THE HYPO-EUCLIDEAN NORM OF AN N −TUPLE
OF VECTORS IN INNER PRODUCT SPACES AND
APPLICATIONS
Hypo-Euclidean Norm
S.S. Dragomir
vol. 8, iss. 2, art. 52, 2007
S.S. DRAGOMIR
School of Computer Science and Mathematics
Victoria University
PO Box 14428, Melbourne City
8001, VIC, Australia
EMail: sever.dragomir@vu.edu.au
URL: http://rgmia.vu.edu.au/dragomir
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Received:
19 March, 2007
J
Accepted:
28 April, 2007
Page 1 of 43
Communicated by:
J.M. Rassias
2000 AMS Sub. Class.:
Primary 47C05, 47C10; Secondary 47A12.
Key words:
Inner product spaces, Norms, Bessel’s inequality, Boas-Bellman and Bombieri
inequalities, Bounded linear operators, Numerical radius.
Abstract:
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 Boas-Bellman [1]-[3] and Bombieri [2] type inequalities are provided.
Applications for n−tuples of bounded linear operators defined on Hilbert spaces
are also given.
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Contents
1
Introduction
3
2
Fundamental Properties
6
3
Upper Bounds via the Boas-Bellman and Bombieri Type Inequalities
14
Hypo-Euclidean Norm
4
Various Inequalities for the Hypo-Euclidean Norm
20
S.S. Dragomir
vol. 8, iss. 2, art. 52, 2007
5
Reverse Inequalities
28
6
Applications for n−Tuples of Operators
32
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7
A Norm on B (H)
40
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1.
Introduction
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
Hypo-Euclidean Norm
S.S. Dragomir
vol. 8, iss. 2, art. 52, 2007
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! 12
|λk |2
Contents
.
k=1
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
( nk=1 kxk kp ) p
if p ∈ [1, ∞);
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
λj x j ,
(1.3)
kXkh,n :=
sup
(λ1 ,...,λn )∈B (k·k )
n
j=1
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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
that generates the hypo-Euclidean norm on E n , i.e.,
n
X
(1.5)
kXkh,e :=
sup
λ
x
j j .
(λ1 ,...,λn )∈Bn
j=1
Hypo-Euclidean Norm
S.S. Dragomir
vol. 8, iss. 2, art. 52, 2007
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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
λj x 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
Pby the fact that the hypo-Euclidean
norm is closely related to the quadratic form nj=1 |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 hypo-Euclidean 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 hypoEuclidean 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.
Hypo-Euclidean Norm
S.S. Dragomir
vol. 8, iss. 2, art. 52, 2007
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2.
Fundamental Properties
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
S.S. Dragomir
which generates the Euclidean norm k·k2 on H n , i.e.,
(2.2)
kXk2 :=
Hypo-Euclidean Norm
n
X
! 12
kxj k2
,
X ∈ H n.
vol. 8, iss. 2, art. 52, 2007
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Contents
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)
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i.e., k·k2 and k·ke are equivalent norms on H n .
Close
Proof. By the Cauchy-Bunyakovsky-Schwarz inequality we have
! 21
! 12
n
n
n
X
X
X
2
2
(2.4)
λj x j ≤
|λj |
kxj k
j=1
j=1
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
positive Borel measure on the
P
