
2006.
SEVER S. DRAGOMIR
School of Computer Science and Mathematics
Victoria University
PO Box 14428, MCMC 8001
VIC, Australia.
EMail: sever@csm.vu.edu.au
URL: http://rgmia.vu.edu.au/dragomir
Received 21 March, 2006;
accepted 12 July, 2006.
Communicated by: F. Zhang
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
088-06
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Abstract
Some norm inequalities for sequences of linear operators defined on Hilbert
spaces that are related to the classical Schwarz inequality are given. Applications for vector inequalities are also provided.
2000 Mathematics Subject Classification: Primary 47A05, 47A12.
Key words: Bounded linear operators, Hilbert spaces, Schwarz inequality, Cartesian
decomposition of operators.
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
Some General Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3
Other Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
4
Vector Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
References
Norm Inequalities for
Sequences of Operators
Related to the Schwarz
Inequality
Sever S. Dragomir
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1.
Introduction
Let (H; h·, ·i) be a real or complex Hilbert space and B (H) the Banach algebra
of all bounded linear operators that map H into H.
In many estimates one needs to use upper bounds for the norm of the linear
combination of bounded linear operators A1 , . . . , An with the scalars α1 , . . . , αn ,
where separate information for scalars and operators are provided. In this situation, the classical approach is to use a Hölder type inequality as stated below
!
n
n
X
X
αi Ai ≤
|αi | kAi k
i=1
i=1
P

max {|αi |} ni=1 kAi k ;


1≤i≤n


 P
1 P
1
( ni=1 |αi |p ) p ( ni=1 kAi kq ) q if p > 1, p1 + 1q = 1;
≤



P

 max {kAi k} n |αi | .
i=1
1≤i≤n
Notice that, the case when p = q = 2, which provides the Cauchy-BunyakovskySchwarz inequality
! 21
! 12
n
n
n
X
X
X
(1.1)
αi Ai ≤
|αi |2
kAi k2
i=1
i=1
Norm Inequalities for
Sequences of Operators
Related to the Schwarz
Inequality
Sever S. Dragomir
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is of special interest and of larger utility.
In the previous paper [1], in order to improve (1.1), we have established the
following norm inequality for the operators A1 , . . . , An ∈ B (H) and scalars
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α1 , . . . , αn ∈ K:
(1.2)
2
n
X
αi Ai i=1
+

P
max |αi |2 ni=1 kAi k2 ;



1≤i≤n



1
1

 Pn
2p p Pn
2q q
|α
|
kA
k
i
i
i=1
i=1
≤
1
1
if
p
>
1,
+
= 1;

p
q




P


 ni=1 |αi |2 max kAi k2
1≤i≤n
P
max {|αi | |αj |} 1≤i6=j≤n Ai A∗j ;



1≤i6=j≤n




hP
i1 
1
Pn

n
r 2
2r r P

Ai A∗j s s
 ( i=1 |αi | ) − i=1 |αi |
1≤i6=j≤n


if r > 1, 1r +




hP
i


P

 ( ni=1 |αi |)2 − ni=1 |αi |2 max Ai A∗j ,
1
s
Norm Inequalities for
Sequences of Operators
Related to the Schwarz
Inequality
Sever S. Dragomir
= 1;
1≤i6=j≤n
where (1.2) should be seen as all the 9 possible configurations.
Some particular inequalities of interest that can be obtained from (1.2) and
provide alternative bounds for the classical Cauchy-Bunyakovsky-Schwarz (CBS)
inequality are the following [1]:
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(1.3)
n
X
αi Ai ≤ max |αi |
1≤i≤n
i=1
n
X
i,j=1
! 12
Ai A∗j ,
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(1.4)
2
n
n
X
X
2
2
∗
αi Ai ≤
|αi | max kAi k + (n − 1) max Ai Aj ,
1≤i≤n
1≤i6=j≤n
i=1
(1.5)
i=1
! 21 

2
n
n
X
X
αi Ai ≤
|αi |2  max kAi k2 +
1≤i≤n
i=1
X Ai A∗j 2
i=1

1≤i6=j≤n
Norm Inequalities for
Sequences of Operators
Related to the Schwarz
Inequality
and
(1.6)
2
n
X
αi Ai ≤
i=1
n
X
! p1 
|αi |2p
n
X

! 1q
kAi k2q
Sever S. Dragomir
i=1
i=1
! 1q 
+ (n − 1)
1
p
X Ai A∗j q
,
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1≤i6=j≤n
where p > 1,
(1.7)
1
p
+
1
q
= 1. In particular, for p = q = 2, we have from (1.6)
n
2
X
αi Ai ≤
i=1
JJ
J
n
X
i=1
! 21 
4
|αi |
n
X

