Volume 8 (2007), Issue 2, Article 52, 22 pp.
THE HYPO-EUCLIDEAN NORM OF AN n−TUPLE OF VECTORS IN INNER
PRODUCT SPACES AND APPLICATIONS
S.S. DRAGOMIR
S CHOOL OF C OMPUTER S CIENCE AND M ATHEMATICS
V ICTORIA U NIVERSITY
PO B OX 14428, M ELBOURNE C ITY
8001, VIC, AUSTRALIA
sever.dragomir@vu.edu.au
URL: http://rgmia.vu.edu.au/dragomir
Received 19 March, 2007; accepted 28 April, 2007
Communicated by J.M. Rassias
A BSTRACT. The concept of hypo-Euclidean norm for an n−tuple of vectors in inner product
spaces is introduced. Its fundamental properties are established. Upper bounds via the BoasBellman [1]-[3] and Bombieri [2] type inequalities are provided. Applications for n−tuples of
bounded linear operators defined on Hilbert spaces are also given.
Key words and phrases: Inner product spaces, Norms, Bessel’s inequality, Boas-Bellman and Bombieri inequalities, Bounded
linear operators, Numerical radius.
2000 Mathematics Subject Classification. Primary 47C05, 47C10; Secondary 47A12.
1. I NTRODUCTION
Let (E, k·k) be a normed linear space over the real or complex number field K. On Kn
endowed with the canonical linear structure we consider a norm k·kn and the unit ball
B (k·kn ) := {λ = (λ1 , . . . , λn ) ∈ Kn | kλkn ≤ 1} .
As an example of such norms we should mention the usual p−norms
(
max {|λ1 | , . . . , |λn |} if p = ∞;
(1.1)
kλkn,p :=
1
P
if p ∈ [1, ∞).
( nk=1 |λk |p ) p
The Euclidean norm is obtained for p = 2, i.e.,
kλkn,2 =
n
X
k=1
084-07
! 12
|λk |2
.
2
S.S. D RAGOMIR
It is well known that on E n := E × · · · × E endowed with the canonical linear structure we can
define the following p−norms:
(
max {kx1 k , . . . , kxn k} if p = ∞;
(1.2)
kXkn,p :=
1
P
if p ∈ [1, ∞);
( nk=1 kxk kp ) p
where X = (x1 , . . . , xn ) ∈ E n .
For a given norm k·kn on Kn we define the functional k·kh,n : E n → [0, ∞) given by
n
X
(1.3)
kXkh,n :=
sup
λj x j ,
(λ1 ,...,λn )∈B (k·k )
n
j=1
where X = (x1 , . . . , xn ) ∈ E n .
It is easy to see that:
(i) kXkh,n ≥ 0 for any X ∈ E n ;
(ii) kX + Y kh,n ≤ kXkh,n + kY kh,n for any X, Y ∈ E n ;
(iii) kαXkh,n = |α| kXkh,n for each α ∈ K and X ∈ E n ;
and therefore k·kh,n is a semi-norm on E n . This will be called the hypo-semi-norm generated
by the norm k·kn on X n .
P
We observe that kXkh,n = 0 if and only if nj=1 λj xj = 0 for any (λ1 , . . . , λn ) ∈ B (k·kn ) .
If there exists λ01 , . . . , λ0n 6= 0 such that (λ01 , 0, . . . , 0) , (0, λ02 , . . . , 0) , . . . , (0, 0, . . . , λ0n ) ∈
B (k·kn ) then the semi-norm generated by k·kn is a norm on E n .
If by Bn,p with p ∈ [1, ∞] we denote the balls generated by the p−norms k·kn,p on Kn , then
we can obtain the following hypo-p-norms on X n :
n
X
(1.4)
kXkh,n,p :=
sup
λj x j ,
(λ1 ,...,λn )∈Bn,p j=1
with p ∈ [1, ∞] .
For p = 2, we have the Euclidean ball in Kn , which we denote by Bn ,
)
(
n
X
Bn = λ = (λ1 , . . . , λn ) ∈ Kn |λi |2 ≤ 1
i=1
n
that generates the hypo-Euclidean norm on E , i.e.,
(1.5)
kXkh,e
n
X
λj x j .
:=
sup
(λ1 ,...,λn )∈Bn
j=1
Moreover, if E = H, H is a Hilbert space over K, then the hypo-Euclidean norm on H n will
be denoted simply by
n
X
(1.6)
k(x1 , . . . , xn )ke :=
sup
λ
x
j j ,
(λ1 ,...,λn )∈Bn
j=1
and its properties will be extensively studied in the present paper.
Both the notation in (1.6) and the necessity of investigating its main properties are motivated
by the recent work of G. Popescu [9] who introduced a similar norm on the Cartesian product
of Banach algebra B (H) of all bounded linear operators on H and used it to investigate various properties of n−tuple of operators in Multivariable Operator Theory. The study is also
motivated by the fact that the hypo-Euclidean norm is closely related to the quadratic form
J. Inequal. Pure and Appl. Math., 8(2) (2007), Art. 52, 22 pp.
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H YPO -E UCLIDEAN N ORM
3
Pn
|hx, xj i|2 (see the representation Theorem 2.2) that plays a key role in many problems
arising in the Theory of Fourier expansions in Hilbert spaces.
The paper is structured as follows: in Section 2 we establish the equivalence of the hypoEuclidean norm with the usual Euclidean norm on H n , provide a representation result and
obtain some lower bounds for it. In Section 3, on utilising the classical results of Boas-Bellman
and Bombieri as well as some recent similar results obtained by the author, we give various
upper bounds for the hypo-Euclidean norm. These are complemented in Section 4 with other
inequalities between p−norms and the hypo-Euclidean norm. Section 5 is devoted to the presentation of some conditional reverse inequalities between the hypo-Euclidean norm and the
norm of the sum of the vectors involved. In Section 6, the natural connection between the
hypo-Euclidean norm and the operator norm k(·, . . . , ·)ke introduced by Popescu in [9] is investigated. A representation result is obtained and some applications for operator inequalities
are pointed out. Finally, in Section 7, a new norm for operators is introduced and some natural
inequalities are obtained.
j=1
2. F UNDAMENTAL P ROPERTIES
Let (H; h·, ·i) be a Hilbert space over K and n ∈ N, n ≥ 1. In the Cartesian product H n :=
H × · · · × H, for the n−tuples of vectors X = (x1 , . . . , xn ), Y = (y1 , . . . , yn ) ∈ H n , we can
define the inner product h·, ·i by
(2.1)
hX, Y i :=
n
X
X, Y ∈ H n ,
hxj , yj i ,
j=1
which generates the Euclidean norm k·k2 on H n , i.e.,
(2.2)
kXk2 :=
n
X
! 21
kxj k2
X ∈ H n.
,
j=1
The following result connects the usual Euclidean norm k·k with the hypo-Euclidean norm
k·ke .
Theorem 2.1. For any X ∈ H n we have the inequalities
1
kXk2 ≥ kXke ≥ √ kXk2 ,
n
(2.3)
i.e., k·k2 and k·ke are equivalent norms on H n .
Proof. By the Cauchy-Bunyakovsky-Schwarz inequality we have
(2.4)
n
X
λj x j ≤
j=1
n
X
j=1
! 12
|λj |2
n
X
! 12
kxj k2
j=1
for any (λ1 , . . . , λn ) ∈ Kn . Taking the supremum over (λ1 , . . . , λn ) ∈ Bn in (2.4) we obtain the
first inequality in (2.3).
If by σ we denote the rotation-invariant
normalised
Pn
positive Borel measure on the unit sphere
2
n
∂Bn ∂Bn = (λ1 , . . . , λn ) ∈ K
i=1 |λi | = 1 whose existence and properties have been
J. Inequal. Pure and Appl. Math., 8(2) (2007), Art. 52, 22 pp.
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4
S.S. D RAGOMIR
pointed out in [10], then we can state that
Z
1
(2.5)
|λk |2 dσ (λ) =
and
n
∂Bn
Z
λk λj dσ (λ) = 0 if k 6= j, k, j = 1, . . . , n.
∂Bn
Utilising these properties, we have
2
#
" n
n
X
X
λk x k =
sup
λk λj hxk , xj i
kXk2e =
sup
(λ1 ,...,λn )∈Bn k,j=1
(λ1 ,...,λn )∈Bn k=1
#
Z "X
n
n Z
X
λk λj hxk , xj i dσ (λ) =
λk λj hxk , xj i dσ (λ)
≥
∂Bn
k,j=1
k,j=1
∂Bn
n
=
1X
1
kxk k2 = kXk22 ,
n k=1
n
from where we deduce the second inequality in (2.3).
The following representation result for the hypo-Euclidean norm plays a key role in obtaining
various bounds for this norm:
Theorem 2.2. For any X ∈ H n with X = (x1 , . . . , xn ) , we have
! 12
n
X
(2.6)
kXke = sup
|hx, xj i|2
.
kxk=1
j=1
Proof. We use the following well known representation result for scalars:
2
n
n
X
X
(2.7)
|zj |2 =
sup
λj zj ,
(λ1 ,...,λn )∈Bn j=1
j=1
n
where (z1 , . . . , zn ) ∈ K .
Utilising this property, we thus have
! 12
n
X
(2.8)
|hx, xj i|2
=
j=1
*
+
n
X
sup
λj x j x,
(λ1 ,...,λn )∈Bn
j=1
for any x ∈ H.
Now, taking the supremum over kxk = 1 in (2.8) we get
! 21
"
n
X
sup
|hx, xj i|2
= sup
sup
kxk=1
j=1
*
+#
n
X
x,
λ
x
j j kxk=1 (λ1 ,...,λn )∈Bn
j=1
*
"
+#
n
X
=
sup
sup x,
λj x j (λ1 ,...,λn )∈Bn kxk=1 j=1
n
X
=
sup
λ
x
j j ,
(λ1 ,...,λn )∈Bn
j=1
since, in any Hilbert space we have that supkuk=1 |hu, vi| = kvk for each v ∈ H.
J. Inequal. Pure and Appl. Math., 8(2) (2007), Art. 52, 22 pp.
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H YPO -E UCLIDEAN N ORM
5
Corollary 2.3. If X = (x1 , . . . , xn ) is an n−tuple of orthonormal vectors, i.e., we recall that
kxk k = 1 and hxk , xj i = 0 for k, j ∈ {1, . . . , n} with k 6= j, then kXke ≤ 1.
The proof is obvious by Bessel’s inequality.
The next proposition contains two lower bounds for the hypo-Euclidean norm that are sometimes better than the one in (2.3), as will be shown by some examples later.
Proposition 2.4. For any X = (x1 , . . . , xn ) ∈ H n \ {0} we have