unit sphere ∂Bn ∂Bn = (λ1 , . . . , λn ) ∈ Kn ni=1 |λi |2 = 1 whose existence and
properties have been pointed out in [10], then we can state that
Z
1
|λk |2 dσ (λ) =
(2.5)
and
n
∂Bn
Z
λk λj dσ (λ) = 0 if k 6= j, k, j = 1, . . . , n.
Hypo-Euclidean Norm
S.S. Dragomir
vol. 8, iss. 2, art. 52, 2007
∂Bn
Utilising these properties, we have
2
" n
#
n
X
X
2
kXke =
sup
λk x k =
sup
λk λj hxk , xj i
(λ1 ,...,λn )∈Bn k=1
(λ1 ,...,λn )∈Bn k,j=1
#
Z "X
n
n Z
X
≥
λk λj hxk , xj i dσ (λ) =
λk λj hxk , xj i dσ (λ)
∂Bn
=
1
n
n
X
k=1
k,j=1
kxk k2 =
k,j=1
∂Bn
1
kXk22 ,
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:
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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
(2.7)
|zj | =
sup
λj zj ,
(λ1 ,...,λn )∈Bn j=1
j=1
where (z1 , . . . , zn ) ∈ Kn .
Utilising this property, we thus have
! 21
n
X
|hx, xj i|2
=
(2.8)
j=1
Hypo-Euclidean Norm
S.S. Dragomir
vol. 8, iss. 2, art. 52, 2007
Title Page
*
+
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
*
! 12
"
+#
n
n
X
X
sup
|hx, xj i|2
= sup
sup
x,
λ
x
j j kxk=1
kxk=1 (λ1 ,...,λn )∈Bn
j=1
j=1
*
+#
"
n
X
=
sup
sup x,
λj x j (λ1 ,...,λn )∈Bn kxk=1 j=1
n
X
=
sup
λj x j ,
(λ1 ,...,λn )∈Bn j=1
since, in any Hilbert space we have that supkuk=1 |hu, vi| = kvk for each v ∈ H.
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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 j=1 xj .
n
Proof. By the definition of the hypo-Euclidean norm we have that, if (λ01 , . . . , λ0n ) ∈
Bn , then obviously
n
X
kXke ≥ λ0j xj .
j=1
The choice
λ0j :=
kxj k
,
kXk2
j ∈ {1, . . . , n} ,
S.S. Dragomir
vol. 8, iss. 2, art. 52, 2007
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(λ01 , . . . , λ0n )
which satisfies the condition
while the selection
1
λ0j = √ ,
n
will give the second inequality in (2.9).
Hypo-Euclidean Norm
∈ Bn will produce the first inequality
j ∈ {1, . . . , n} ,
Remark 1. 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| ,
x, y ∈ C.
2
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 1 and Figure 2, respectively, show
that the bound B1 is not always better than B2 or B3 . However, since the plot of
Hypo-Euclidean Norm
S.S. Dragomir
vol. 8, iss. 2, art. 52, 2007
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D3 (x, y) := B2 (x, y) − B3 (x, y) (see Figure 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.
Hypo-Euclidean Norm
10
S.S. Dragomir
8
vol. 8, iss. 2, art. 52, 2007
6
4
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0
-2
-4
10
5
5
0
x
-5
0
-5
y
10
-10
-10
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Figure 1: The behaviour of D1 (x, y)
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Hypo-Euclidean Norm
S.S. Dragomir
vol. 8, iss. 2, art. 52, 2007
10
8
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6
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4
2
-10
0
-5
-2
0
-4
10
5
5
0
x
-5
10
-10
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Figure 2: The behaviour of D2 (x, y)
Close
Hypo-Euclidean Norm
S.S. Dragomir
vol. 8, iss. 2, art. 52, 2007
3.5
3
2.5
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1.5
1
0.5
0
10
5
0
-5
-10
-5
0 y
5
10
-10
x
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Figure 3: The behaviour of D3 (x, y)
Close
3.
Upper Bounds via the Boas-Bellman and Bombieri Type
Inequalities
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]):

