(1)THE HYPO-EUCLIDEAN NORM OF AN n−TUPLE OF VECTORS IN INNER PRODUCT SPACES AND APPLICATIONS S.S

THE HYPO-EUCLIDEAN NORM OF AN n−TUPLE OF VECTORS IN INNER PRODUCT SPACES AND APPLICATIONS

S.S. DRAGOMIR

SCHOOL OFCOMPUTERSCIENCE ANDMATHEMATICS

VICTORIAUNIVERSITY

PO BOX14428, MELBOURNECITY

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

ABSTRACT. The concept of hypo-Euclidean norm for ann−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 forn−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. 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 normk·k_nand the unit ball

B(k·k_n) :={λ= (λ₁, . . . , λ_n)∈Kⁿ| kλk_n≤1}. As an example of such norms we should mention the usualp−norms

(1.1) kλk_n,p:=

( max{|λ1|, . . . ,|λn|} if p=∞;

(Pn

k=1|λk|^p)^1p if p∈[1,∞).

The Euclidean norm is obtained forp= 2, i.e., kλk_n,2 =

n

X

k=1

|λ_k|²

!¹₂ .

084-07

It is well known that onEⁿ:=E× · · · ×Eendowed with the canonical linear structure we can define the followingp−norms:

(1.2) kXk_n,p:=

( max{kx₁k, . . . ,kx_nk} if p=∞;

(Pn

k=1kx_kk^p)^1p if p∈[1,∞);

whereX = (x₁, . . . , x_n)∈Eⁿ.

For a given normk·k_nonKⁿwe define the functionalk·k_h,n:Eⁿ→[0,∞)given by

(1.3) kXk_h,n:= sup

(λ1,...,λn)∈B(^k·kn)

n

X

j=1

λ_jx_j , whereX = (x₁, . . . , x_n)∈Eⁿ.

It is easy to see that:

(i) kXk_h,n≥0for anyX ∈Eⁿ;

(ii) kX+Yk_h,n≤ kXk_h,n+kYk_h,nfor anyX, Y ∈Eⁿ; (iii) kαXk_h,n =|α| kXk_h,nfor eachα∈KandX ∈Eⁿ;

and therefore k·k_h,nis a semi-norm on Eⁿ. This will be called the hypo-semi-norm generated by the normk·k_nonXⁿ.

We observe thatkXk_h,n= 0if and only ifPn

j=1λjxj = 0for any(λ1, . . . , λn)∈B(k·k_n). If there exists λ⁰₁, . . . , λ⁰_n 6= 0 such that (λ⁰₁,0, . . . ,0), (0, λ⁰₂, . . . ,0), . . . , (0,0, . . . , λ⁰_n) ∈ B(k·k_n)then the semi-norm generated byk·k_nis a norm onEⁿ.

If byBn,pwithp∈[1,∞]we denote the balls generated by the p−normsk·k_n,ponKⁿ,then we can obtain the following hypo-p-norms onXⁿ:

(1.4) kXk_h,n,p := sup

(λ1,...,λn)∈Bn,p

n

X

j=1

λ_jx_j , withp∈[1,∞].

Forp= 2,we have the Euclidean ball inKⁿ,which we denote byBⁿ, Bn=

(

λ= (λ₁, . . . , λ_n)∈Kⁿ

n

X

i=1

|λ_i|² ≤1 )

that generates the hypo-Euclidean norm onEⁿ,i.e.,

(1.5) kXk_h,e := sup

(λ1,...,λn)∈Bn

n

X

j=1

λ_jx_j .

Moreover, ifE =H, H is a Hilbert space overK, then the hypo-Euclidean norm onHⁿwill be denoted simply by

(1.6) k(x₁, . . . , x_n)k_e := sup

(λ1,...,λn)∈Bn

n

X

j=1

λ_jx_j , 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 on H and used it to investigate var- ious 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

Pn

j=1|hx, x_ji|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 Hⁿ, 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 betweenp−norms and the hypo-Euclidean norm. Section 5 is devoted to the pre- sentation 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 normk(·, . . . ,·)k_e introduced by Popescu in [9] is in- vestigated. 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.

