VECTORS IN INNER PRODUCT SPACES AND APPLICATIONS
S.S. DRAGOMIR
Abstract. The concept of hypo-Euclidean norm for ann−tuple of vectors in inner product spaces is introduced. Its fundamental properties are es-tablished. Applications forn−tuples of bounded linear operators defined on Hilbert spaces are also given.
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)∈K n| kλk
n≤1}. As an example of such norms we should mention the usualp−norms
(1.1) kλkn,p:=
max{|λ1|, . . . ,|λn|} if p=∞; (Pn
k=1|λk| p
)1p if p∈[1,∞).
TheEuclidean norm is obtained forp= 2, i.e., kλkn,2=
n
X
k=1 |λk|2
!12 .
It is well known that on En := E× · · · ×E endowed with the canonical linear structure we can define the followingp−norms:
(1.2) kXkn,p:=
max{kx1k, . . . ,kxnk} if p=∞; (Pn
k=1kxkk p
) 1
p if p∈[1,∞);
whereX = (x1, . . . , xn)∈En.
For a given normk·knonKn we define the functionalk·kh,n:E
n→[0,∞) given by
(1.3) kXkh,n:= sup (λ1,...,λn)∈B(k·kn)
n
X
j=1 λjxj
,
whereX = (x1, . . . , xn)∈En. It is easy to see that:
Date: 09 October, 2006.
2000Mathematics Subject Classification. Primary 47C05, 47C10; Secondary 47A12.
Key words and phrases. Inner porduct spaces, Norms, Bessel’s inequality, Boas-Bellman and Bombieri inequalities, Bounded linear operators, Numerical radius.
(i) kXkh,n≥0 for anyX∈En;
(ii) kX+Ykh,n≤ kXkh,n+kYkh,nfor anyX, Y ∈En; (iii) kαXkh,n=|α| kXkh,nfor eachα∈KandX ∈En;
and therefore k·kh,n is a semi-norm on En. This will be called the hypo-semi-norm generated by the normk·kn onXn.
We observe thatkXkh,n= 0 if and only ifPn
j=1λjxj = 0 for any (λ1, . . . , λn)∈ B(k·kn).If there existsλ01, . . . , λ0n6= 0 such that λ01,0, . . . ,0
, 0, λ02, . . . ,0
, . . . , 0,0, . . . , λ0n
∈B(k·kn) then the semi-norm generated byk·kn is anormonEn. If byBn,p withp∈[1,∞] we denote the balls generated by thep−normsk·kn,p onKn,then we can obtain the followinghypo-p-norms onXn:
(1.4) kXkh,n,p:= sup (λ1,...,λn)∈Bn,p
n
X
j=1 λjxj
,
withp∈[1,∞].
For p = 2, we have the Euclidean ball in Kn, which we denote by
Bn, Bn =
n
λ= (λ1, . . . , λn)∈Kn
Pn
i=1|λi| 2
≤1o that generates the hypo-Euclidean norm
onEn,i.e.,
(1.5) kXkh,e:= sup (λ1,...,λn)∈Bn
n
X
j=1 λjxj
.
Moreover, ifE=H, His a Hilbert space overK, then thehypo-Euclidean norm
onHn will be denoted simply by
(1.6) k(x1, . . . , xn)ke:= sup (λ1,...,λn)∈Bn
n
X
j=1 λjxj
,
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 algebraB(H) of all bounded linear operators onH and used it to investigate various properties ofn−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 formPn
j=1|hx, xji| 2
(see the representation Theorem 2) that plays a key role in many problems arising in the Theory of Fourier expansions in Hilbert spaces.
investigated. A representation result is obtained and some applications for opera-tor inequalities are pointed out. Finally, in Section 7, a new norm for operaopera-tors is introduced and some natural inequalities are obtained.
2. Fundamental Properties
Let (H;h·,·i) be a Hilbert space over K and n ∈ N, n ≥ 1. In the Cartesian product Hn :=H × · · · ×H, for the n−tuples of vectors X = (x
1, . . . , xn), Y = (y1, . . . , yn)∈Hn, we can define the inner producth·,·iby
(2.1) hX, Yi:= n
X
j=1
hxj, yji, X, Y ∈Hn,
which generates the Euclidean normk·k2 onHn,i.e.,
(2.2) kXk2:=
n
X
j=1 kxjk2
1 2
, X ∈Hn.
The following result connects the usual Euclidean norm k·k with the hypo-Euclidean normk·ke.
Theorem 1. For anyX ∈Hn we have the inequalities
(2.3) kXk2≥ kXke≥√1
nkXk2,
i.e., k·k2 andk·ke are equivalent norms onHn.