P
n
1 

 kXk2 j=1 kxj k xj ,
(2.9)
kXke ≥
P


n
 √1 x
j=1 j .
n
Proof. By the definition of the hypo-Euclidean norm we have that, if (λ01 , . . . , λ0n ) ∈ Bn , then
obviously
n
X
0 kXke ≥ λj x j .
j=1
The choice
kxj k
,
j ∈ {1, . . . , n} ,
kXk2
which satisfies the condition (λ01 , . . . , λ0n ) ∈ Bn will produce the first inequality while the selection
1
λ0j = √ ,
j ∈ {1, . . . , n} ,
n
will give the second inequality in (2.9).
λ0j :=
Remark 2.5. For n = 2, the hypo-Euclidean norm on H 2
1
k(x, y)ke = sup kλx + µyk = sup |hz, xi|2 + |hz, yi|2 2
(λ,µ)∈B2
kzk=1
is bounded below by
1
1
B1 (x, y) := √ kxk2 + kyk2 2 ,
2
kkxk x + kyk yk
B2 (x, y) :=
1
kxk2 + kyk2 2
and
1
B3 (x, y) := √ kx + yk .
2
If H = C endowed with the canonical inner product hx, yi := xȳ where x, y ∈ C, then
1
1
B1 (x, y) = √ |x|2 + |y|2 2 ,
2
||x| x + |y| y|
B2 (x, y) =
1
|x|2 + |y|2 2
and
1
B3 (x, y) = √ |x + y| ,
2
J. Inequal. Pure and Appl. Math., 8(2) (2007), Art. 52, 22 pp.
x, y ∈ C.
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6
S.S. D RAGOMIR
The plots of the differences D1 (x, y) := B1 (x, y) − B2 (x, y) and D2 (x, y) := B1 (x, y) −
B3 (x, y) which are depicted in Figure 2.1 and Figure 2.2, respectively, show that the bound B1
is not always better than B2 or B3 . However, since the plot of D3 (x, y) := B2 (x, y) − B3 (x, y)
(see Figure 2.3) appears to indicate that, at least in the case of C2 , it may be possible that the
bound B2 is always better than B3 , hence we can ask in general which bound from (2.6) is better
for a given n ≥ 2? This is an open problem that will be left to the interested reader for further
investigation.
10
8
6
4
2
0
-2
-4
10
5
5
0
-5
x
-5
y
0
-10
10
-10
Figure 2.1: The behaviour of D1 (x, y)
10
8
6
4
2
-10
0
-5
-2
-4
10
0
y
5
5
0
x
-5
10
-10
Figure 2.2: The behaviour of D2 (x, y)
J. Inequal. Pure and Appl. Math., 8(2) (2007), Art. 52, 22 pp.
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H YPO -E UCLIDEAN N ORM
7
3.5
3
2.5
2
1.5
1
0.5
0
10
5
0
-5
-10
-5
0 y
5
10
-10
x
Figure 2.3: The behaviour of D3 (x, y)
3. U PPER B OUNDS VIA
THE
B OAS -B ELLMAN AND B OMBIERI T YPE I NEQUALITIES
In 1941, R.P. Boas [3] and in 1944, independently, R. Bellman [1] proved the following
generalisation of Bessel’s inequality that can be stated for any family of vectors {y1 , . . . , yn }(see
also [8, p. 392] or [5, p. 125]):