2. FUNDAMENTALPROPERTIES

Let(H;h·,·i)be a Hilbert space overKandn ∈ N,n ≥ 1.In the Cartesian productHⁿ :=

H× · · · ×H,for then−tuples of vectorsX = (x₁, . . . , x_n), Y = (y₁, . . . , y_n) ∈Hⁿ,we can define the inner producth·,·iby

(2.1) hX, Yi:=

n

X

j=1

hx_j, y_ji, X, Y ∈Hⁿ, which generates the Euclidean normk·k₂ onHⁿ,i.e.,

(2.2) kXk₂ :=

n

X

j=1

kx_jk²

!¹₂

, X ∈Hⁿ.

The following result connects the usual Euclidean norm k·kwith the hypo-Euclidean norm k·k_e.

Theorem 2.1. For anyX ∈Hⁿwe have the inequalities

(2.3) kXk₂ ≥ kXk_e≥ 1

√nkXk₂, i.e.,k·k₂andk·k_e are equivalent norms onHⁿ.

Proof. By the Cauchy-Bunyakovsky-Schwarz inequality we have

(2.4)

n

X

j=1

λ_jx_j

≤

n

X

j=1

|λ_j|²

!¹_{2 n} X

j=1

kx_jk²

!¹₂

for any(λ₁, . . . , λ_n)∈Kⁿ.Taking the supremum over(λ₁, . . . , λ_n)∈Bnin (2.4) we obtain the first inequality in (2.3).

If byσwe denote the rotation-invariant normalised positive Borel measure on the unit sphere

∂Bⁿ ∂Bⁿ = (λ₁, . . . , λ_n)∈Kⁿ

Pn

i=1|λ_i|² = 1

whose existence and properties have been

pointed out in [10], then we can state that Z

∂Bn

|λ_k|²dσ(λ) = 1 n and (2.5)

Z

∂Bn

λ_kλ_jdσ(λ) = 0 if k 6=j, k, j= 1, . . . , n.

Utilising these properties, we have kXk²_e = sup

(λ1,...,λn)∈Bn

n

X

k=1

λ_kx_k

2

= sup

(λ1,...,λn)∈Bn

" _n X

k,j=1

λ_kλ_jhx_k, x_ji

#

≥ Z

∂Bn

" _n X

k,j=1

λ_kλ_jhx_k, x_ji

#

dσ(λ) =

n

X

k,j=1

Z

∂Bn

λ_kλ_jhx_k, x_ji dσ(λ)

= 1 n

n

X

k=1

kx_kk² = 1

nkXk²₂,

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 anyX ∈HⁿwithX = (x1, . . . , xn),we have

(2.6) kXk_e = sup

kxk=1 n

X

j=1

|hx, x_ji|²

!¹₂ .

Proof. We use the following well known representation result for scalars:

(2.7)

n

X

j=1

|z_j|² = sup

(λ1,...,λn)∈Bn

n

X

j=1

λ_jz_j

2

, where(z₁, . . . , z_n)∈Kⁿ.

Utilising this property, we thus have (2.8)

n

X

j=1

|hx, x_ji|²

!¹₂

= sup

(λ1,...,λn)∈Bn

* x,

n

X

j=1

λ_jx_j +

for anyx∈H.

Now, taking the supremum overkxk= 1in (2.8) we get sup

kxk=1 n

X

j=1

|hx, x_ji|²

!¹₂

= sup

kxk=1

"

sup

(λ1,...,λn)∈Bn

* x,

n

X

j=1

λ_jx_j +

#

= sup

(λ1,...,λn)∈Bn

"

sup

kxk=1

* x,

n

X

j=1

λ_jx_j +

#

= sup

(λ1,...,λn)∈Bn

n

X

j=1

λ_jx_j ,

since, in any Hilbert space we have thatsup_kuk=1|hu, vi|=kvkfor eachv ∈H.