Proof. By the Cauchy-Bunyakovsky-Schwarz inequality we have
(2.4)
n
X
j=1 λjxj
≤
n
X
j=1 |λj|2
1 2
n
X
j=1 kxjk2
1 2
for any (λ1, . . . , λn)∈Kn.Taking the supremum over (λ1, . . . , λn)∈
Bnin (2.4) we obtain the first inequality in (2.3).
If byσ we denote the rotation-invariant normalised positive Borel measure on the unit sphere ∂Bn
∂Bn= (λ1, . . . , λn)∈Kn
Pn
i=1|λi| 2
= 1 whose existence and properties have been pointed out in [10], then we can state that
Z
∂Bn
|λk|2dσ(λ) = 1 n and (2.5)
Z
∂Bn
Utilising these properties, we have
kXk2e= sup (λ1,...,λn)∈Bn
n X k=1 λkxk
2 = sup (λ1,...,λn)∈Bn
n
X
k,j=1
λkλjhxk, xji
≥
Z
∂Bn
n
X
k,j=1
λkλjhxk, xji
dσ(λ) =
n
X
k,j=1
Z
∂Bn
λkλjhxk, xji
dσ(λ) = 1 n n X k=1
kxkk2= 1 nkXk
2 2,
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. For anyX ∈Hn with X= (x
1, . . . , xn), we have
(2.6) kXke= sup
kxk=1
n
X
j=1
|hx, xji|2
1 2 .
Proof. We use the following well known representation result for scalars:
(2.7)
n
X
j=1
|zj|2= sup (λ1,...,λn)∈Bn
n X j=1 λjzj 2 ,
where (z1, . . . , zn)∈Kn.
Utilising this property, we thus have
(2.8) n X j=1
|hx, xji|2
1 2
= sup (λ1,...,λn)∈Bn
* x, n X j=1 λjxj +
for anyx∈H.
Now, taking the supremum overkxk= 1 in (2.8) we get
sup
kxk=1
n
X
j=1
|hx, xji|2
1 2
= sup
kxk=1
sup
(λ1,...,λn)∈Bn
* x, n X j=1 λjxj
+ = sup (λ1,...,λn)∈Bn
sup kxk=1
* x, n X j=1 λjxj
+ = sup (λ1,...,λn)∈Bn
n X j=1 λjxj ,
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 1. For anyX = (x1, . . . , xn)∈Hn\ {0} we have
(2.9) kXke≥
1
kXk2
Pn
j=1kxjkxj
,
1
√
n
Pn
j=1xj
.
Proof. By the definition of the hypo-Euclidean norm we have that, if λ01, . . . , λ 0 n
∈
Bn,then obviously
kXke≥
n
X
j=1 λ0jxj
. The choice
λ0j := kxjk
kXk2, j∈ {1, . . . , n}, which satisfies the condition λ01, . . . , λ0n
∈ Bn will produce the first inequality while the selection
λ0j =√1
n, j∈ {1, . . . , n}, will give the second inequality in (2.9).
Remark 1. Forn= 2, the hypo-Euclidean norm onH2 k(x, y)ke= sup
(λ,µ)∈B2
kλx+µyk= sup
kzk=1
h
|hz, xi|2+|hz, yi|2i 1 2
is bounded below by
B1(x, y) :=√1 2
kxk2+kyk2 1 2 , B2(x, y) := kkxkx+kykyk
kxk2+kyk2 1 2
and
B3(x, y) :=√1
2kx+yk.
If H =C endowed with the canonical inner product hx, yi:= x¯y where x, y ∈ C,
then
B1(x, y) =√1 2
|x|2+|y|2 1 2 , B2(x, y) = ||x|x+|y|y|
|x|2+|y|2 1 2
and
B3(x, y) = √1
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
D3(x, y) :=B2(x, y)−B3(x, y)(see Figure 3) appears to indicate that, at least in
the case ofC2,it may be possible that the boundB
2 is always better thanB3,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 -5 10
-4 -2
0 0
5 2
y 4
6
x 8
0 10
5 -5 -1010
Figure 1. The behaviour ofD1(x, y)
3. Upper Bounds via the Boas-Bellman and Bombieri Type Inequalities
In 1941, R.P. Boas [2] 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]):
(3.1)
n
X
j=1
|hx, yji| 2
≤ kxk2
1max≤j≤nkyjk
2 +
X
1≤j6=k≤n
|hyk, yji| 2
1 2
for any x, y1. . . , yn vectors in the real or complex inner product space (H;h·,·i). This results is known in the literature as theBoas-Bellman inequality.