! 12 
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


max kxj k2 +
|hxk , xj i|2
,
2
1≤j≤n
(3.2)
kXke ≤
1≤j6=k≤n


 max kxj k2 + (n − 1) max |hxk , xj i| ;
1≤j≤n
Hypo-Euclidean Norm
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vol. 8, iss. 2, art. 52, 2007
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1≤j6=k≤n
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"
(3.3)
kXk2e
≤
max kxj k
1≤j≤n
2
n
X
Close
kxj k2
j=1
# 21
+ max {kxj k kxk k}
1≤j6=k≤n
X
1≤j6=k≤n
|hxj , xk i|
and
(3.4) kXk4e ≤

n
2 P


max
kx
k
kxj k2 + (n − 1) kXk2e max |hxj , xk i| ,

j
1≤j≤n
1≤j6=k≤n
j=1
2


max kxj k2 + max {kxj k kxk k}
 kXke 1≤j≤n
1≤j6=k≤n
P
|hxj , xk i| .
1≤j6=k≤n
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


j=1
|cj |2 max kyj k2 ,

1≤j≤n
j=1
P

max {|cj ck |}
|hyj , yk i| ,

 1≤j6=k≤n
1≤j6=k≤n
2
+ kxk ×
n
P

 (n − 1)
|cj |2 max |hyj , yk i| ,
j=1
1≤j6=k≤n
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


j=1
|hx, yj i|2 max kyj k2 ,

j=1
1≤j≤n
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vol. 8, iss. 2, art. 52, 2007
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2
+ kxk ×



P
max {|hx, yj i| |hx, yk i|}
1≤j6=k≤n
|hyj , yk i| ,
1≤j6=k≤n
n
P

 (n − 1)
|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

2
! uq  u1
n
n
n
n
X
X
X
X
1
1
αj zj ≤ n p + t −1
|αk |2 
|hzj , zk i|q  ,
j=1
k=1
S.S. Dragomir
vol. 8, iss. 2, art. 52, 2007
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k=1
In [6] (see also [5, p. 138]), we have established the following norm inequalities:
(3.9)
Hypo-Euclidean Norm
k=1
j=1
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
n
2
! 21
n
n
X
X
X
2
2
(3.10)
|αk |
|hzj , zk i|
.
αj zj ≤
j=1
k=1
j,k=1
Other similar inequalities for norms are the following ones [6] (see also [5, pp.
139-140]):
"
n
2
# 1q 
n
n
X


X
X
1
|αk |2 max
|hzj , zk i|q
,
(3.11)
αj zj ≤ n p
1≤j≤n 

j=1
k=1
k=1
where
(3.14)
1
m
+
k=1
j=1
1
l
= 1. For m = l = 2, we get
2
" n # 12
n
n
X
X
X
√
αj zj ≤ n
|αk |2
max |hzj , zk i|2
.
1≤k≤n
j=1
k=1
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k=1
Also, if 1 < m ≤ 2, then [6]:
2
( n ) 1l
n
n
X
X
X
1
(3.13)
αj zj ≤ n m
|αk |2
max |hzj , zk i|l
,
1≤k≤n
j=1
S.S. Dragomir
vol. 8, iss. 2, art. 52, 2007
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
# 12
" n
n
n
X
X
X
√
(3.12)
αj zj ≤ n
|αk |2 max
|hzj , zk i|2 .
1≤j≤n
j=1
Hypo-Euclidean Norm
j=1
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Finally, we can also state the inequality [6]:
2
n
n
X
X
(3.15)
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:
Hypo-Euclidean Norm
S.S. Dragomir
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
vol. 8, iss. 2, art. 52, 2007
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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