! 21 
n
X
X
(3.1)
|hx, yj i|2 ≤ kxk2  max kyj k2 +
|hyk , yj i|2 
1≤j≤n
j=1
1≤j6=k≤n
for any x, y1 . . . , yn vectors in the real or complex inner product space (H; h·, ·i) . This result is
known in the literature as the Boas-Bellman inequality.
The following result provides various upper bounds for the hypo-Euclidean norm:
Theorem 3.1. For any X = (x1 , . . . , xn ) ∈ H n , we have

! 12

P

2
2
 max kx k +
,
|hxk , xj i|
j
1≤j≤n
(3.2)
kXk2e ≤
1≤j6=k≤n


 max kxj k2 + (n − 1) max |hxk , xj i| ;
1≤j≤n
"
(3.3)
kXk2e ≤
1≤j6=k≤n
n
X
2
max kxj k
kxj k2 + max {kxj k kxk k}
1≤j≤n
j=1
1≤j6=k≤n
# 12
X
|hxj , xk i|
1≤j6=k≤n
and
(3.4)

n
2 P

kxj k2 + (n − 1) kXk2e max |hxj , xk i| ,
 max kxj k
1≤j≤n
1≤j6=k≤n
4
j=1
kXke ≤
P
2
2

|hxj , xk i| .
 kXke max kxj k + max {kxj k kxk k}
1≤j≤n
J. Inequal. Pure and Appl. Math., 8(2) (2007), Art. 52, 22 pp.
1≤j6=k≤n
1≤j6=k≤n
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8
S.S. D RAGOMIR
Proof. Taking the supremum over kxk = 1 in (3.1) and utilising the representation (2.6), we
deduce the first inequality in (3.2).
In [4], we proved amongst others the following inequalities

n
2 P
2


max
|c
|
kyj k2 ,
n
X
 1≤j≤n j
j=1
(3.5) cj hx, yj i ≤ kxk2 ×
n
P
2


|cj | max kyj k2 ,
j=1

1≤j≤n
j=1
+ kxk2 ×



P
max {|cj ck |}
1≤j6=k≤n

 (n − 1)
|hyj , yk i| ,
1≤j6=k≤n
n
P
|cj |2 max |hyj , yk i| ,
1≤j6=k≤n
j=1
for any y1 , . . . , yn , x ∈ H and c1 , . . . , cn ∈ K, where (3.5) should be seen as all possible
configurations.
The choice cj = hx, yj i, j ∈ {1, . . . , n} will produce the following four inequalities:

n
2 P

" n
#2

max
|hx,
y
i|
kyj k2 ,
j
 1≤j≤n
X
j=1
(3.6)
|hx, yj i|2 ≤ kxk2 ×
n
P
2


|hx, yj i| max kyj k2 ,
j=1

1≤j≤n
j=1
2
+ kxk ×



P
max {|hx, yj i| |hx, yk i|}
1≤j6=k≤n

 (n − 1)
|hyj , yk i| ,
1≤j6=k≤n
n
P
|hx, yj i|2 max |hyj , yk i| .
1≤j6=k≤n
j=1
Taking the supremum over kxk = 1 and utilising the representation (2.6) we easily deduce the
rest of the four inequalities.
A different generalisation of Bessel’s inequality for non-orthogonal vectors is the Bombieri
inequality (see [2] or [8, p. 397] and [5, p. 134]):
( n
)
n
X
X
(3.7)
|hx, yj i|2 ≤ kxk2 max
|hyj , yk i| ,
1≤j≤n
j=1
k=1
for any x ∈ H, where y1 , . . . , yn are vectors in the real or complex inner product space
(H; h·, ·i) .
Note that, the Bombieri inequality was not stated in the general case of inner product spaces
in [2]. However, the inequality presented there easily leads to (3.7) which, apparently, was
firstly mentioned as is in [8, p. 394].
On utilising the Bombieri inequality (3.7) and the representation Theorem 2.2, we can state
the following simple upper bound for the hypo-Euclidean norm k·ke .
Theorem 3.2. For any X = (x1 , . . . , xn ) ∈ H n , we have
( n
)
X
2
(3.8)
kXke ≤ max
|hxj , xk i| .
1≤j≤n
k=1
In [6] (see also [5, p. 138]), we have established the following norm inequalities:

n
2
! uq  u1
n
n
n
X
X
X
X
1
1
|αk |2 
|hzj , zk i|q  ,
(3.9)
αj zj ≤ n p + t −1
j=1
k=1
J. Inequal. Pure and Appl. Math., 8(2) (2007), Art. 52, 22 pp.
k=1
j=1
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H YPO -E UCLIDEAN N ORM
9
where p1 + 1q = 1, 1t + u1 = 1 and 1 < p ≤ 2, 1 < t ≤ 2 and αj ∈ C, zj ∈ H, j ∈ {1, . . . , n} .
An interesting particular case of (3.9) obtained for p = q = 2, t = u = 2 is incorporated in
2
n
n
X
X
α
z
≤
|αk |2
j j
(3.10)
j=1
k=1
n
X
! 12
|hzj , zk i|2
.
j,k=1
Other similar inequalities for norms are the following ones [6] (see also [5, pp. 139-140]):
"
2
# 1q 
n
n
n