-10
-5
0 y
5 -4
10 -2
5
0 0
-5 10
2
x -10
4 6 8 10
Figure 2. The behaviour ofD2(x, y)
Theorem 3. For anyX = (x1, . . . , xn)∈Hn,we have
(3.2) kXk2e≤
max 1≤j≤nkxjk
2
+ P
1≤j6=k≤n
|hxk, xji| 2
!12
,
max 1≤j≤nkxjk
2
+ (n−1) max
1≤j6=k≤n|hxk, xji|;
(3.3) kXk2e≤
max
1≤j≤nkxjk 2
n
X
j=1 kxjk
2
+ max
1≤j6=k≤n{kxjk kxkk}
X
1≤j6=k≤n
|hxj, xki|
1 2
and
(3.4) kXk4e≤
max 1≤j≤nkxjk
2 Pn
j=1 kxjk
2
+ (n−1)kXk2e max
1≤j6=k≤n|hxj, xki|, kXk2e max
1≤j≤nkxjk 2
+ max
1≤j6=k≤n{kxjk kxkk}
P
1≤j6=k≤n
|hxj, xki|.
-10 -5
0 y
5
10 0
x 5
0.5
0 -5
1
10 -10 1.5
2 2.5
3 3.5
Figure 3. The behaviour ofD3(x, y)
In [4], we proved amongst others the following inequalities
(3.5)
n
X
j=1
cjhx, yji
2
≤ kxk2×
max 1≤j≤n|cj|
2Pn
j=1 kyjk2,
n
P
j=1
|cj|2 max 1≤j≤nkyjk
2 ,
+kxk2×
max
1≤j6=k≤n{|cjck|}
P
1≤j6=k≤n
|hyj, yki|,
(n−1) n
P
j=1 |cj|
2 max
1≤j6=k≤n|hyj, yki|,
for any y1, . . . , yn, x ∈ H and c1, . . . , cn ∈ K, where (3.5) should be seen as all
The choicecj =hx, yji, j ∈ {1, . . . , n} will produce the following four inequali-ties:
(3.6)
n
X
j=1
|hx, yji|2
2
≤ kxk2×
max
1≤j≤n|hx, yji| 2 Pn
j=1 kyjk
2 ,
n
P
j=1
|hx, yji|2 max 1≤j≤nkyjk
2 ,
+kxk2×
max
1≤j6=k≤n{|hx, yji| |hx, yki|}
P
1≤j6=k≤n
|hyj, yki|,
(n−1) n
P
j=1
|hx, yji| 2
max
1≤j6=k≤n|hyj, yki|.
Taking the supremum overkxk= 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]):
(3.7)
n
X
j=1
|hx, yji|2≤ kxk2 max 1≤j≤n
( n
X
k=1
|hyj, yki|
)
,
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, we can state the following simple upper bound for the hypo-Euclidean normk·ke. Theorem 4. For anyX = (x1, . . . , xn)∈Hn,we have
(3.8) kXk2e≤ max 1≤j≤n
( n
X
k=1
|hxj, xki|
)
.
In [6] (see also [5, p. 138]), we have established the following norm inequalities:
(3.9)
n
X
j=1 αjzj
2
≤n1p+
1
t−1
n
X
k=1 |αk|2
n
X
k=1
n
X
j=1
|hzj, zki|q
u q
1
u
,
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
(3.10)
n
X
j=1 αjzj
2
≤ n
X
k=1 |αk|
2
n
X
j,k=1
|hzj, zki| 2
Other similar inequalities for norms are the following ones [6] (see also [5, pp. 139-140]): (3.11) n X j=1 αjzj 2
≤np1
n
X
k=1
|αk|2 max 1≤j≤n
" n X k=1
|hzj, zki|q
#1q
,
provided that 1 < p≤2 and 1 p +
1
q = 1, αj ∈ C, zj ∈ H, j ∈ {1, . . . , n}. In the particular casep=q= 2,we have
(3.12) n X j=1 αjzj 2
≤√n n
X
k=1
|αk|2 max 1≤j≤n
" n
X
k=1
|hzj, zki|2
#12 .
Also, if 1< m≤2, then [6]:
(3.13) n X j=1 αjzj
2
≤nm1
n
X
k=1 |αk|
2 n X j=1 max
1≤k≤n|hzj, zki| l 1 l ,
where m1 +1l = 1.Form=l= 2,we get
(3.14) n X j=1 αjzj 2
≤√n n X k=1 |αk|2 n X j=1 max
1≤k≤n|hzj, zki| 2 1 2 .
Finally, we can also state the inequality [6]:
(3.15) n X j=1 αjzj
2 ≤n n X k=1 |αk|
2 max
1≤j,k≤n|hzj, zki|.