" 
# 12


n
P
√

2


max |hxj , xk i|
.
 n
j=1
1≤k≤n
Hypo-Euclidean Norm
S.S. Dragomir
vol. 8, iss. 2, art. 52, 2007
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4.
Various Inequalities for the Hypo-Euclidean Norm
For an n−tuple X = (x1 , . . . , xn ) of vectors in H, we consider the usual p−norms:
! p1
n
X
p
kXkp :=
kxj k
,
j=1
P
where p ∈ [1, ∞), and denote with S the sum nj=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.
Hypo-Euclidean Norm
S.S. Dragomir
vol. 8, iss. 2, art. 52, 2007
Theorem 4.1. For any X = (x1 , . . . , xn ) ∈ H n , we have
(4.1)
(0 ≤) kXk22 − kXk2e ≤ kXk21 − kSk2 .
Contents
If
kXk2(2)
n
X
xj + xk 2
:=
2 ,
j,k=1
then also
(4.2)
Title Page
(0 ≤) kXk22 − kXk2e ≤ kXk2(2) − kSk2
( ≤ n kXk22 − kSk2 ).
Proof. We observe, for any x ∈ H, that
n
2
n
n
n
n
X
X
X
X
X
(4.3)
hx, xk i
hx, xj i =
hx, xj i
hx, xk i = hx, xj 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
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≤
≤
n
X
k=1
n
X
X
|hx, xk i|2 + 1≤j6=k≤n
X
|hx, xk i|2 +
k=1
hx, xj i hxk , xi
|hx, xj i| |hxk , xi| .
1≤j6=k≤n
Taking the supremum over kxk = 1, we get
2
n
X
(4.4) sup hx, xj i
kxk=1
Hypo-Euclidean Norm
S.S. Dragomir
vol. 8, iss. 2, art. 52, 2007
j=1
≤ sup
n
X
sup |hx, xj i| · sup |hxk , xi| .
1≤j6=k≤n kxk=1
kxk=1 k=1
However,
X
|hx, xk i|2 +
kxk=1
2
*
+2
n
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
and
kxk=1
sup |hx, xk i| = kxk k
kxk=1
for j, k ∈ {1, . . . , n} , and by (4.4) we get
X
kSk2 ≤ kXk2e +
= kXk2e +
=
+
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kxj k kxj k
1≤j6=k≤n
n
X
kxj k kxk k −
kXk21
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j,k=1
kXk2e
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n
X
k=1
−
kXk22
,
Close
kxk k2
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
hx, xj i = Re
(4.5)
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
Re (uv̄) ≤
(4.6)
we can state that
X
1
|u + v|2 ,
4
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] ≤
and by (4.5) we get
2
n
n
X
X
(4.7)
hx, xj i ≤
|hx, xk i|2 +
for any x ∈ H.
S.S. Dragomir
vol. 8, iss. 2, art. 52, 2007
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Contents
1≤k6=j≤n
j=1
u, v ∈ C,
Hypo-Euclidean Norm
k=1
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2
x
+
x
j
k
x,
2
1≤k6=j≤n
X
Close
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
kxk k2
= kXke +
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
k=1
j,k=1
Hypo-Euclidean Norm
S.S. Dragomir
vol. 8, iss. 2, art. 52, 2007
Title Page
Contents
and the last part of (4.2) is obvious.
JJ
II
Remark 2. For n = 2, X = (x, y) ∈ H 2 we have the upper bounds
J
I
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 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.
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200
Hypo-Euclidean Norm
100
S.S. Dragomir
vol. 8, iss. 2, art. 52, 2007
-10
0
-5
Title Page
-100
0
y
Contents
5
-200
10
5
0
x
-5
10
-10
Figure 4: The behaviour of ∆ (x, y)
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From a different view-point we can state the following result:
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n
Theorem 4.2. For any X = (x1 , . . . , xn ) ∈ H , we have

! 21 
n
X
(4.8)
kSk2 ≤ kXke kXke +
kS − xk k2 
k=1
Close
and
(4.9)
kSk2


≤ kXke kXke +


max kS − xk k2 +
1≤k≤n
X
|hS − xk , S − xl i|2
! 21  12


1≤k6=l≤n


,
Hypo-Euclidean Norm
respectively.
S.S. Dragomir
Proof. Utilising the identity (4.5) above we have
n
2
*
n
X
X
2
(4.10)
hx, xj i =
|hx, xk i| + Re x,
j=1
k=1
vol. 8, iss. 2, art. 52, 2007
+
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
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Utilising the Cauchy-Bunyakovsky-Schwarz inequality we have
! 12
! 21
n
n
n
X
X
X
kS − xk k2
(4.12)
hx, xk i (S − xk ) ≤
|hx, xk i|2
k=1
k=1
k=1
and then by (4.10) – (4.12) we can state the inequality:
(4.13)
Hypo-Euclidean Norm
n
2
X
hx, xj i
S.S. Dragomir
vol. 8, iss. 2, art. 52, 2007
j=1
≤
n
X
! 12 
|hx, xk i|2
n
X