Corollary 2.3. If X = (x₁, . . . , x_n)is ann−tuple of orthonormal vectors, i.e., we recall that kx_kk= 1andhx_k, x_ji= 0fork, j ∈ {1, . . . , n}withk 6=j,thenkXk_e≤1.

The proof is obvious by Bessel’s inequality.

The next proposition contains two lower bounds for the hypo-Euclidean norm that are some- times better than the one in (2.3), as will be shown by some examples later.

Proposition 2.4. For anyX = (x₁, . . . , x_n)∈Hⁿ\ {0}we have

(2.9) kXk_e≥











1 kXk₂

Pn

j=1kxjkxj

,

√1 n

Pn j=1x_j

.

Proof. By the definition of the hypo-Euclidean norm we have that, if(λ⁰₁, . . . , λ⁰_n) ∈ Bn,then obviously

kXk_e≥

n

X

j=1

λ⁰_jx_j . The choice

λ⁰_j := kxjk

kXk₂, j ∈ {1, . . . , n},

which satisfies the condition(λ⁰₁, . . . , λ⁰_n) ∈ Bn will produce the first inequality while the se- lection

λ⁰_j = 1

√n, j ∈ {1, . . . , n},

will give the second inequality in (2.9).

Remark 2.5. Forn= 2,the hypo-Euclidean norm onH² k(x, y)k_e = sup

(λ,µ)∈B2

kλx+µyk= sup

kzk=1

|hz, xi|²+|hz, yi|²¹₂ is bounded below by

B₁(x, y) := 1

√2 kxk²+kyk²¹₂ , B₂(x, y) := kkxkx+kykyk

kxk²+kyk²¹₂ and

B₃(x, y) := 1

√2kx+yk.

IfH =Cendowed with the canonical inner producthx, yi:=x¯ywherex, y ∈C, then B₁(x, y) = 1

√2 |x|²+|y|²¹₂ , B₂(x, y) = ||x|x+|y|y|

|x|²+|y|²¹₂ and

B₃(x, y) = 1

√2|x+y|, x, y ∈C.

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 boundB₁ is not always better thanB₂orB₃.However, since the plot ofD₃(x, y) :=B₂(x, y)−B₃(x, y) (see Figure 2.3) appears to indicate that, at least in the case ofC²,it may be possible that the boundB₂is always better thanB₃,hence we can ask in general which bound from (2.6) is better for a givenn ≥ 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 -10

10

Figure 2.1: The behaviour ofD₁(x, y)

-10 -5 0 y -4 5

10 -2

5 0

0

-5 10

2

x -10

4 6 8 10

Figure 2.2: The behaviour ofD2(x, y)

-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 2.3: The behaviour ofD3(x, y)

3. UPPERBOUNDS VIA THEBOAS-BELLMAN ANDBOMBIERITYPE 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, . . . , y_n}(see also [8, p. 392] or [5, p. 125]):

(3.1)

n

X

j=1

|hx, y_ji|² ≤ kxk²



max

1≤j≤nky_jk²+ X

1≤j6=k≤n

|hy_k, y_ji|²

!¹₂



for anyx, y₁. . . , y_nvectors 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 anyX = (x1, . . . , xn)∈Hⁿ,we have

(3.2) kXk²_e ≤











1≤j≤nmax kx_jk²+ P

1≤j6=k≤n

|hx_k, x_ji|²

!¹₂ ,

1≤j≤nmax kx_jk²+ (n−1) max

1≤j6=k≤n|hx_k, x_ji|;

(3.3) kXk²_e ≤

"

1≤j≤nmax kxjk²

n

X

j=1

kxjk²+ max

1≤j6=k≤n{kxjk kxkk} X

1≤j6=k≤n

|hxj, xki|

#¹₂

and

(3.4) kXk⁴_e ≤







1≤j≤nmax kx_jk²

n

P

j=1

kx_jk²+ (n−1)kXk²_e max

1≤j6=k≤n|hx_j, x_ki|, kXk²_e max

1≤j≤nkx_jk²+ max

1≤j6=k≤n{kx_jk kx_kk} P

1≤j6=k≤n

|hx_j, x_ki|.