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 and 4:
Theorem 5. For anyX = (x1, . . . , xn)∈Hn,we have
(3.16) kXk2e≤
np1+
1
t−1
n P k=1 n P j=1
|hxj, xki| q
!uq
1 u where 1 p+ 1 q = 1, 1
t+ 1
u = 1 and 1< p≤2, 1< t≤2;
np1 max
1≤j≤n
" n P j=1
|hxj, xki|q
#1q
where 1p+1q = 1
and 1< p≤2;
nm1
( n P j=1 max
1≤k≤n|hxj, xki| l
)1l
where m1 +1l = 1
and 1< m≤2; n max
and, in particular,
(3.17) kXk2e≤
"
n
P
j,k=1
|hxj, xki|2
#12 ;
√ n max
1≤j≤n
n
P
k=1
|hxj, xki| 2
1 2 ;
√ n
"
n
P
j=1
max 1≤k≤n
n
|hxj, xki|2o
#
1 2 .
4. Various Inequalities for the Hypo-Euclidean Norm
For ann−tupleX = (x1, . . . , xn) of vectors inH,we consider the usualp−norms:
kXkp:=
n
X
j=1 kxjkp
1
p
,
wherep∈[1,∞),and denote withS the sum Pn
j=1xj.
With these notations we can state the following reverse of the inequalitykXk2≥ kXke, that has been pointed out in Theorem 1.
Theorem 6. For anyX = (x1, . . . , xn)∈Hn,we have
(4.1) (0≤)kXk22− kXk2e≤ kXk21− kSk2.
If
kXk2(2):= n
X
j,k=1
xj+xk 2
2 ,
then also
(0≤)kXk22− kXk2e≤ kXk2(2)− kSk2 (4.2)
(≤nkXk22− kSk2).
Proof. We observe, for anyx∈H,that
n
X
j=1 hx, xji
2
= n
X
j=1 hx, xji
n
X
k=1
hx, xki=
n
X
j=1 hx, xji
n
X
k=1 hx, xki
(4.3)
=
n
X
k=1
|hx, xki| 2
+ X 1≤j6=k≤n
hx, xji hxk, xi
≤ n
X
k=1
|hx, xki|2+
X
1≤j6=k≤n
hx, xji hxk, xi
≤ n
X
k=1
|hx, xki|2+ X 1≤j6=k≤n
Taking the supremum overkxk= 1,we get
(4.4) sup
kxk=1
n
X
j=1 hx, xji
2
≤ sup
kxk=1 n
X
k=1
|hx, xki| 2
+ X 1≤j6=k≤n
sup
kxk=1
|hx, xji| · sup
kxk=1
|hxk, xi|.
However,
sup
kxk=1
n
X
j=1 hx, xji
2
= sup
kxk=1
*
x, n
X
j=1 xj
+
2
=kSk2,
sup
kxk=1
|hx, xji|=kxjk and sup
kxk=1
|hx, xki|=kxkk
forj, k∈ {1, . . . , n},and by (4.4) we get kSk2≤ kXk2e+ X
1≤j6=k≤n
kxjk kxjk
=kXk2e+ n
X
j,k=1
kxjk kxkk − n
X
k=1 kxkk2
=kXk2e+kXk21− kXk22, which is clearly equivalent with (4.1).
Further on, we also observe that, for anyx∈H we have the identity:
n
X
j=1 hx, xji
2
= Re
n
X
k,j=1
hx, xji hxk, xi
(4.5)
= n
X
k=1
|hx, xki|2+ X 1≤j6=k≤n
Re [hx, xji hxk, xi].
Utilising the elementary inequality for complex numbers
(4.6) Re (u¯v)≤ 1 4|u+v|
2
, u, v∈C,
we can state that
X
1≤k6=j≤n
Re [hx, xji hxk, xi]≤ 1 4
X
1≤k6=j≤n
|hx, xji+hx, xki| 2
= X 1≤k6=j≤n
x,xj+xk 2
2 ,
and by (4.5) we get
(4.7)
n
X
j=1 hx, xji
2
≤ n
X
k=1
|hx, xki|2+ X 1≤k6=j≤n
x,xj+xk 2
2
Taking the supremum overkxk= 1 in (4.7) we deduce
n
X
j=1 xj
2
≤ kXk2e+ X 1≤k6=j≤n
xj+xk 2
2
=kXk2e+ n
X
k,j=1
xj+xk 2
2 −
n
X
k=1 kxkk
2
which provides the first inequality in (4.2). By the convexity ofk·k2we have
n
X
j,k=1
xj+xk 2
2 ≤1
2 n
X
j,k=1
h
kxjk2+kxkk 2i
=n n
X
k=1 kxkk2
and the last part of (4.2) is obvious.
Remark 2. Forn= 2, X = (x, y)∈H2 we have the bounds
B1(x, y) :=kxk2+kyk2− kx+yk2 = 2 (kxk kyk −Rehx, yi)
and
B2(x, y) :=kxk 2
+kyk2
for the differencekXk22−kXk2e, X∈H2as provided by (4.1) and (4.2) respectively.