X
X
X
1
|αk |2 max
|hzj , zk i|q
,
αj zj ≤ n p
1≤j≤n 

(3.11)
j=1
k=1
k=1
provided that 1 < p ≤ 2 and p1 + 1q = 1, αj ∈ C, zj ∈ H, j ∈ {1, . . . , n} . In the particular case
p = q = 2, we have
2
" n
# 21
n
n
X
X
X
√
αj zj ≤ n
|αk |2 max
|hzj , zk i|2 .
1≤j≤n
(3.12)
j=1
k=1
k=1
Also, if 1 < m ≤ 2, then [6]:
2
( n ) 1l
n
n
X
X
X
1
2
l
αj zj ≤ n m
|αk |
max |hzj , zk i|
,
1≤k≤n
(3.13)
j=1
where
(3.14)
1
m
+
1
l
j=1
k=1
= 1. For m = l = 2, we get
n
2
" n # 12
n
X
X
X
√
.
αj zj ≤ n
|αk |2
max |hzj , zk i|2
1≤k≤n
j=1
j=1
k=1
Finally, we can also state the inequality [6]:
(3.15)
n
2
n
X
X
αj zj ≤ n
|αk |2 max |hzj , zk i| .
1≤j,k≤n
j=1
k=1
Utilising the above norm-inequalities and the definition of the hypo-Euclidean norm, we can
state the following result which provides other upper bounds than the ones outlined in Theorem
3.1 and 3.2:
J. Inequal. Pure and Appl. Math., 8(2) (2007), Art. 52, 22 pp.
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10
S.S. D RAGOMIR
Theorem 3.3. For any X = (x1 , . . . , xn ) ∈ H n , we have


! uq  u1





n
n
P
P
1
1

q
+ t −1

p
 n
|hxj , xk i|
where p1 + 1q = 1,





j=1
k=1



1


+ u1 = 1 and 1 < p ≤ 2, 1 < t ≤ 2;

t



"

# 1q 





n

P
1
q

 n p max
|hx
,
x
i|
where p1 + 1q = 1

j
k

1≤j≤n  j=1

(3.16)
kXk2e ≤

and 1 < p ≤ 2;






( 
) 1l

n

P
1


max |hxj , xk i|l
where m1 + 1l = 1
 nm

1≤k≤n

j=1




and 1 < m ≤ 2;







 n max |hxk , zj i| ;
1≤k≤n
and, in particular,
(3.17)
 "
# 12

n
P



|hxj , xk i|2 ;



j,k=1



n
21
 √
P
2
2
kXke ≤
n max
|hxj , xk i|
;
1≤j≤n k=1




" 
# 21


n
P
√



max |hxj , xk i|2
.
 n
1≤k≤n
j=1
4. VARIOUS I NEQUALITIES FOR
THE
H YPO -E UCLIDEAN N ORM
For an n−tuple X = (x1 , . . . , xn ) of vectors in H, we consider the usual p−norms:
! p1
n
X
kXkp :=
kxj kp
,
j=1
Pn
where p ∈ [1, ∞), and denote with S the sum j=1 xj .
With these notations we can state the following reverse of the inequality kXk2 ≥ kXke , that
has been pointed out in Theorem 2.1.
Theorem 4.1. For any X = (x1 , . . . , xn ) ∈ H n , we have
(4.1)
(0 ≤) kXk22 − kXk2e ≤ kXk21 − kSk2 .
If
kXk2(2)
n X
xj + xk 2
:=
2 ,
j,k=1
then also
(4.2)
(0 ≤) kXk22 − kXk2e ≤ kXk2(2) − kSk2
( ≤ n kXk22 − kSk2 ).
J. Inequal. Pure and Appl. Math., 8(2) (2007), Art. 52, 22 pp.
http://jipam.vu.edu.au/
H YPO -E UCLIDEAN N ORM
11
Proof. We observe, for any x ∈ H, that
2
n
n
n
n
n
X
X
X
X
X
(4.3)
hx, xj i =
hx, xj i
hx, xk i = hx, xj i
hx, xk i
j=1
j=1
j=1
k=1
k=1
n
X
X
=
|hx, xk i|2 +
hx, xj i hxk , xi
k=1
1≤j6=k≤n
n
X
X
2
≤
|hx, xk i| + hx, xj i hxk , xi
k=1
n
X
≤
1≤j6=k≤n
X
|hx, xk i|2 +
k=1
Taking the supremum over kxk = 1, we get
2
n
n
X
X
(4.4)
sup hx, xj i ≤ sup
|hx, xk i|2 +
kxk=1 kxk=1
j=1
|hx, xj i| |hxk , xi| .
1≤j6=k≤n
k=1
X
sup |hx, xj i| · sup |hxk , xi| .
1≤j6=k≤n kxk=1
kxk=1
However,
n
2
*
+2
n
X
X
sup hx, xj i = sup x,
xj = kSk2 ,
kxk=1 kxk=1 j=1
j=1
sup |hx, xj i| = kxj k
sup |hx, xk i| = kxk k
and
kxk=1
kxk=1
for j, k ∈ {1, . . . , n} , and by (4.4) we get
X
kSk2 ≤ kXk2e +
= kXk2e +
kxj k kxj k
1≤j6=k≤n
n
X
kxj k kxk k −
j,k=1
=
kXk2e
+
kXk21
n
X
kxk k2
k=1
−
kXk22
,
which is clearly equivalent with (4.1).
Further on, we also observe that, for any x ∈ H we have the identity:
2
" n
#
n
X
X
(4.5)
hx, xj i = Re
hx, xj i hxk , xi
j=1
k,j=1
=
n
X
X
|hx, xk i|2 +
k=1
Re [hx, xj i hxk , xi] .
1≤j6=k≤n
Utilising the elementary inequality for complex numbers
(4.6)
Re (uv̄) ≤
1
|u + v|2 ,
4
J. Inequal. Pure and Appl. Math., 8(2) (2007), Art. 52, 22 pp.
u, v ∈ C,
http://jipam.vu.edu.au/
12
S.S. D RAGOMIR
we can state that
X
1 X
|hx, xj i + hx, xk i|2
4 1≤k6=j≤n
X xj + xk 2
x,
,
=
2
1≤k6=j≤n
Re [hx, xj i hxk , xi] ≤
1≤k6=j≤n
and by (4.5) we get
2
n
n
X
X
(4.7)
|hx, xk i|2 +
hx, xj i ≤
j=1
k=1
2
x
+
x
j
k
x,
2
1≤k6=j≤n
X
for any x ∈ H.
Taking the supremum over kxk = 1 in (4.7) we deduce
2
n
X
X xj + xk 2
2
xj ≤ kXke +
2 j=1
1≤k6=j≤n
n n
X
xj + xk 2 X
2
= kXke +
kxk k2
2 −
k,j=1
k=1
which provides the first inequality in (4.2).
By the convexity of k·k2 we have
n n
n
X
X
xj + xk 2 1 X
2
2
≤
kxj k + kxk k = n
kxk k2
2 2
j,k=1
j,k=1
k=1
and the last part of (4.2) is obvious.
Remark 4.2. For n = 2, X = (x, y) ∈ H 2 we have the upper bounds
B1 (x, y) := kxk2 + kyk2 − kx + yk2
= 2 (kxk kyk − Re hx, yi)
and
B2 (x, y) := kxk2 + kyk2
for the difference kXk22 − kXk2e , X ∈ H 2 as provided by (4.1) and (4.2) respectively. If
H = R then B1 (x, y) = 2 (|xy| − xy) , B2 (x, y) = x2 + y 2 . If we consider the function
∆ (x, y) = B2 (x, y) − B1 (x, y) then the plot of ∆ (x, y) depicted in Figure 4.1 shows that the
bounds provided by (4.1) and (4.2) cannot be compared in general, meaning that sometimes the
first is better than the second and vice versa.
From a different view-point we can state the following result:
Theorem 4.3. For any X = (x1 , . . . , xn ) ∈ H n , we have