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

(3.5)

n

X

j=1

c_jhx, y_ji

2

≤ kxk²×











1≤j≤nmax |c_j|²

n

P

j=1

ky_jk²,

n

P

j=1

|c_j|² max

1≤j≤nky_jk², +kxk²×







1≤j6=k≤nmax {|c_jc_k|} P

1≤j6=k≤n

|hy_j, y_ki|, (n−1)

n

P

j=1

|cj|² max

1≤j6=k≤n|hyj, yki|, for any y₁, . . . , y_n, x ∈ H and c₁, . . . , c_n ∈ K, where (3.5) should be seen as all possible configurations.

The choicec_j =hx, y_ji, j ∈ {1, . . . , n}will produce the following four inequalities:

(3.6)

" _n X

j=1

|hx, y_ji|²

#2

≤ kxk²×











1≤j≤nmax |hx, y_ji|²

n

P

j=1

ky_jk²,

n

P

j=1

|hx, y_ji|² max

1≤j≤nky_jk², +kxk²×







1≤j6=k≤nmax {|hx, y_ji| |hx, y_ki|} P

1≤j6=k≤n

|hy_j, y_ki|, (n−1)

n

P

j=1

|hx, y_ji|² max

1≤j6=k≤n|hy_j, y_ki|.

Taking the supremum overkxk = 1and 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, y_ji|² ≤ kxk² max

1≤j≤n

( _n X

k=1

|hy_j, y_ki|

) ,

for any x ∈ H, where y₁, . . . , y_n 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 normk·k_e.

Theorem 3.2. For anyX = (x₁, . . . , x_n)∈Hⁿ,we have

(3.8) kXk²_e ≤ max

1≤j≤n

( _n X

k=1

|hx_j, x_ki|

) .

In [6] (see also [5, p. 138]), we have established the following norm inequalities:

(3.9)

n

X

j=1

α_jz_j

2

≤n^1p+1t−1

n

X

k=1

|α_k|²





n

X

k=1 n

X

j=1

|hz_j, z_ki|^q

!^u_q



1 u

,

where ¹_p + ¹_q = 1, ¹_t +¹_u = 1and1< p ≤2,1< t≤2andα_j ∈C,z_j ∈H, j ∈ {1, . . . , n}. An interesting particular case of (3.9) obtained forp=q = 2, t=u= 2is incorporated in

(3.10)

n

X

j=1

α_jz_j

2

≤

n

X

k=1

|α_k|²

n

X

j,k=1

|hz_j, z_ki|²

!¹₂ .

Other similar inequalities for norms are the following ones [6] (see also [5, pp. 139-140]):

(3.11)

n

X

j=1

αjzj

2

≤n^1p

n

X

k=1

|αk|² max

1≤j≤n







" _n X

k=1

|hzj, zki|^q

#¹_q



 ,

provided that1< p≤2and ¹_p +¹_q = 1, α_j ∈C,z_j ∈H, j ∈ {1, . . . , n}.In the particular case p=q = 2,we have

(3.12)

n

X

j=1

α_jz_j

2

≤√ n

n

X

k=1

|α_k|² max

1≤j≤n

" _n X

k=1

|hz_j, z_ki|²

#¹₂ .

Also, if1< m≤2,then [6]:

(3.13)

n

X

j=1

α_jz_j

2

≤n^m1

n

X

k=1

|α_k|² ( _n

X

j=1

1≤k≤nmax |hz_j, z_ki|^l )¹_l

,

where _m¹ + ¹_l = 1.Form=l= 2,we get

(3.14)

n

X

j=1

α_jz_j

2

≤√ n

n

X

k=1

|α_k|²

" _n X

j=1

1≤k≤nmax |hz_j, z_ki|² #¹₂

.