If H =R then B1(x, y) = 2 (|xy| −xy), B2(x, y) = x2+y2. 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.
From a different view-point we can state the following result:
Theorem 7. For anyX = (x1, . . . , xn)∈Hn,we have
(4.8) kSk2≤ kXke
kXke+
n
X
k=1
kS−xkk 2
!12
and
(4.9) kSk2
≤ kXke
kXke+
max
1≤k≤nkS−xkk 2
+
X
1≤k6=l≤n
|hS−xk, S−xli| 2
1 2
1
l
,
respectively.
Proof. Utilising the identity (4.5) above we have
(4.10)
n
X
j=1 hx, xji
2
= n
X
k=1
|hx, xki|2+ Re
*
x, X
1≤j6=k≤n
hx, xkixj
+
-10
-5
0 y
5 -200
10 5 0 10
-5 -10
-100
x 0
100 200
Figure 4. The behaviour of ∆ (x, y)
By the Schwarz inequality in the inner product space (H,h·,·i), we have that
Re
*
x, X
1≤j6=k≤n
hx, xkixj
+
≤ kxk
X
1≤j6=k≤n
hx, xkixj
(4.11)
=kxk
n
X
j,k=1
hx, xkixj− n
X
k=1
hx, xkixk
=kxk
*
x, n
X
k=1 xk
+ n
X
j=1 xj−
n
X
k=1
hx, xkixk
=kxk
n
X
k=1
hx, xki(S−xk)
.
Utilising the Cauchy-Bunyakovsky-Schwarz inequality we have
(4.12)
n
X
k=1
hx, xki(S−xk)
≤ n
X
k=1
|hx, xki|2
!12 n
X
k=1
kS−xkk2
and then by (4.10) – (4.12) we can state the inequality: (4.13) n X j=1 hx, xji
2 ≤ n X k=1
|hx, xki|2
!12
n
X
k=1
|hx, xki| 2
!12 +
n
X
k=1
kS−xkk 2
!12
for anyx∈H,kxk= 1.Taking the supremum overkxk= 1 we deduce the desired result (4.8).
Now, following the above argument, we can also state that
(4.14) * x, n X j=1 xj + 2 ≤ n X k=1
|hx, xki| 2 +kxk n X k=1
hx, xki(S−xk)
for anyx∈H.
Utilising the inequality
(4.15) n X j=1 αjzj
2 ≤ n X j=1 |αj|2
max 1≤j≤nkzjk
2 +
X
1≤j6=k≤n
|hzj, zki| 2 1 2 ,
whereαj∈C,zj ∈H, j∈ {1, . . . , n},that has been obtained in [4], see also [5, p.
128], we can state that
(4.16) n X k=1
hx, xki(S−xk)
≤ n X k=1
|hx, xki|2
!12
max
1≤k≤nkS−xkk 2
+
X
1≤k6=l≤n
|hS−xk, S−xli|2
1 2 1 2
for anyx∈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:
(4.17) n X j=1 αjzj
2 ≤ n X j=1 |αj|
2 max 1≤k≤nkzkk
2
+ (n−1) max
1≤k6=l≤n|hzk, zli|
,
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
1≤k≤nkS−xkk 2
+ (n−1) max
1≤k6=l≤n|hS−xk, S−xli| 2
1 2#
Other similar results may be stated by making use of the results from [6]. The details are left to the interested reader.
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:
Ifγ,Γ∈K(K=C,R) andαj ∈K,j∈ {1, . . . , n} with the property that
0≤Re [(Γ−αj) (αj−¯γ)] (5.1)
= (Re Γ−Reαj) (Reαj−Reγ) + (Im Γ−Imαj) (Imαj−Imγ) or, equivalently,
(5.2)
αj−γ+ Γ 2
≤ 1 2|Γ−γ| for eachj∈ {1, . . . , n},then (see for instance [5, p. 9])
(5.3) n n
X
j=1 |αj|2−
n
X
j=1 αj
2
≤ 1 4·n
2|Γ−γ|2 .
In addition, if Re (Γ¯γ)>0,then (see for example [5, p. 26]):
n n
X
j=1 |αj|
2 ≤ 1
4·
n
Reh Γ + ¯¯ γ Pnj=1αjio 2
Re (Γ¯γ) (5.4)
≤ 1 4·
|Γ +γ|2 Re (Γ¯γ)
n
X
j=1 αj
2
and
(5.5) n n
X
j=1 |αj|2−
n
X
j=1 αj
2
≤1 4 ·
|Γ−γ|2 Re (Γ¯γ)
n
X
j=1 αj
2
.