! 12
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kAi k
Close
i=1
! 12 
+ (n − 1)
1
2
X Ai A∗j 2
.
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The aim of thePpresent paper is to establish other upper bounds of interest
for the quantity k ni=1 αi Ai k , where, as above, α1 , . . . , αn are real or complex
numbers, while A1 , . . . , An are bounded linear operators on the Hilbert space
(H; h·, ·i) . These are compared with the (CBS) inequality (1.1) and shown that
some times they are better. Applications for vector inequalities are also given.
Norm Inequalities for
Sequences of Operators
Related to the Schwarz
Inequality
Sever S. Dragomir
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2.
Some General Results
The following result containing 9 different inequalities may be stated:
Theorem 2.1. Let α1 , . . . , αn ∈ K and A1 , . . . , An ∈ B (H) . Then
2 
n
 A
X
B
(2.1)
αi Ai ≤

i=1
C
where
A :=

2 Pn
Ai A∗j ,

max
|α
k|

i,j=1

1≤k≤n




P 
s 1s
1

n
 max |αk | (Pn |αi |r ) r Pn
Ai A∗j ,
i=1
i=1
j=1
Norm Inequalities for
Sequences of Operators
Related to the Schwarz
Inequality
Sever S. Dragomir
1≤k≤n

where r > 1, 1r +





P 
Pn

n

Ai A∗j ,
 max |αk | i=1 |αi | max
j=1
1≤k≤n
1
s
= 1;
1≤i≤n

P 1
Pn

p p1 Pn
n
∗ q q

max
|α
|
(
|α
|
)
A
A

i
k
i j
k=1
i=1
j=1

1≤i≤n





P uq u1


q
n
 Pn |α |t 1t (Pn |α |p ) p1 Pn
∗
i
k
i=1
k=1
i=1
j=1 Ai Aj
(2.2) B :=


where t > 1, 1t + u1 = 1;







q 1q
1
P
P
P

n
p
n
n
∗

,
 i=1 |αi | ( k=1 |αk | ) p max
j=1 Ai Aj
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1≤i≤n
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+ 1q = 1 and

Pn
Pn
Ai A∗j ,
max
|α
|
|α
|
max

i
k
k=1
i=1 1≤j≤n

1≤i≤n




"

l # 1l


1 P
P
P

n
 ( n |α |m ) m n |α |
max Ai A∗j ,
k
i
k=1
i=1 1≤j≤n
i=1
C :=



where m, l > 1, m1 + 1l = 1;





P


 ( nk=1 |αk |)2 max Ai A∗j .
for p > 1,
1
p
1≤i,j≤n
Proof. We observe, in the operator partial order of B(H), we have that
! n
!∗
n
X
X
(2.3)
0≤
αi Ai
αi Ai
=
i=1
n
X
n
X
i=1
j=1
αj A∗j =
n X
n
X
αi αj Ai A∗j .
i=1 j=1
Taking the norm in (2.3) and noticing that kU U ∗ k = kU k2 for any U ∈ B (H) ,
we have:
n
2 n n
X
X X
αi Ai = αi αj Ai A∗j i=1
≤
i=1 j=1
n X
n
X
|αi | |αj | Ai A∗j i=1 j=1
Sever S. Dragomir
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i=1
αi Ai
Norm Inequalities for
Sequences of Operators
Related to the Schwarz
Inequality
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=
n
X
i=1
|αi |
n
X
!
|αj | Ai A∗j =: M.
j=1
Utilising Hölder’s discrete inequality we have that
n
X
|αj | Ai A∗j j=1

Pn Ai A∗j ,
max
|α
|

k
j=1

1≤k≤n



 P
q 1q
n
p p1 Pn
∗
≤
where p > 1,
( k=1 |αk | )
j=1 Ai Aj



P


 nk=1 |αk | max Ai A∗j ,
1
p
+
1
q
= 1;
Norm Inequalities for
Sequences of Operators
Related to the Schwarz
Inequality
Sever S. Dragomir
1≤j≤n
for any i ∈ {1, . . . , n} .
This provides the following inequalities:
P 
Pn
n
∗

max |αk | i=1 |αi |
=: M1

j=1 Ai Aj

1≤k≤n





1
1

 (Pn |αk |p ) p Pn |αi | Pn Ai A∗ q q := Mp
j
k=1
i=1
j=1
M≤
where p > 1, p1 + 1q = 1;