! 12
|hx, xk i|2
+
k=1
k=1
n
X
! 12 
kS − xk k2

k=1
Contents
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
*
+2
n
n
n
X
X
X
2
(4.14)
xj ≤
|hx, xk i| + kxk hx, xk i (S − xk )
x,
j=1
k=1
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for any x ∈ H.
Utilising the inequality

2
n
n
X

X
2
(4.15)
αj zj ≤
|αj |
max kzj k2 +
1≤j≤n
j=1
j=1
Close
X
1≤j6=k≤n
|hzj , zk i|2
! 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 
! 21  12
n


X
X
≤
|hx, xk i|2
max kS − xk k2 +
|hS − xk , S − xl i|2
1≤k≤n

k=1
1≤k6=l≤n
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 3. 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:
"
≤ kXke kXke +
S.S. Dragomir
vol. 8, iss. 2, art. 52, 2007
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kSk2
(4.18)
Hypo-Euclidean Norm
max kS − xk k2 + (n − 1) max |hS − xk , S − xl i|2
1≤k≤n
1 #
2
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.
Close
5.
Reverse Inequalities
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)
Hypo-Euclidean Norm
S.S. Dragomir
vol. 8, iss. 2, art. 52, 2007
or, equivalently,
(5.2)
1
γ
+
Γ
αj −
≤ |Γ − γ|
2 2
for each j ∈ {1, . . . , n} , then (see for instance [5, p. 9])
2
n
n
X
X
1
2
(5.3)
n
|αj | − αj ≤ · n2 |Γ − γ|2 .
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)
n
|αj |2 ≤ ·
4
Re (Γγ̄)
j=1
2
n
1 |Γ + γ|2 X ≤ ·
αj 4 Re (Γγ̄) j=1
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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
−
αj ≤ n ·
.
(5.6)
n
|αj |
4
|Γ + γ|
j=1
j=1
Hypo-Euclidean Norm
S.S. Dragomir
vol. 8, iss. 2, art. 52, 2007
Finally, from [7] we can also state that
n
n
n
X 2
h
i X
X
p
2
(5.7)
n
|αj | − 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
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Page 29 of 43
(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 :
Theorem 5.1. Let ϕ, φ ∈ K and X = (x1 , . . . , xn ) ∈ H n such that either:
1
ϕ
+
φ
hx, xj i −
≤ |φ − ϕ|
(5.9)
2 2
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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)
Hypo-Euclidean Norm
S.S. Dragomir
1 |φ + ϕ|2
≤
·
kSk2
4n Re (φϕ)
vol. 8, iss. 2, art. 52, 2007
Title Page
and
kXk2e
(5.13)
h
i
p
1
2
≤ kSk + |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
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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 4. 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
1
ϕ
+
φ
xj −
≤ |φ − ϕ|
·
x
2
2
Hypo-Euclidean Norm
S.S. Dragomir
vol. 8, iss. 2, art. 52, 2007
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for each j ∈ {1, . . . , n} and x ∈ H, kxk = 1.
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6.
Applications for n−Tuples of Operators
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
Hypo-Euclidean Norm
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
S.S. Dragomir
vol. 8, iss. 2, art. 52, 2007
Title Page
(6.2)
k(T1 , . . . , Tn )ke = k(T1∗ , . . . , Tn∗ )ke ,
Contents
where Ti∗ is the adjoint operator of Ti , i ∈ {1, . . . , n} .
It has been shown in [9] that the following inequality holds true:
1
1
n
n
2
X
2
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
(6.4)
we (T1 , . . . , Tn ) := sup
kxk=1
n
X
j=1
! 12
|hTj x, xi|2
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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:
n
1
n
1
2
X
2
1 X
∗
∗
√ Tj Tj (6.6)
Tj Tj ≤ we (T1 , . . . , Tn ) ≤ 2 n
j=1
Hypo-Euclidean Norm
S.S. Dragomir
vol. 8, iss. 2, art. 52, 2007
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 hypo-Euclidean 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) .
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Theorem 6.1. For any (T1 , . . . , Tn ) ∈ B (n) (H) we have
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(6.7)
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Page 33 of 43
k(T1 , . . . , Tn )ke = sup k(T1 y, . . . , Tn y)ke
kyk=1
=
n
X
sup
kyk=1,kxk=1
k(T1 , . . . , Tn )ke =
sup
(λ1 ,...,λn )∈Bn
|hTj y, xi|2
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.
j=1
Proof. By the definition of the k·ke −norm on B
norm on H n , we have:
"
(6.8)
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! 12
(n)
Close
(H) and the hypo-Euclidean
#
sup k(λ1 T1 + · · · + λn Tn ) yk
kyk=1
"
#
= sup
kyk=1
sup
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
! 12
n
X
(6.9)
k(T1 y, . . . , Tn y)ke = sup
|hTj y, xi|2
.
kxk=1
Remark 5. Utilising Theorem 2.1, we have
! 21
n
X
1
≥ k(T1 y, . . . , Tn y)ke ≥ √
(6.10)
kTj yk2
n
j=1
for any y ∈ H, kyk = 1.
Since
* n
+
n
X
X
kTj yk2 =
Tj∗ Tj y, y ,
n
X
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! 21
kTj yk2
j=1
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kyk = 1
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j=1
j=1
Contents
Page 34 of 43
hence, on taking the supremum over kyk = 1 in (6.10) and on observing that
* n
+
! n
n
n
X
X
X
X
∗
∗
∗
∗
sup
Tj Tj y, y = w
Tj Tj = Tj Tj = Tj Tj ,
kyk=1
j=1
S.S. Dragomir
vol. 8, iss. 2, art. 52, 2007
j=1
Making use of (6.8) and (6.9) we deduce the desired equality (6.7).
j=1
Hypo-Euclidean Norm
j=1
j=1
we deduce the inequality (6.3) that has been established in [9] by a different argument.
Close
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 6.2. For any (T1 , . . . , Tn ) ∈ B (n) (H) , we have
(6.11)
1
k(T1 , . . . , Tn )ke ≥ √ kT1 + · · · + Tn k .
n
Proof. Utilising Proposition 2.4 and Theorem 6.1 we have:
Hypo-Euclidean Norm
S.S. Dragomir
vol. 8, iss. 2, art. 52, 2007
Title Page
k(T1 , . . . , Tn )ke = sup k(T1 y, . . . , Tn y)ke
Contents
kyk=1
1
≥ √ sup kT1 y + · · · + Tn yk
n kyk=1
1
= √ kT1 + · · · + Tn k
n
which is the desired inequality (6.11).
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We can state the following results concerning various upper bounds for the operator norm k(., . . . , .)ke :
Close
Theorem 6.3. 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