Also, if Γ6=−γ,then (see for instance [5, p. 32]):
(5.6)
n
n
X
j=1 |αj|
2
1 2
−
n
X
j=1 αj
≤ 1 4n·
|Γ−γ|2 |Γ +γ|. Finally, from [7] we can also state that
(5.7) n n
X
j=1 |αj|2−
n
X
j=1 αj
2
≤nh|Γ +γ| −2pRe (Γ¯γ)i
n
X
j=1 αj
,
provided Re (Γ¯γ)>0.
We can state and prove the following conditional inequalities for the hypo-Euclidean normk·ke:
Theorem 8. Let ϕ, φ∈KandX = (x1, . . . , xn)∈Hn such that either:
(5.9)
hx, xji − ϕ+φ
2
≤ 1 2|φ−ϕ|
or, equivalently,
(5.10) Re [(φ− hx, xji) (hxj, xi −ϕ)]¯ ≥0
for eachj ∈ {1, . . . , n} and for anyx∈H,kxk= 1. Then
(5.11) kXk2e≤ 1 nkSk
2 +1
4n|φ−ϕ| 2
.
Moreover, if Re (φ¯ϕ)>0,then
(5.12) kXk2e≤ 1 4n·
|φ+ϕ|2 Re (φϕ)kSk
2
and
(5.13) kXk2e≤ 1 nkSk
2
+h|φ+ϕ| −2pRe (φ¯ϕ)ikSk.
If φ6=−ϕ,then
(5.14) kXke≤ 1 nkSk+
1 4n·
|φ−ϕ|2 |φ+ϕ| ,
whereS=Pn
j=1xj.
Proof. We only prove the inequality (5.11).
Let x∈ H, kxk = 1. Then, on applying the inequality (5.3) for αj = hx, xji, j∈ {1, . . . , n} and Γ =φ, γ=ϕ,we can state that
(5.15)
n
X
j=1
|hx, xji| 2
≤ 1 n
*
x, n
X
j=1 xj
+
2
+1
4n|φ−ϕ| 2
.
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
hx, xji −ϕ+φ 2
≤
xj−ϕ+φ 2 ·x
for any j∈ {1, . . . , n} andx∈H,kxk= 1, then a sufficient condition for (5.9) to hold is that
xj−ϕ+φ 2 ·x
≤1 2|φ−ϕ|
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 spaceH:
(6.1) k(T1, . . . , Tn)ke:= sup (λ1,...,λn)∈Bn
kλ1T1+· · ·+λnTnk,
where (T1, . . . , Tn) ∈ B(n)(H) and Bn :=
n
(λ1, . . . , λn)∈Cn
Pn
i=1|λi| 2
≤1o is the Euclidean closed ball inCn.It is clear thatk·keis a norm onB
(n)(H) and for any (T1, . . . , Tn)∈B(n)(H) we have
(6.2) k(T1, . . . , Tn)ke=k(T
∗
1, . . . , Tn∗)ke, whereTi∗ is the adjoint operator ofTi, i∈ {1, . . . , n}.
It has been shown in [9] that the following inequality holds true:
(6.3) √1 n
n
X
j=1 TjTj∗
1 2
≤ k(T1, . . . , Tn)ke≤
n
X
j=1 TjTj∗
1 2
for any n−tuple (T1, . . . , Tn) ∈ B(n)(H) and the constants √1
n and 1 are best possible.
In the same paper [9] the author has introduced the Euclidean operator radius
of ann−tuple of operators (T1, . . . , Tn) by
(6.4) we(T1, . . . , Tn) := sup
kxk=1
n
X
j=1
|hTjx, xi| 2
1 2
and proved thatwe(·) is a norm onB(n)(H) and satisfies the double inequality: (6.5) 1
2k(T1, . . . , Tn)ke≤we(T1, . . . , Tn)≤ k(T1, . . . , Tn)ke for eachn−tuple (T1, . . . , Tn)∈B(n)(H).
As pointed out in [9], the Euclidean numerical radius also satisfies the double inequality:
(6.6) 1 2√n
n
X
j=1 TjTj∗
1 2
≤we(T1, . . . , Tn)≤
n
X
j=1 TjTj∗
1 2
for any (T1, . . . , Tn)∈B(n)(H) and the constants 2√1n and 1 are best possible. We are now able to establish the following natural connections that exists be-tween the hypo-Euclidean norm of vectors in a Cartesian product of Hilbert spaces and the normk·keforn−tuples of operators in the Banach algebraB(H). Theorem 9. For any(T1, . . . , Tn)∈B(n)(H)we have
k(T1, . . . , Tn)ke= sup
kyk=1
k(T1y, . . . , Tny)ke (6.7)
= sup
kyk=1,kxk=1
n
X
j=1
|hTjy, xi| 2
Proof. By the definition of the k·ke−norm on B(n)(H) and the hypo-Euclidean norm onHn,we have:
k(T1, . . . , Tn)ke= sup (λ1,...,λn)∈Bn
"
sup
kyk=1
k(λ1T1+· · ·+λnTn)yk
#
(6.8)
= sup
kyk=1
"
sup (λ1,...,λn)∈Bn
kλ1T1y+· · ·+λnTnyk
#
= sup
kyk=1
k(T1y, . . . , Tny)ke.