Pn
Pn


∗
 k=1 |αk | i=1 |αi | max Ai Aj
:= M∞ .
1≤j≤n
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Utilising Hölder’s inequality for r, s > 1,
n
X
i=1
1
r
+
1
s
= 1, we have:
!
n
X
Ai A∗j |αi |
j=1

Pn Ai A∗j max
|α
|

i
i,j=1

1≤i≤n




P h

s i 1s

n
 (Pn |αi |r ) r1 Pn
Ai A∗j ,
i=1
i=1
j=1
≤
1
1

where r > 1, r + s = 1;




P 

P
n

Ai A∗j ,
 ni=1 |αi | max
j=1
1≤i≤n
and thus we can state that

2 Pn
Ai A∗j ;

max
|α
k|

i,j=1

1≤k≤n




P 
s 1s
1

n
 max |αk | (Pn |αi |r ) r Pn
Ai A∗j ,
i=1
i=1
j=1
1≤k≤n
M1 ≤

where r > 1, 1r + 1s = 1;





P 
Pn

n
∗

,
 max |αk | i=1 |αi | max
j=1 Ai Aj
1≤k≤n
1≤i≤n
and the first part of the theorem is proved.
Norm Inequalities for
Sequences of Operators
Related to the Schwarz
Inequality
Sever S. Dragomir
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By Hölder’s inequality we can also have that (for p > 1,
Mp ≤
1
p
+
1
q
= 1)

1
Pn Pn 
∗ q q

max
|α
|
A
A
;

i
i j
i=1
j=1

1≤i≤n




1

1

!
q uq u
1t Pn Pn Pn

n
p
t

∗
X
,
i=1 |αi |
i=1
j=1 Ai Aj
|αk |p
×


where t > 1, 1t + u1 = 1;
k=1







q 1q
P
P

n
n
∗

,
 i=1 |αi | max
j=1 Ai Aj
1≤i≤n
and the second part of (2.1) is proved.
Finally, we may state that

Pn
Ai A∗j 
max
|α
|
max
i

i=1 1≤j≤n
 1≤i≤n



"

l # 1l


1

P
P
n
n
 ( n |α |m ) m
X
max Ai A∗j i
i=1
i=1 1≤j≤n
M∞ ≤
|αk | ×


k=1

where m, l > 1, m1 + 1l = 1;





Pn


 i=1 |αi | max Ai A∗j ,
1≤i,j≤n
giving the last part of (2.1).
Remark 1. It is obvious that out of (2.1) one can obtain various particular
inequalities. For instance, the choice t = 2, p = 2 (therefore u = q = 2) in the
Norm Inequalities for
Sequences of Operators
Related to the Schwarz
Inequality
Sever S. Dragomir
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B−branch of (2.2) gives:
(2.4)
2
n
n
X
X
αi Ai ≤
|αi |2
i=1
=
n
X
Ai A∗j 2
i=1
i,j=1
n
X
n
X
|αi |2
i=1
! 12
! 12
X Ai A∗j .
kAi k4 +
i=1
1≤i6=j≤n
If we consider now the usual Cauchy-Bunyakovsky-Schwarz (CBS) inequality
2
n
n
n
X
X
X
(2.5)
αi Ai ≤
|αi |2
kAi k2 ,
i=1
i=1
i,j=1
Sever S. Dragomir
i=1
and observe that
n
X
Ai A∗j 2
Norm Inequalities for
Sequences of Operators
Related to the Schwarz
Inequality
Title Page
!
1
2
≤
n
X
i,j=1
!
2
kAi k A∗j 2
1
2
=
n
X
Contents
2
kAi k ,
i=1
then we can conclude that (2.4) is a refinement of the (CBS) inequality (2.5).
Corollary 2.2. Let α1 , . . . , αn ∈ K and A1 , . . . , An ∈ B (H) so that Ai A∗j = 0
with i 6= j. Then
2  Ã
n
X

(2.6)
α
A
≤
B̃
i i

i=1
C̃
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where
à :=

P

max |αk |2 ni=1 kAi k2 ;


1≤k≤n




1
1

 max |αk | (Pn |αi |r ) r Pn kAi k2s s ,
i=1
i=1
1≤k≤n

where r > 1, 1r + 1s = 1;





P


 max |αk | ni=1 |αi | max kAi k2 ,
1≤i≤n
1≤k≤n

Pn
p p1 Pn

kAi k2 ;
max
|α
|
(
|α
|
)

i
k
i=1
k=1

1≤i≤n


1
1

Pn

t t Pn
p p1 Pn
2u u
 v
(
|α
|
)
kA
k
,
|α
|
k
i
i
k=1
i=1
i=1
1
1
B̃ :=
where t > 1, t + u = 1;