2


max kTj k + (n − 1) max {w (Tk∗ Tj )} ;


1≤j≤n
1≤j6=k≤n

 
2
2
(6.12) k(T1 , . . . , Tn )ke ≤
P
n

2 ∗ 

max
kT
k
T
T

j
j

1≤j≤n
j=1 j 




# 12



P


+ max {kTj k kTk k}
w Tk Tj∗
.


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 6.4. For any (T1 , . . . , Tn ) ∈ B (n) (H) , we have:
n

P

∗

max
w (Tk Tj ) ;


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
Hypo-Euclidean Norm
S.S. Dragomir
vol. 8, iss. 2, art. 52, 2007
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The results from Section 5 can be also naturally used to provide some reverse
inequalities that are of interest.
Theorem 6.5. 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)
2
n
X
1
1
k(T1 , . . . , Tn )k2e ≤ Tj + |φ − ϕ|2 .
n j=1 n
Hypo-Euclidean Norm
S.S. Dragomir
vol. 8, iss. 2, art. 52, 2007
Title Page
In addition, if Re (φϕ̄) > 0, then
(6.16)
n
2
1 |φ + ϕ|2 X 2
·
Tj k(T1 , . . . , Tn )ke ≤
4n Re (φϕ̄) j=1 n
n
X 2 h
i X
p
1
Tj .
k(T1 , . . . , Tn )k2e ≤ Tj + |φ + ϕ| − 2 Re (φϕ̄) n j=1 j=1
If φ 6= −ϕ, then also
(6.18)
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Page 37 of 43
and
(6.17)
Contents
2
n
X
1 1 √ |φ − ϕ|2
k(T1 , . . . , Tn )ke ≤ √ Tj +
n·
.
4
|φ + ϕ|
n j=1 Go Back
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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
1
X
k(T1 y, . . . , Tn y)ke ≤ Tj y + n |φ − ϕ|2
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. 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
Hypo-Euclidean Norm
S.S. Dragomir
vol. 8, iss. 2, art. 52, 2007
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hold is then:
(6.20)