! 12 
n
X
(4.8)
kSk2 ≤ kXke kXke +
kS − xk k2 
k=1
and


(4.9) kSk2 ≤ kXke kXke +


max kS − xk k2 +
1≤k≤n
J. Inequal. Pure and Appl. Math., 8(2) (2007), Art. 52, 22 pp.
X
1≤k6=l≤n
|hS − xk , S − xl i|2
! 12  12




,
http://jipam.vu.edu.au/
H YPO -E UCLIDEAN N ORM
13
200
100
-10
0
-5
-100
0
y
5
-200
10
5
0
x
-5
10
-10
Figure 4.1: The behaviour of ∆ (x, y)
respectively.
Proof. Utilising the identity (4.5) above we have
2
*
n
n
X
X
(4.10)
hx, xj i =
|hx, xk i|2 + Re x,
j=1
k=1
+
X
hx, xk i xj
1≤j6=k≤n
for any x ∈ H.
By the Schwarz inequality in the inner product space (H, h·, ·i), we have that
*
+
X
X
(4.11)
Re x,
hx, xk i xj ≤ kxk hx, xk i xj 1≤j6=k≤n
1≤j6=k≤n
n
n
X
X
= kxk hx, xk i xj −
hx, xk i xk j,k=1
k=1
*
+ n
n
n
X
X
X
= kxk x,
xk
xj −
hx, xk i xk j=1
k=1
k=1
n
X
= kxk hx, xk i (S − xk ) .
k=1
Utilising the Cauchy-Bunyakovsky-Schwarz inequality we have
! 12
! 21
n
n
n
X
X
X
2
2
kS − xk k
(4.12)
hx, xk i (S − xk ) ≤
|hx, xk i|
k=1
J. Inequal. Pure and Appl. Math., 8(2) (2007), Art. 52, 22 pp.
k=1
k=1
http://jipam.vu.edu.au/
14
S.S. D RAGOMIR
and then by (4.10) – (4.12) we can state the inequality:
2
! 12
! 12  n
n
n
X
X
X

|hx, xk i|2
+
(4.13)
hx, xj i ≤
|hx, xk i|2
j=1
k=1
k=1
n
X
! 12 
kS − xk k2

k=1
for any x ∈ H, kxk = 1. Taking the supremum over kxk = 1 we deduce the desired result
(4.8).
Now, following the above argument, we can also state that
*
n
+2
n
n
X
X
X
2
(4.14)
x,
x
≤
|hx,
x
i|
+
kxk
hx,
x
i
(S
−
x
)
j k
k
k j=1
k=1
for any x ∈ H.
Utilising the inequality

n
2
n
X

X
2
(4.15)
αj zj ≤
|αj |
max kzj k2 +
1≤j≤n
j=1
j=1
k=1
X
|hzj , zk i|2
1≤j6=k≤n
! 21 

,

where αj ∈ C, zj ∈ H, j ∈ {1, . . . , n} , that has been obtained in [4], see also [5, p. 128], we
can state that
n
X
(4.16) hx, xk i (S − xk )
k=1
! 21  12
! 21 
n