1
P
P


 ni=1 |αi | ( nk=1 |αk |p ) p max kAi k2 ,
1≤i≤n
where p > 1 and
Norm Inequalities for
Sequences of Operators
Related to the Schwarz
Inequality
Sever S. Dragomir
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
P
P

max |αi | nk=1 |αk | ni=1 kAi k2 ;


1≤i≤n



1


 (Pn |α |m ) m1 Pn |α | Pn kA k2l l ,
k
i
i
i=1
k=1
i=1
C̃ :=
1

where m, l > 1, m + 1l = 1;





P


 ( nk=1 |αk |)2 max kAi k2 .
1≤i,j≤n
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3.
Other Results
A different approach is embodied in the following theorem:
Theorem 3.1. If α1 , . . . , αn ∈ K and A1 , . . . , An ∈ B (H) , then
2
n
n
n
X
X
X
2
Ai A∗j (3.1)
α
A
≤
|α
|
i i
i
i=1
i=1
j=1
hP  P
i
n
2
n
Ai A∗j ;

|α
|
max
i

i=1
j=1

1≤i≤n




P 1 h

q i 1q
 Pn
n
2p p Pn
∗
i=1
j=1 Ai Aj
i=1 |αi |
≤


where p > 1, p1 + 1q = 1;





P

 max |αi |2 ni,j=1 Ai A∗j .
1≤i≤n
Proof. From the proof of Theorem 2.1 we have that
2
n
n X
n
X
X
αi Ai ≤
|αi | |αj | Ai A∗j .
i=1
i=1 j=1
1
|αi |2 + |αj |2 ,
2
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|αi | |αj | ≤
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i, j ∈ {1, . . . , n} ,
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we have
n X
n
1X
2
∗
|αi | |αj | Ai Aj ≤
|αi | + |αj |2 Ai A∗j 2 i=1 j=1
j=1
n X
n
X
i=1
=
=
n
n
1 XX 2
|αi | Ai A∗j + |αj |2 Ai A∗j 2 i=1 j=1
n X
n
X
|αi |2 Ai A∗j ,
i=1 j=1
which proves the first inequality in (3.1).
The second part follows by Hölder’s inequality and the details are omitted.
Remark 2. If in (3.1) we choose α1 = · · · = αn = 1, then we get
n
X
Ai ≤
i=1
n
X
i=1
kAi k2 +
! 12
n
n
X
X
Ai A∗j ≤
kAi k ,
1≤i6=j≤n
i=1
which is a refinement for the generalised triangle inequality.
The following corollary may be stated:
Corollary 3.2. If A1 , . . . , An ∈ B (H) are such that Ai A∗j = 0 for i 6= j,
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i, j ∈ {1, . . . , n} , then
n
2
n
X
X
(3.2)
αi Ai ≤
|αi |2 kAi k2
i=1
i=1

Pn
2

max kAi k2 ;

i=1 |αi | 1≤i≤n




i1
h


 Pn |α |2p p1 Pn kA k2q q
i
i
i=1
j=1
≤
1

where p > 1, p + 1q = 1;





P


 max |αi |2 ni=1 kAi k2 .
1≤i≤n
Finally, the following result may be stated as well:
Theorem 3.3. If α1 , . . . , αn ∈ K and A1 , . . . , An ∈ B (H) , then

P

max |αi |2 ni,j=1 Ai A∗j ;


1≤i≤n



n
2 
1
 Pn
p p2 Pn
X

∗ q q
( i=1 |αi | )
i,j=1 Ai Aj
(3.3)
αi Ai ≤

where p > 1, p1 + 1q = 1;

i=1




P


 ( ni=1 |αi |)2 max Ai A∗j .
1≤i,j≤n
Proof. We know that
2
n
n X
n
X
X
αi Ai ≤
|αi | |αj | Ai A∗j =: P.
i=1
i=1 j=1
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Firstly, we obviously have that
P ≤ max {|αi | |αj |}
1≤i,j≤n
n
n
X
X
Ai A∗j = max |αi |2
Ai A∗j .
1≤i≤n
i,j=1
i,j=1
Secondly, by the Hölder inequality for double sums, we obtain
"
P ≤
n
X
# p1
n
X
Ai A∗j q
(|αi | |αj |)p
i,j=1
=
=
i,j=1
n
X
n
X
p
|αi |
|αj |p
i=1
j=1
n
X
! p1
n
X
Ai A∗j q
i,j=1
! p2
n
X
Ai A∗j q
|αi |p
i=1
1
p
! 1q
,
where p > 1, + = 1.
Finally, we have
n
X
∗
P ≤ max
Ai Aj
|αi | |αj |
=
i=1
i,j=1
!2
|αi |
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1
q
n
X
Sever S. Dragomir
! 1q
i,j=1
1≤i,j≤n
! 1q
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and the theorem is proved.
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Corollary 3.4. If α1 , . . . , αn ∈ K and A1 , . . . , An ∈ B (H) are such that
Ai A∗j = 0 for i, j ∈ {1, . . . , n} with i 6= j, then