Utilising the representation of the hypo-Euclidean norm on Hn from Theorem 2, we have
(6.9) k(T1y, . . . , Tny)ke= sup
kxk=1
n
X
j=1
|hTjy, xi|2
1 2 .
Making use of (6.8) and (6.9) we deduce the desired equality (6.7).
Remark 5. Utilising Theorem 1, we have
(6.10)
n
X
j=1 kTjyk
2
1 2
≥ k(T1y, . . . , Tny)ke≥ 1 √ n
n
X
j=1 kTjyk
2
1 2
for any y∈H,kyk= 1.
Since
n
X
j=1
kTjyk2=
* n X
j=1
Tj∗Tjy, y
+
, kyk= 1
hence, on taking the supremum over kyk= 1in (6.10) and on observing that
sup
kyk=1
* n
X
j=1
Tj∗Tjy, y
+
=w
n
X
j=1 Tj∗Tj
=
n
X
j=1 Tj∗Tj
=
n
X
j=1 TjTj∗
,
we deduce the inequality (6.3) that has been established in [9] by a different argu-ment.
We observe that, due to the representation Theorem 9, some inequalities ob-tained for the hypo-Euclidean norm can be utilised in obtaining various new in-equalities for the operator normk·keby employing a standard approach consisting in taking the supremum overkyk= 1,as described in the above remark.
The following different lower bound for the Euclidean operator normk·kecan be stated:
Proposition 2. For any(T1, . . . , Tn)∈B(n)(H), we have
(6.11) k(T1, . . . , Tn)ke≥√1
Proof. Utilising Proposition 1 and Theorem 9 we have:
k(T1, . . . , Tn)ke= sup
kyk=1
k(T1y, . . . , Tny)ke
≥ √1 nksupyk=1
kT1y+· · ·+Tnyk
= √1
nkT1+· · ·+Tnk which is the desired inequality (6.11).
We can state the following results concerning various upper bounds for the op-erator normk(., . . . , .)ke:
Theorem 10. For any (T1, . . . , Tn)∈B(n)(H), we have the inequalities:
(6.12) k(T1, . . . , Tn)k2e≤
max 1≤j≤n
n
kTjk2o+
" P
1≤j6=k≤n
w2(Tk∗Tj)
#12 ;
max 1≤j≤n
n
kTjk 2o
+ (n−1) max
1≤j6=k≤n{w(T
∗
kTj)};
max
1≤j≤n
n
kTjk 2o
n
P
j=1 Tj∗Tj
2
+ max
1≤j6=k≤n{kTjk kTkk}
P
1≤j6=k≤n
w TkTj∗
#12 .
The proof follows by Theorem 3 and Theorem 9 and the details are omitted. On utilising the inequalities (3.8) and (3.17) we can state the following result as well:
Theorem 11. For any (T1, . . . , Tn)∈B(n)(H), we have:
(6.13) k(T1, . . . , Tn)k2e≤
max 1≤j≤n
n
P
k=1 w(T∗
kTj)
;
"
n
P
j,k=1
w2(Tk∗Tj)
#12 ;
n max 1≤j≤n
n
P
k=1 w2(T∗
kTj)
12 ;
n
"
n
P
j=1 max 1≤k≤n
w2(T∗
kTj)
#12 .
Theorem 12. Let (T1, . . . , Tn)∈B(n)(H)andϕ, φ∈Ksuch that
(6.14)
Tjy−ϕ+φ 2 ·x
≤ 1
2|φ−ϕ| for any kxk=kyk= 1
and for each j∈ {1, . . . , n}.Then
(6.15) k(T1, . . . , Tn)k2e≤ 1 n
n
X
j=1 Tj
2
+ 1
n|φ−ϕ| 2
.
In addition, ifRe (φ¯ϕ)>0,then
(6.16) k(T1, . . . , Tn)k2e≤ 1 4n·
|φ+ϕ|2 Re (φ¯ϕ)
n
X
j=1 Tj
2
and
(6.17) k(T1, . . . , Tn)k2e≤ 1 n
n
X
j=1 Tj
2
+h|φ+ϕ| −2pRe (φ¯ϕ)i
n
X
j=1 Tj
.