X
X
≤
max kS − xk k2 +
|hS − xk , S − xl i|2
|hx, xk i|2
1≤k≤n

1≤k6=l≤n
k=1
for any x ∈ H.
Now, by the use of (4.14) – (4.16) we deduce the desired result (4.9). The details are omitted.
Remark 4.4. On utilising the inequality:
2
n
n
X
X
2
2
(4.17)
αj zj ≤
|αj | max kzk k + (n − 1) max |hzk , zl i| ,
1≤k≤n
1≤k6=l≤n
j=1
j=1
where αj ∈ C, zj ∈ H, j ∈ {1, . . . , n} , that has been obtained in [4], (see also [5, p. 130])
in place of (4.15) above, we can state the following inequality for the hypo-Euclidean norm as
well:
(4.18)
kSk2
"
≤ kXke kXke +
max kS − xk k2 + (n − 1) max |hS − xk , S − xl i|2
1≤k≤n
21 #
1≤k6=l≤n
for any X = (x1 , . . . , xn ) ∈ H n .
Other similar results may be stated by making use of the results from [6]. The details are left
to the interested reader.
J. Inequal. Pure and Appl. Math., 8(2) (2007), Art. 52, 22 pp.
http://jipam.vu.edu.au/
H YPO -E UCLIDEAN N ORM
15
5. R EVERSE I NEQUALITIES
Before we proceed with establishing some reverse inequalities for the hypo-Euclidean norm,
we recall some reverse results of the Cauchy-Bunyakovsky-Schwarz inequality for real or complex numbers as follows:
If γ, Γ ∈ K (K = C, R) and αj ∈ K, j ∈ {1, . . . , n} with the property that
(5.1)
0 ≤ Re [(Γ − αj ) (αj − γ̄)]
= (Re Γ − Re αj ) (Re αj − Re γ) + (Im Γ − Im αj ) (Im αj − Im γ)
or, equivalently,
αj − γ + Γ ≤ 1 |Γ − γ|
2 2
(5.2)
for each j ∈ {1, . . . , n} , then (see for instance [5, p. 9])
2
n
n
X
X
1
|αj |2 − αj ≤ · n2 |Γ − γ|2 .
(5.3)
n
4
j=1
j=1
In addition, if Re (Γγ̄) > 0, then (see for example [5, p. 26]):
n h
io2
Pn
n
Re
Γ̄
+
γ̄
α
X
j=1 j
1
(5.4)
|αj |2 ≤ ·
n
4
Re (Γγ̄)
j=1
n
2
1 |Γ + γ|2 X ≤ ·
αj 4 Re (Γγ̄) j=1
and
2
n
X
1 |Γ − γ|2
n
|αj |2 − αj ≤ ·
4 Re (Γγ̄)
j=1
j=1
n
X
(5.5)
2
n
X
αj .
j=1
Also, if Γ 6= −γ, then (see for instance [5, p. 32]):
! 21 n
n
X 1
X
|Γ − γ|2
2
(5.6)
n
|αj |
−
αj ≤ n ·
.
4
|Γ + γ|
j=1
j=1
Finally, from [7] we can also state that
2
n
n
n
X
h
i X
X
p
(5.7)
n
|αj |2 − αj ≤ n |Γ + γ| − 2 Re (Γγ̄) αj ,
j=1
j=1
j=1
provided Re (Γγ̄) > 0.
We notice that a simple sufficient condition for (5.1) to hold is that
(5.8)
Re Γ ≥ Re αj ≥ Re γ
and
Im Γ ≥ Im αj ≥ Im γ
for each j ∈ {1, . . . , n} .
We can state and prove the following conditional inequalities for the hypo-Euclidean norm
k·ke :
J. Inequal. Pure and Appl. Math., 8(2) (2007), Art. 52, 22 pp.
http://jipam.vu.edu.au/
16
S.S. D RAGOMIR
Theorem 5.1. Let ϕ, φ ∈ K and X = (x1 , . . . , xn ) ∈ H n such that either:
ϕ + φ 1
(5.9)
hx, xj i − 2 ≤ 2 |φ − ϕ|
or, equivalently,
Re [(φ − hx, xj i) (hxj , xi − ϕ̄)] ≥ 0
(5.10)
for each j ∈ {1, . . . , n} and for any x ∈ H, kxk = 1. Then
kXk2e ≤
(5.11)
1
1
kSk2 + n |φ − ϕ|2 .
n
4
Moreover, if Re (φϕ̄) > 0, then
kXk2e
(5.12)
1 |φ + ϕ|2
≤
·
kSk2
4n Re (φϕ)
and
kXk2e ≤
(5.13)
h
i
p
1
kSk2 + |φ + ϕ| − 2 Re (φϕ̄) kSk .
n
If φ 6= −ϕ, then
kXke ≤
(5.14)
where S =
Pn
j=1
1
1
|φ − ϕ|2
kSk + n ·
,
n
4
|φ + ϕ|
xj .
Proof. We only prove the inequality (5.11).
Let x ∈ H, kxk = 1. Then, on applying the inequality (5.3) for αj = hx, xj i , j ∈ {1, . . . , n}
and Γ = φ, γ = ϕ, we can state that
*
+2
n
n
X
X
1
1
(5.15)
|hx, xj i|2 ≤ x,
xj + n |φ − ϕ|2 .
n
4
j=1
j=1
Now if in (5.15) we take the supremum over kxk = 1, then we get the desired inequality
(5.11).
The other inequalities follow by (5.4), (5.7) and (5.6) respectively. The details are omitted.
Remark 5.2. Due to the fact that
ϕ
+
φ
ϕ
+
φ
hx, xj i −
≤ xj −
· x
2
2
for any j ∈ {1, . . . , n} and x ∈ H, kxk = 1, then a sufficient condition for (5.9) to hold is that
xj − ϕ + φ · x ≤ 1 |φ − ϕ|
2
2
for each j ∈ {1, . . . , n} and x ∈ H, kxk = 1.
J. Inequal. Pure and Appl. Math., 8(2) (2007), Art. 52, 22 pp.
http://jipam.vu.edu.au/
H YPO -E UCLIDEAN N ORM
17
6. A PPLICATIONS FOR n−T UPLES OF O PERATORS
In [9], the author has introduced the following norm on the Cartesian product B (n) (H) :=
B (H) × · · · × B (H) , where B (H) denotes the Banach algebra of all bounded linear operators
defined on the complex Hilbert space H :
(6.1)
k(T1 , . . . , Tn )ke :=
sup
kλ1 T1 + · · · + λn Tn k ,
(λ1 ,...,λn )∈Bn
P
where (T1 , . . . , Tn ) ∈ B (n) (H) and Bn := (λ1 , . . . , λn ) ∈ Cn ni=1 |λi |2 ≤ 1 is the Euclidean closed ball in Cn . It is clear that k·ke is a norm on B (n) (H) and for any (T1 , . . . , Tn ) ∈
B (n) (H) we have
(6.2)
k(T1 , . . . , Tn )ke = k(T1∗ , . . . , Tn∗ )ke ,
where Ti∗ is the adjoint operator of Ti , i ∈ {1, . . . , n} .
It has been shown in [9] that the following inequality holds true:
1
n
1
n
2
2
X
X
1 ∗
∗
√ (6.3)
Tj Tj ≤ k(T1 , . . . , Tn )ke ≤ Tj Tj n
j=1
j=1
for any n−tuple (T1 , . . . , Tn ) ∈ B (n) (H) and the constants √1n and 1 are best possible.
In the same paper [9] the author has introduced the Euclidean operator radius of an n−tuple
of operators (T1 , . . . , Tn ) by
! 12
n
X
(6.4)
we (T1 , . . . , Tn ) := sup
|hTj x, xi|2
kxk=1
j=1
and proved that we (·) is a norm on B (n) (H) and satisfies the double inequality:
(6.5)
1
k(T1 , . . . , Tn )ke ≤ we (T1 , . . . , Tn ) ≤ k(T1 , . . . , Tn )ke
2
for each n−tuple (T1 , . . . , Tn ) ∈ B (n) (H) .
As pointed out in [9], the Euclidean numerical radius also satisfies the double inequality:
1
1
n
n
2
X
2
X
1 √ (6.6)
Tj Tj∗ ≤ we (T1 , . . . , Tn ) ≤ Tj Tj∗ 2 n
j=1
j=1
for any (T1 , . . . , Tn ) ∈ B (n) (H) and the constants 2√1 n and 1 are best possible.
We are now able to establish the following natural connections that exists between the hypoEuclidean norm of vectors in a Cartesian product of Hilbert spaces and the norm k·ke for
n−tuples of operators in the Banach algebra B (H) .
Theorem 6.1. For any (T1 , . . . , Tn ) ∈ B (n) (H) we have
(6.7)
k(T1 , . . . , Tn )ke = sup k(T1 y, . . . , Tn y)ke
kyk=1
=
sup
kyk=1,kxk=1
J. Inequal. Pure and Appl. Math., 8(2) (2007), Art. 52, 22 pp.
n
X
! 12
|hTj y, xi|2
.
j=1
http://jipam.vu.edu.au/
18
S.S. D RAGOMIR
Proof. By the definition of the k·ke −norm on B (n) (H) and the hypo-Euclidean norm on H n ,
we have:
"
#
(6.8)
k(T1 , . . . , Tn )ke =
sup k(λ1 T1 + · · · + λn Tn ) yk
sup
(λ1 ,...,λn )∈Bn
kyk=1
#
"
= sup
sup
kyk=1
kλ1 T1 y + · · · + λn Tn yk
(λ1 ,...,λn )∈Bn
= sup k(T1 y, . . . , Tn y)ke .
kyk=1
Utilising the representation of the hypo-Euclidean norm on H n from Theorem 2.2, we have
! 21
n
X
2
(6.9)
k(T1 y, . . . , Tn y)ke = sup
|hTj y, xi|
.
kxk=1
j=1
Making use of (6.8) and (6.9) we deduce the desired equality (6.7).
Remark 6.2. Utilising Theorem 2.1, we have
! 21
n
X
1
2
(6.10)
kTj yk
≥ k(T1 y, . . . , Tn y)ke ≥ √
n
j=1
for any y ∈ H, kyk = 1.
Since
n
X
2
kTj yk =
j=1
* n
X
n
X
! 12
kTj yk
2
j=1
+
Tj∗ Tj y, y
kyk = 1
,
j=1
hence, on taking the supremum over kyk = 1 in (6.10) and on observing that
* n
+
! n
n
n
X
X
X
X
Tj∗ Tj = Tj∗ Tj = Tj Tj∗ ,
sup
Tj∗ Tj y, y = w
kyk=1
j=1
j=1
j=1
j=1
we deduce the inequality (6.3) that has been established in [9] by a different argument.
We observe that, due to the representation Theorem 6.1, some inequalities obtained for the
hypo-Euclidean norm can be utilised in obtaining various new inequalities for the operator
norm k·ke by employing a standard approach consisting in taking the supremum over kyk = 1,
as described in the above remark.
The following different lower bound for the Euclidean operator norm k·ke can be stated:
Proposition 6.3. For any (T1 , . . . , Tn ) ∈ B (n) (H) , we have
1
(6.11)
k(T1 , . . . , Tn )ke ≥ √ kT1 + · · · + Tn k .
n
Proof. Utilising Proposition 2.4 and Theorem 6.1 we have:
k(T1 , . . . , Tn )ke = sup k(T1 y, . . . , Tn y)ke
kyk=1
1
≥ √ sup kT1 y + · · · + Tn yk
n kyk=1
1
= √ kT1 + · · · + Tn k
n
which is the desired inequality (6.11).
J. Inequal. Pure and Appl. Math., 8(2) (2007), Art. 52, 22 pp.
http://jipam.vu.edu.au/
H YPO -E UCLIDEAN N ORM
19
We can state the following results concerning various upper bounds for the operator norm
k(., . . . , .)ke :
Theorem 6.4. For any (T1 , . . . , Tn ) ∈ B (n) (H) , we have the inequalities:

"
# 12


P


max kTj k2 +
w2 (Tk∗ Tj ) ;


1≤j≤n

1≤j6=k≤n



 max kT k2 + (n − 1) max {w (T ∗ T )} ;

j

k j

1≤j6=k≤n

 1≤j≤n

2
(6.12)
k(T1 , . . . , Tn )k2e ≤
P
n


 max kTj k2 Tj∗ Tj 

1≤j≤n
j=1





# 12



P


w Tk Tj∗
.
+ max {kTj k kTk k}


1≤j6=k≤n
1≤j6=k≤n
The proof follows by Theorem 3.1 and Theorem 6.1 and the details are omitted.
On utilising the inequalities (3.8) and (3.17) we can state the following result as well:
Theorem 6.5. For any (T1 , . . . , Tn ) ∈ B (n) (H) , we have:
n

P

∗

w (Tk Tj ) ;
max


1≤j≤n k=1




"
# 12


n

P



w2 (Tk∗ Tj ) ;

 j,k=1
(6.13)
k(T1 , . . . , Tn )k2e ≤
n
12

P 2 ∗


n max
w (Tk Tj ) ;


1≤j≤n k=1




"
# 12


n

P


max {w2 (Tk∗ Tj )} .

 n
1≤k≤n
j=1
The results from Section 5 can be also naturally used to provide some reverse inequalities
that are of interest.
Theorem 6.6. Let (T1 , . . . , Tn ) ∈ B (n) (H) and ϕ, φ ∈ K such that
1
ϕ
+
φ
Tj y −
(6.14)
· x
≤ 2 |φ − ϕ| for any kxk = kyk = 1
2
and for each j ∈ {1, . . . , n} . Then
(6.15)
k(T1 , . . . , Tn )k2e
2
n
1
1
X ≤ Tj + |φ − ϕ|2 .
n j=1 n
In addition, if Re (φϕ̄) > 0, then
2
n
X
1 |φ + ϕ| 2
k(T1 , . . . , Tn )ke ≤
·
Tj 4n Re (φϕ̄) j=1 2
(6.16)
and
(6.17)
2
n
n
h
i
X
X
p
1
2
k(T1 , . . . , Tn )ke ≤ Tj + |φ + ϕ| − 2 Re (φϕ̄) Tj .
n
j=1
J. Inequal. Pure and Appl. Math., 8(2) (2007), Art. 52, 22 pp.
j=1
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20
S.S. D RAGOMIR
If φ 6= −ϕ, then also
2
n
X
1 1 √ |φ − ϕ|2
√
k(T1 , . . . , Tn )ke ≤
Tj +
.
n·
4
|φ + ϕ|
n j=1 (6.18)
Proof. For any x, y ∈ H with kxk = kyk = 1 we have
ϕ
+
φ
ϕ
+
φ
hx, Tj yi −
= x, Tj y −
x 2
2
ϕ
+
φ
≤ kxk T
y
−
x
j
2
1
≤ |φ − ϕ|
2
for each j ∈ {1, . . . , n} .
Now, on applying Theorem 5.1 for xj = Tj y, we can write from (5.11) the following inequality
n
1
X
1
Tj y + n |φ − ϕ|2
k(T1 y, . . . , Tn y)ke ≤ 4
n j=1
for each y with kyk = 1.
Taking the supremum over kyk = 1 and utilising Theorem 6.1, we deduce (6.15).
The other inequalities follow by a similar procedure on making use of the inequalities (5.12)
– (5.14) and the details are omitted.
Remark 6.7. The inequality (6.14) is equivalent with
(6.19)
0 ≤ Re [(φ − hx, Tj yi) (hTj y, xi − ϕ̄)]
= (Re (φ) − Re hx, Tj yi) (Re hTj y, xi − Re (ϕ))
+ (Im (φ) − Im hx, Tj yi) (Im hTj y, xi − Im (ϕ))
for each j ∈ {1, . . . , n} and kxk = kyk = 1. A sufficient condition for (6.19) to hold is then:

 Re (ϕ) ≤ Re hx, Tj yi ≤ Re (φ)
(6.20)
 Im (ϕ) ≤ Im hx, T yi ≤ Im (φ)
j
for any x, y ∈ H with kxk = kyk = 1 and j ∈ {1, . . . , n} .
7. A N ORM
ON
B (H)
For an operator A ∈ B (H) we define
(7.1)
δ (A) := k(A, A∗ )ke = sup kλA + µA∗ k ,
(λ,µ)∈B2
where B2 is the Euclidean unit ball in C2 .
The properties of this functional are embodied in the following theorem:
Theorem 7.1. The functional δ is a norm on B (H) and satisfies the double inequality:
(7.2)
kAk ≤ δ (A) ≤ 2 kAk
for any A ∈ B (H) .
J. Inequal. Pure and Appl. Math., 8(2) (2007), Art. 52, 22 pp.
http://jipam.vu.edu.au/
H YPO -E UCLIDEAN N ORM
21
Moreover, we have the inequalities
√
1
1
2
A2 + (A∗ )2 2 ≤ δ (A) ≤ A2 + (A∗ )2 2 ,
(7.3)
2
and
√
1
2
(7.4)
kA + A∗ k ≤ δ (A) ≤ kAk2 + w A2 2
2
for any A ∈ B (H) , respectively.
Proof. First of all, observe, by Theorem 5.1, that we have the representation
1
(7.5)
δ (A) =
sup
|hAy, xi|2 + |hA∗ y, xi|2 2
kxk=1,kyk=1
for each A ∈ B (H) .
Obviously δ (A) ≥ 0 for each A ∈ B (H) and of δ (A) = 0 then, by (7.5), hAy, xi = 0 for
any x, y ∈ H with kxk = kyk = 1 which implies that A = 0. Also, by (7.5), we observe that
1
δ (αA) =
sup
|hαAy, xi|2 + |hᾱA∗ y, xi|2 2
kxk=1,kyk=1
= |α|
sup
1
|hAy, xi|2 + |hA∗ y, xi|2 2
kxk=1,kyk=1
= |α| δ (A)
for any α ∈ R and A ∈ B (H) .
Now, if A, B ∈ B (H) , then
δ (A + B) = sup kλA + µA∗ + λB + µB ∗ k
(λ,µ)∈B2
≤ sup kλA + µA∗ k + sup kλB + µB ∗ k
(λ,µ)∈B2
(λ,µ)∈B2
= δ (A) + δ (B) ,
which proves the triangle inequality.
Also, we observe that
δ (A) ≥
sup
|hAy, xi| = kAk
kxk=1,kyk=1
and
δ (A) ≤
sup
[|hAy, xi| + |hA∗ y, xi|]
kxk=1,kyk=1
≤
sup
|hAy, xi| +
kxk=1,kyk=1
sup
|hA∗ y, xi|
kxk=1,kyk=1
= 2 kAk
and the inequality (7.2) is proved.
The inequality (7.3) follows from (6.3) for n = 2, T1 = A and T2 = A∗ while (7.4) follows
from Proposition 6.3 and the second inequality in (6.12) for the same choices.
Remark 7.2. It is easy to see that
√
1
2
A2 + (A∗ )2 2 ≤ kAk
2
and
2
1 1
A + (A∗ )2 2 , kAk2 + w A2 2 ≤ 2 kAk
J. Inequal. Pure and Appl. Math., 8(2) (2007), Art. 52, 22 pp.
http://jipam.vu.edu.au/
22
S.S. D RAGOMIR
for each A ∈ B (H) . Also, we notice that if A is self-adjoint, then the equality case holds in
the second part of (7.3) and in both
sides of (7.4). However, it is an open question for the author
√
2
which of the lower bounds kAk , 2 kA + A∗ k of the norm δ (A) are better and when. The same
1
1
question applies for the upper bounds A2 + (A∗ )2 2 and kAk2 + w (A2 ) 2 , respectively.
R EFERENCES
[1] R. BELLMAN, Almost orthogonal series, Bull. Amer. Math. Soc., 50 (1944), 517–519.
[2] E. BOMBIERI, A note on the large sieve, Acta Arith., 18 (1971), 401–404.
[3] R.P. BOAS, A general moment problem, Amer. J. Math., 63 (1941), 361–370.
[4] S.S. DRAGOMIR, On the Boas-Bellman inequality in inner product spaces, Bull. Austral. Math.
Soc., 69(2) (2004), 217–225.
[5] S.S. DRAGOMIR, Advances in Inequalities of the Schwarz, Grüss and Bessel Type in Inner Product
Spaces, Nova Science Publishers, Inc., 2005.
[6] S.S. DRAGOMIR, On the Bombieri inequality in inner product spaces, Libertas Math., 25 (2005),
13–26.
[7] S.S. DRAGOMIR, Reverses of the Schwarz inequality generalising the Klamkin-McLeneghan result, Bull. Austral. Math. Soc., 73(1) (2006), 69–78.
[8] D.S. MITRINOVIĆ, J.E. PEČARIĆ AND A.M. FINK, Classical and New Inequalities in Analysis,
Kluwer Academic Publishers, 1993.
[9] G. POPESCU, Unitary
Arχiv.math.04/0410492.
invariants
in
multivariable
operator
theory,
Preprint,
[10] W. RUDIN, Function Theory in the Unit Ball of Cn , Springer Verlag, New York, Berlin, 1980.
J. Inequal. Pure and Appl. Math., 8(2) (2007), Art. 52, 22 pp.
http://jipam.vu.edu.au/