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 3.1 and 3.2:

Theorem 3.3. For anyX = (x₁, . . . , x_n)∈Hⁿ,we have

(3.16) kXk²_e ≤



























































n^1p+1t−1







n

P

k=1 n

P

j=1

|hx_j, x_ki|^q

!^u_q





1 u

where ¹_p +¹_q = 1,

1

t +_u¹ = 1 and 1< p ≤2, 1< t≤2;

n^1p max

1≤j≤n







"

n

P

j=1

|hx_j, x_ki|^q

#¹_q





where ¹_p +¹_q = 1 and 1< p≤2;

n^m1 ( n

P

j=1

1≤k≤nmax |hx_j, x_ki|^l )¹_l

where _m¹ +¹_l = 1 and 1< m≤2;

n max

1≤k≤n|hx_k, z_ji|; and, in particular,

(3.17) kXk²_e ≤



























"

n

P

j,k=1

|hx_j, x_ki|²

#¹₂

;

√n max

1≤j≤n

_n P

k=1

|hx_j, x_ki|^{2 1}₂

;

√n

"

n

P

j=1

1≤k≤nmax

|hx_j, x_ki|² #¹₂

.

4. VARIOUS INEQUALITIES FOR THE HYPO-EUCLIDEANNORM

For ann−tupleX = (x₁, . . . , x_n)of vectors inH,we consider the usualp−norms:

kXk_p :=

n

X

j=1

kxjk^p

!¹_p , wherep∈[1,∞),and denote withSthe sumPn

j=1x_j.

With these notations we can state the following reverse of the inequalitykXk₂ ≥ kXk_e,that has been pointed out in Theorem 2.1.

Theorem 4.1. For anyX = (x₁, . . . , x_n)∈Hⁿ,we have

(4.1) (0≤)kXk²₂ − kXk²_e ≤ kXk²₁− kSk². If

kXk²₍₂₎ :=

n

X

j,k=1

x_j+x_k 2

2

, then also

(0≤)kXk²₂ − kXk²_e ≤ 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, x_ki|²+

X

1≤j6=k≤n

hx, x_ji hx_k, xi

≤

n

X

k=1

|hx, x_ki|²+ X

1≤j6=k≤n

|hx, x_ji| |hx_k, xi|. Taking the supremum overkxk= 1,we get

(4.4) sup

kxk=1

n

X

j=1

hx, x_ji

2

≤ sup

kxk=1 n

X

k=1

|hx, x_ki|²+ X

1≤j6=k≤n

sup

kxk=1

|hx, x_ji| · sup

kxk=1

|hx_k, xi|.

However,

sup

kxk=1

n

X

j=1

hx, x_ji

2

= sup

kxk=1

* x,

n

X

j=1

x_j +

2

=kSk²,

sup

kxk=1

|hx, x_ji|=kx_jk and sup

kxk=1

|hx, x_ki|=kx_kk forj, k ∈ {1, . . . , n},and by (4.4) we get

kSk² ≤ kXk²_e+ X

1≤j6=k≤n

kx_jk kx_jk

=kXk²_e+

n

X

j,k=1

kx_jk kx_kk −

n

X

k=1

kx_kk²

=kXk²_e+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, x_ji

2

= Re

" _n X

k,j=1

hx, x_ji hx_k, xi

# (4.5)

=

n

X

k=1

|hx, x_ki|²+ X

1≤j6=k≤n

Re [hx, x_ji hx_k, 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, x_ji hx_k, xi]≤ 1 4

X

1≤k6=j≤n

|hx, x_ji+hx, x_ki|²

= X

1≤k6=j≤n

x,x_j+x_k 2

2

, and by (4.5) we get

(4.7)

n

X

j=1

hx, x_ji

2

≤

n

X

k=1

|hx, x_ki|²+ X

1≤k6=j≤n

x,x_j+x_k 2

2

for anyx∈H.

Taking the supremum overkxk= 1in (4.7) we deduce

n

X

j=1

x_j

2

≤ kXk²_e+ X

1≤k6=j≤n

x_j +x_k 2

2

=kXk²_e+

n

X

k,j=1

x_j +x_k 2

2

−

n

X

k=1

kx_kk² which provides the first inequality in (4.2).

By the convexity ofk·k² we have

n

X

j,k=1

x_j +x_k 2

2

≤ 1 2

n

X

j,k=1

kx_jk²+kx_kk²

=n

n

X

k=1

kx_kk²

and the last part of (4.2) is obvious.