If φ6=−ϕ,then also
(6.18) k(T1, . . . , Tn)ke≤√1 n
n
X
j=1 Tj
2
+1 4
√
n·|φ−ϕ| 2
|φ+ϕ| .
Proof. For anyx, y∈H withkxk=kyk= 1 we have
hx, Tjyi − ϕ+φ
2
=
x, Tjy− ϕ+φ
2 x
≤ kxk
Tjy− ϕ+φ
2 x
≤1 2|φ−ϕ| for eachj∈ {1, . . . , n}.
Now, on applying Theorem 8 forxj=Tjy,we can write from (5.11) the following inequality
k(T1y, . . . , Tny)ke≤ 1 n
n
X
j=1 Tjy
+1
4n|φ−ϕ| 2
for eachywithkyk= 1.
Taking the supremum overkyk= 1 and utilising Theorem 9, we deduce (6.15). The other inequalities follow by a similar procedure on making use of the in-equalities (5.12) – (5.14) and the details are omitted.
Remark 6. The inequality (6.14) is equivalent with
0≤Re [(φ− hx, Tjyi) (hTjy, xi −ϕ)]¯ (6.19)
for each j ∈ {1, . . . , n} and kxk = kyk = 1. A sufficient condition for (6.19) to hold is then:
(6.20)
Re (ϕ)≤Rehx, Tjyi ≤Re (φ) Im (ϕ)≤Imhx, Tjyi ≤Im (φ)
for any x, y∈H withkxk=kyk= 1andj∈ {1, . . . , n}.
7. A Norm onB(H) For an operatorA∈B(H) we define
(7.1) δ(A) :=k(A, A∗)ke= sup (λ,µ)∈B2
kλA+µA∗k,
whereB2 is the Euclidean unit ball inC2.
The properties of this functional are embodied in the following theorem:
Theorem 13. The functional δ is a norm on B(H) and satisfies the double inequality:
(7.2) kAk ≤δ(A)≤2kAk
for any A∈B(H).
Moreover, we have the inequalities
(7.3)
√ 2 2
A
2+ (A∗)2
1 2
≤δ(A)≤ A
2+ (A∗)2
1 2 ,
and
(7.4)
√ 2
2 kA+A
∗k ≤δ(A)≤hkAk2
+w A2i
1 2
for any A∈B(H), respectively.
Proof. First of all, observe, by Theorem 8, that we have the representation
(7.5) δ(A) = sup
kxk=1,kyk=1
h
|hAy, xi|2+|hA∗y, xi|2i 1 2
for eachA∈B(H).
Obviously δ(A) ≥ 0 for each A ∈ B(H) and of δ(A) = 0 then, by (7.5), hAy, xi= 0 for anyx, y∈H with kxk=kyk = 1 which implies thatA= 0.Also, by (7.5), we observe that
δ(αA) = sup
kxk=1,kyk=1
h
|hαAy, xi|2+|h¯αA∗y, xi|2i 1 2
=|α| sup
kxk=1,kyk=1
h
|hAy, xi|2+|hA∗y, xi|2i 1 2
Now, ifA, B∈B(H),then δ(A+B) = sup
(λ,µ)∈B2
kλA+µA∗+λB+µB∗k ≤ sup
(λ,µ)∈B2
kλA+µA∗k+ sup (λ,µ)∈B2
kλB+µB∗k =δ(A) +δ(B),
which proves the triangle inequality. Also, we observe that
δ(A)≥ sup
kxk=1,kyk=1
|hAy, xi|=kAk
and
δ(A)≤ sup
kxk=1,kyk=1
[|hAy, xi|+|hA∗y, xi|]
≤ sup
kxk=1,kyk=1
|hAy, xi|+ sup
kxk=1,kyk=1
|hA∗y, xi| = 2kAk
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 2 and the second inequality in (6.12) for the same choices.
Remark 7. It is easy to see that
√ 2 2
A
2+ (A∗)2
1 2
≤ kAk
and
A
2+ (A∗)2
1 2
,hkAk2+w A2i 1 2
≤2kAk
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 which of the lower bounds kAk,
√
2
2 kA+A
∗k of the
norm δ(A) are better and when. The same question applies for the upper bounds
A
2+ (A∗)2
1 2
andhkAk2+w A2i 1 2
, respectively.
References
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[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, Gr¨uss 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.
[9] G. POPESCU, Unitary invariants in multivariable operator theory, Preprint, Arχiv.math.04/0410492.
[10] W. RUDIN,Function Theory in the Unit Ball of Cn, Springer Verlag, New York, Berlin, 1980.
School of Computer Science and Mathematics, Victoria University, PO Box 14428, Melbourne City, 8001, VIC, Australia
E-mail address:sever.dragomir@vu.edu.au