 Re (ϕ) ≤ Re hx, Tj yi ≤ Re (φ)
 Im (ϕ) ≤ Im hx, T yi ≤ Im (φ)
j
for any x, y ∈ H with kxk = kyk = 1 and j ∈ {1, . . . , n} .
Hypo-Euclidean Norm
S.S. Dragomir
vol. 8, iss. 2, art. 52, 2007
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7.
A Norm 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:
Hypo-Euclidean Norm
S.S. Dragomir
vol. 8, iss. 2, art. 52, 2007
kAk ≤ δ (A) ≤ 2 kAk
(7.2)
Title Page
for any A ∈ B (H) .
Moreover, we have the inequalities
√
1
1
2
A2 + (A∗ )2 2 ≤ δ (A) ≤ A2 + (A∗ )2 2 ,
(7.3)
2
and
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Page 40 of 43
√
(7.4)
Contents
1
2
kA + A∗ k ≤ δ (A) ≤ kAk2 + w A2 2
2
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for any A ∈ B (H) , respectively.
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Proof. First of all, observe, by Theorem 5.1, that we have the representation
(7.5)
δ (A) =
sup
kxk=1,kyk=1
for each A ∈ B (H) .
1
|hAy, xi|2 + |hA∗ y, xi|2 2
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
Hypo-Euclidean Norm
kxk=1,kyk=1
S.S. Dragomir
= |α| δ (A)
vol. 8, iss. 2, art. 52, 2007
for any α ∈ R and A ∈ B (H) .
Now, if A, B ∈ B (H) , then
Title Page
δ (A + B) = sup kλA + µA∗ + λB + µB ∗ k
Contents
(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
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Page 41 of 43
|hAy, xi| = kAk
kxk=1,kyk=1
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and
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δ (A) ≤
sup
∗
[|hAy, xi| + |hA y, xi|]
kxk=1,kyk=1
≤
sup
kxk=1,kyk=1
= 2 kAk
|hAy, xi| +
sup
kxk=1,kyk=1
|hA∗ y, xi|
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 6.2 and the second inequality in (6.12) for the same
choices.
Remark 7. It is easy to see that
√
1
2
A2 + (A∗ )2 2 ≤ kAk
2
and
Hypo-Euclidean Norm
S.S. Dragomir
vol. 8, iss. 2, art. 52, 2007
2
1 1
A + (A∗ )2 2 , kAk2 + w A2 2 ≤ 2 kAk
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
√
2
open question for the author which of the lower bounds kAk , 2 kA + A∗ k of the
norm δ (A) are better and when. The same question applies for the upper bounds
2
1
1
A + (A∗ )2 2 and kAk2 + w (A2 ) 2 , respectively.
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References
[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.
Hypo-Euclidean Norm
[4] S.S. DRAGOMIR, On the Boas-Bellman inequality in inner product spaces,
Bull. Austral. Math. Soc., 69(2) (2004), 217–225.
vol. 8, iss. 2, art. 52, 2007
[5] S.S. DRAGOMIR, Advances in Inequalities of the Schwarz, Grüss and Bessel
Type in Inner Product Spaces, Nova Science Publishers, Inc., 2005.
Title Page
[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 invariants in multivariable operator theory, Preprint,
Arχiv.math.04/0410492.
n
[10] W. RUDIN, Function Theory in the Unit Ball of C , Springer Verlag, New
York, Berlin, 1980.
S.S. Dragomir
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