Remark 4.2. Forn= 2, X = (x, y)∈H² we have the upper bounds B₁(x, y) :=kxk²+kyk²− kx+yk²

= 2 (kxk kyk −Rehx, yi) and

B₂(x, y) :=kxk²+kyk²

for the difference kXk²₂ − kXk²_e, X ∈ H² as 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 4.1 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 anyX = (x₁, . . . , x_n)∈Hⁿ,we have

(4.8) kSk² ≤ kXk_e



kXk_e+

n

X

k=1

kS−xkk²

!¹₂



and

(4.9) kSk² ≤ kXk_e





kXk_e+







1≤k≤nmax kS−xkk²+ X

1≤k6=l≤n

|hS−xk, S−xli|²

!¹₂





1 2



,

-10 -5 0 y 5 -200

10 5 10

0 -5 -10

-100

x 0

100 200

Figure 4.1: The behaviour of∆ (x, y)

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

+

for anyx∈H.

By the Schwarz inequality in the inner product space(H,h·,·i), we have that Re

*

x, X

1≤j6=k≤n

hx, x_kix_j +

≤ kxk

X

1≤j6=k≤n

hx, x_kix_j (4.11)

=kxk

n

X

j,k=1

hx, x_kix_j −

n

X

k=1

hx, x_kix_k

=kxk

* x,

n

X

k=1

x_k + _n

X

j=1

x_j −

n

X

k=1

hx, x_kix_k

=kxk

n

X

k=1

hx, x_ki(S−x_k) . Utilising the Cauchy-Bunyakovsky-Schwarz inequality we have

(4.12)

n

X

k=1

hx, x_ki(S−x_k)

≤

n

X

k=1

|hx, x_ki|²

!¹_{2 n} X

k=1

kS−x_kk²

!¹₂

and then by (4.10) – (4.12) we can state the inequality:

(4.13)

n

X

j=1

hx, x_ji

2

≤

n

X

k=1

|hx, x_ki|²

!¹₂ 



n

X

k=1

|hx, x_ki|²

!¹₂ +

n

X

k=1

kS−x_kk²

!¹₂



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 (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

α_jz_j

2

≤

n

X

j=1

|α_j|²







1≤j≤nmax kz_jk² + X

1≤j6=k≤n

|hz_j, z_ki|²

!¹₂



 ,

whereα_j ∈ C,z_j ∈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, x_ki(S−x_k)

≤

n

X

k=1

|hx, x_ki|²

!¹₂ 





1≤k≤nmax kS−x_kk² + X

1≤k6=l≤n

|hS−x_k, S−x_li|²

!¹₂





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 4.4. On utilising the inequality:

(4.17)

n

X

j=1

α_jz_j

2

≤

n

X

j=1

|α_j|²

1≤k≤nmax kz_kk²+ (n−1) max

1≤k6=l≤n|hz_k, z_li|

,

whereα_j ∈ C, z_j ∈ 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

≤ kXk_e

"

kXk_e+

1≤k≤nmax kS−x_kk²+ (n−1) max

1≤k6=l≤n|hS−x_k, S−x_li|^{2 1}₂#

for anyX= (x₁, . . . , x_n)∈Hⁿ.

Other similar results may be stated by making use of the results from [6]. The details are left to the interested reader.

5. REVERSEINEQUALITIES

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 com- plex numbers as follows:

Ifγ,Γ∈K(K=C,R)andα_j ∈K,j ∈ {1, . . . , n}with the property that 0≤Re [(Γ−α_j) (α_j−γ)]¯

(5.1)

= (Re Γ−Reα_j) (Reα_j −Reγ) + (Im Γ−Imα_j) (Imα_j−Imγ) or, equivalently,

(5.2)

α_j − 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

|α_j|²−

n

X

j=1

α_j

2

≤ 1

4 ·n²|Γ−γ|². In addition, ifRe (Γ¯γ)>0,then (see for example [5, p. 26]):

n

n

X

j=1

|α_j|² ≤ 1 4·

n Re

h Γ + ¯¯ γ Pn j=1αj

io2

Re (Γ¯γ) (5.4)