P
max |αi |2 ni=1 kAi k2 ;



 1≤i≤n
n
2 


X
 (Pn |α |p ) p2 Pn kA k2q 1q ,
i
i
i=1
i=1
(3.4)
αi Ai ≤
1
1
where
p
>
1,
+
= 1;

p
q

i=1



P


 ( ni=1 |αi |)2 max kAi k2 .
1≤i≤n
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4.
Vector Inequalities
As pointed out in our previous paper [1], the operator inequalities obtained
above may provide various vector inequalities of interest.
If by M (α, A) we denote
P any of 2the bounds provided by (2.1), (2.4), (3.1)
or (3.3) for the quantity k ni=1 αi Ai k , then we may state the following general
fact:
Under the assumptions of Theorem 2.1, we have:
2
n
X
2
α
A
x
i i ≤ kxk M (α, A) .
(4.1)
i=1
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for any x ∈ H and
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(4.2)
2
n
X
αi hAi x, yi ≤ kxk2 kyk2 M (α, A) .
i=1
for any x, y ∈ H, respectively.
The proof follows by the Schwarz inequality in the Hilbert space (H, h·, ·i),
see for instance [1], and the details are omitted.
Now, we consider the non zero vectors y1 , . . . , yn ∈ H. Define the operators
[1]
hx, yi i
Ai : H → H, Ai x =
· yi , i ∈ {1, . . . , n} .
kyi k
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Since
kAi k = kyi k , i ∈ {1, . . . , n}
then Ai are bounded linear operators in H. Also, since
|hx, yi i|2
hAi x, xi =
≥ 0, x ∈ H, i ∈ {1, . . . , n}
kyi k
and
hx, yi i hyi , zi
hAi x, zi =
,
kyi k
hx, yi i hyi , zi
,
hx, Ai zi =
kyi k
giving
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hAi x, zi = hx, Ai zi , x, z ∈ H, i ∈ {1, . . . , n} ,
we may conclude that Ai (i = 1, . . . , n) are positive self-adjoint operators on
H.
Since, for any x ∈ H, one has
k(Ai Aj ) (x)k =
|hx, yj i| |hyj , yi i|
, i, j ∈ {1, . . . , n} ,
kyj k
then we deduce that
kAi Aj k = |(yi , yj )| ; i, j ∈ {1, . . . , n} .
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If (yi )i=1,n is an orthonormal family on H, then kAi k = 1 and Ai Aj = 0 for
i, j ∈ {1, . . . , n} , i 6= j.
Now, utilising, for instance, the inequalities in Theorem 3.1 we may state
that:
(4.3)
2
n
X
hx, yi i yi αi
kyi k i=1
≤ kxk
2
n
X
i=1
2
|αi |
n
X
|hyi , yj i|
j=1
 P
hP
i
n
2
n

|α
|
max
|hy
,
y
i|
;

i
i j
i=1
j=1

1≤i≤n




P
q i 1q
 P
1 h
n
n
2p p Pn
2
≤ kxk ×
i=1
j=1 |(yi , yj )|
i=1 |αi |



where p > 1, p1 + 1q = 1;


P


 v max |αi |2 ni,j=1 |hyi , yj i| .
1≤i≤n
for any x, y1 , . . . , yn ∈ H and αn , . . . , αn ∈ K.
ii
The proof follows on choosing Ai = h·,y
y in Theorem 3.1 and taking into
kyi k i
account that kAi k = kyi k ,
Ai A∗j = |hyi , yj i| , i, j ∈ {1, . . . , n} .
We omit the details.
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The choice αi = kyi k (i = 1, . . . , n) will produce some interesting bounds
for the norm of the Fourier series
n
X
hx, yi i yi .
i=1
Notice that the vectors yi (i = 1, . . . , n) are not necessarily orthonormal.
Similar inequalities may be stated if one uses the other two main theorems.
For the sake of brevity, they will not be stated here.
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References
[1] S.S. DRAGOMIR, Some Schwarz type inequalities for sequences of operators in Hilbert spaces, Bull. Austral. Math. Soc., 73 (2006), 17–26.
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