≤ 1

4· |Γ +γ|² Re (Γ¯γ)

n

X

j=1

α_j

2

and

(5.5) n

n

X

j=1

|α_j|²−

n

X

j=1

αj

2

≤ 1

4· |Γ−γ|² Re (Γ¯γ)

n

X

j=1

αj

2

. Also, ifΓ6=−γ,then (see for instance [5, p. 32]):

(5.6) n

n

X

j=1

|α_j|²

!¹₂

−

n

X

j=1

α_j

≤ 1

4n·|Γ−γ|²

|Γ +γ|. Finally, from [7] we can also state that

(5.7) n

n

X

j=1

|α_j|²−

n

X

j=1

α_j

2

≤nh

|Γ +γ| −2p

Re (Γ¯γ)i

n

X

j=1

α_j , providedRe (Γ¯γ)>0.

We notice that a simple sufficient condition for (5.1) to hold is that (5.8) Re Γ≥Reα_j ≥Reγ and Im Γ≥Imα_j ≥Imγ for eachj ∈ {1, . . . , n}.

We can state and prove the following conditional inequalities for the hypo-Euclidean norm k·k_e :

Theorem 5.1. Letϕ, φ∈KandX = (x₁, . . . , x_n)∈Hⁿsuch that either:

(5.9)

hx, xji − ϕ+φ 2

≤ 1

2|φ−ϕ|

or, equivalently,

(5.10) Re [(φ− hx, x_ji) (hx_j, xi −ϕ)]¯ ≥0 for eachj ∈ {1, . . . , n}and for anyx∈H,kxk= 1.Then

(5.11) kXk²_e ≤ 1

n kSk²+ 1

4n|φ−ϕ|². Moreover, ifRe (φϕ)¯ >0,then

(5.12) kXk²_e ≤ 1

4n ·|φ+ϕ|² Re (φϕ) kSk² and

(5.13) kXk²_e ≤ 1

nkSk²+h

|φ+ϕ| −2p

Re (φϕ)¯ i kSk. Ifφ 6=−ϕ,then

(5.14) kXk_e ≤ 1

n kSk+ 1

4n· |φ−ϕ|²

|φ+ϕ|, whereS=Pn

j=1x_j.

Proof. We only prove the inequality (5.11).

Letx∈H,kxk= 1.Then, on applying the inequality (5.3) forα_j =hx, x_ji, j ∈ {1, . . . , n}

andΓ =φ, γ =ϕ,we can state that (5.15)

n

X

j=1

|hx, x_ji|² ≤ 1 n

* x,

n

X

j=1

x_j +

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 5.2. Due to the fact that

hx, x_ji −ϕ+φ 2

≤

x_j − ϕ+φ 2 ·x

for anyj ∈ {1, . . . , n}andx∈H,kxk= 1,then a sufficient condition for (5.9) to hold is that

x_j − ϕ+φ 2 ·x

≤ 1

2|φ−ϕ|

for eachj ∈ {1, . . . , n}andx∈H,kxk= 1.

6. APPLICATIONS FORn−TUPLES OFOPERATORS

In [9], the author has introduced the following norm on the Cartesian product B⁽ⁿ⁾(H) :=

B(H)× · · · ×B(H),whereB(H)denotes the Banach algebra of all bounded linear operators defined on the complex Hilbert spaceH :

(6.1) k(T₁, . . . , T_n)k_e := sup

(λ1,...,λn)∈Bn

kλ₁T₁ +· · ·+λ_nT_nk, where (T₁, . . . , T_n) ∈ B⁽ⁿ⁾(H) and Bn :=

(λ₁, . . . , λ_n)∈Cⁿ

Pn

i=1|λ_i|² ≤1 is the Eu- clidean closed ball inCⁿ.It is clear thatk·k_eis a norm onB⁽ⁿ⁾(H)and for any(T₁, . . . , T_n)∈ B⁽ⁿ⁾(H)we have

(6.2) k(T₁, . . . , T_n)k_e =k(T₁^∗, . . . , T_n^∗)k_e, whereT_i^∗ is the adjoint operator ofT_i, i∈ {1, . . . , n}.

It has been shown in [9] that the following inequality holds true:

(6.3) 1

√n

n

X

j=1

TjT_j^∗

1 2

≤ k(T1, . . . , Tn)k_e ≤

n

X

j=1

TjT_j^∗

1 2

for anyn−tuple(T₁, . . . , T_n)∈B⁽ⁿ⁾(H)and the constants ^√1_n and1are best possible.

In the same paper [9] the author has introduced the Euclidean operator radius of ann−tuple of operators(T₁, . . . , T_n)by

(6.4) we(T1, . . . , Tn) := sup

kxk=1 n

X

j=1

|hTjx, xi|²

!¹₂

and proved thatw_e(·)is a norm onB⁽ⁿ⁾(H)and satisfies the double inequality:

(6.5) 1

2k(T₁, . . . , T_n)k_e≤w_e(T₁, . . . , T_n)≤ k(T₁, . . . , T_n)k_e for eachn−tuple(T₁, . . . , T_n)∈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

TjT_j^∗

1 2

≤we(T1, . . . , Tn)≤

n

X

j=1

TjT_j^∗

1 2

for any(T₁, . . . , T_n)∈B⁽ⁿ⁾(H)and the constants ₂^√1_n and1are 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·k_e for n−tuples of operators in the Banach algebraB(H).

Theorem 6.1. For any(T₁, . . . , T_n)∈B⁽ⁿ⁾(H)we have k(T1, . . . , Tn)k_e = sup

kyk=1

k(T1y, . . . , Tny)k_e (6.7)

= sup

kyk=1,kxk=1 n

X

j=1

|hT_jy, xi|²

!¹₂ .

Proof. By the definition of thek·k_e−norm onB⁽ⁿ⁾(H)and the hypo-Euclidean norm on Hⁿ, we have:

k(T₁, . . . , T_n)k_e = sup

(λ1,...,λn)∈Bn

"

sup

kyk=1

k(λ₁T₁+· · ·+λ_nT_n)yk

# (6.8)

= sup

kyk=1

"

sup

(λ1,...,λn)∈Bn

kλ1T1y+· · ·+λnTnyk

#

= sup

kyk=1

k(T₁y, . . . , T_ny)k_e.

Utilising the representation of the hypo-Euclidean norm onHⁿfrom Theorem 2.2, we have (6.9) k(T1y, . . . , Tny)k_e = sup

kxk=1 n

X

j=1

|hTjy, xi|²

!¹₂ .

Making use of (6.8) and (6.9) we deduce the desired equality (6.7).

Remark 6.2. Utilising Theorem 2.1, we have (6.10)

n

X

j=1

kT_jyk²

!¹₂

≥ k(T₁y, . . . , T_ny)k_e≥ 1

√n

n

X

j=1

kT_jyk²

!¹₂

for anyy∈H,kyk= 1.

Since

n

X

j=1

kT_jyk² =

* _n X

j=1

T_j^∗T_jy, y +

, kyk= 1

hence, on taking the supremum overkyk= 1in (6.10) and on observing that sup

kyk=1

* _n X

j=1

T_j^∗Tjy, y +

=w

n

X

j=1

T_j^∗Tj

!

=

n

X

j=1

T_j^∗Tj

=

n

X

j=1

TjT_j^∗ , 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 normk·k_eby 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·k_ecan be stated:

Proposition 6.3. For any(T₁, . . . , T_n)∈B⁽ⁿ⁾(H),we have (6.11) k(T₁, . . . , T_n)k_e ≥ 1

√nkT₁+· · ·+T_nk. Proof. Utilising Proposition 2.4 and Theorem 6.1 we have:

k(T₁, . . . , T_n)k_e= sup

kyk=1

k(T₁y, . . . , T_ny)k_e

≥ 1

√n sup

kyk=1

kT₁y+· · ·+T_nyk

= 1

√nkT₁+· · ·+T_nk

which is the desired inequality (6.11).