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EXTRACTA

MATHEMATICAE

Volumen 33, Número 2, 2018

instituto de investigación de matemáticas de la

universidad de extremadura

EXTRACTA MATHEMATICAE

Vol. 34, Num. 2 (2019), 201 – 235

doi:10.17398/2605-5686.34.2.201

Available online June 3, 2019

Hypo-q-norms on cartesian products of algebras of
bounded linear operators on Hilbert spaces

S.S. Dragomir 1,2

1 Mathematics, College of Engineering & Science
Victoria University, Melbourne City 8001, Australia

2 DST-NRF Centre of Excellence in the Mathematical and Statistical Sciences
School of Computer Science & Applied Mathematics

University of the Witwatersrand, Johannesburg 2050, South Africa

sever.dragomir@vu.edu.au , http://rgmia.org/dragomir

Received March 6, 2019 Presented by Horst Martini
Accepted May 14, 2019

Abstract: In this paper we introduce the hypo-q-norms on a Cartesian product of algebras of bounded
linear operators on Hilbert spaces. A representation of these norms in terms of inner products, the

equivalence with the q-norms on a Cartesian product and some reverse inequalities obtained via the

scalar reverses of Cauchy-Buniakowski-Schwarz inequality are also given. Several bounds for the
norms δp, ϑp and the real norms ηr,p and θr,p are provided as well.

Key words: Hilbert spaces, bounded linear operators, operator norm and numerical radius, n-tuple

of operators, operator inequalities.

AMS Subject Class. (2010): 46C05, 26D15.

1. Introduction

In [13], 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:

‖(T1, . . . ,Tn)‖n,e := sup
(λ1,...,λn)∈Bn

‖λ1T1 + · · · + λnTn‖, (1.1)

where (T1, . . . ,Tn) ∈ B(n)(H) and

Bn :=
{

(λ1, . . . ,λn) ∈ Cn :
∑n

i=1 |λi|
2 ≤ 1

}
is the Euclidean closed ball in Cn. It is clear that ‖·‖n,e is a norm on B(n)(H)
and for any (T1, . . . ,Tn) ∈ B(n)(H) we have

‖(T1, . . . ,Tn)‖n,e = ‖(T∗1 , . . . ,T
∗
n)‖n,e , (1.2)

ISSN: 0213-8743 (print), 2605-5686 (online)

https://doi.org/10.17398/2605-5686.34.2.201
mailto:sever.dragomir@vu.edu.au
http://rgmia.org/dragomir
https://www.eweb.unex.es/eweb/extracta/
https://creativecommons.org/licenses/by-nc/3.0/


202 s.s. dragomir

where T∗i is the adjoint operator of Ti, i ∈{1, . . . ,n}.
It has been shown in [13] that the following inequality holds true:

1
√
n

∥∥∥∥∥
n∑
j=1

TjT
∗
j

∥∥∥∥∥
1
2

≤‖(T1, . . . ,Tn)‖n,e ≤

∥∥∥∥∥
n∑
j=1

TjT
∗
j

∥∥∥∥∥
1
2

(1.3)

for any n-tuple (T1, . . . ,Tn) ∈ B(n)(H) and the constants 1√n and 1 are best
possible.

In the same paper [13] the author has introduced the Euclidean operator
radius of an n-tuple of operators (T1, . . . ,Tn) by

wn,e (T1, . . . ,Tn) := sup
‖x‖=1

(
n∑
j=1

|〈Tjx,x〉|2
)1

2

(1.4)

and proved that wn,e (·) is a norm on B(n)(H) and satisfies the double inequal-
ity:

1

2
‖(T1, . . . ,Tn)‖n,e ≤ wn,e (T1, . . . ,Tn) ≤‖(T1, . . . ,Tn)‖n,e (1.5)

for each n-tuple (T1, . . . ,Tn) ∈ B(n)(H).
As pointed out in [13], the Euclidean numerical radius also satisfies the

double inequality:

1

2
√
n

∥∥∥∥∥
n∑
j=1

TjT
∗
j

∥∥∥∥∥
1
2

≤ wn,e (T1, . . . ,Tn) ≤

∥∥∥∥∥
n∑
j=1

TjT
∗
j

∥∥∥∥∥
1
2

(1.6)

for any (T1, . . . ,Tn) ∈ B(n)(H) and the constants 12√n and 1 are best possible.
Now, let (E,‖·‖) be a normed linear space over the complex number field

C. On Cn endowed with the canonical linear structure we consider a norm
‖ ·‖n. As an example of such norms we should mention the usual p-norms

‖λ‖n,p :=




max
{
|λ1|, . . . , |λn|

}
if p = ∞,(∑n

k=1 |λk|
p
)1

p
if p ∈ [1,∞).

The Euclidean norm is obtained for p = 2, i.e.,

‖λ‖n,2 :=

(
n∑
k=1

|λk|2
)1

2

.



hypo-q-norms on cartesian products 203

It is well known that on En := E×···×E endowed with the canonical linear
structure we can define the following p-norms:

‖x‖n,p :=




max
{
|x1|, . . . , |xn|

}
if p = ∞,(∑n

k=1 |xk|
p
)1

p
if p ∈ [1,∞).

where x = (x1, . . . ,xn) ∈ En.
Following the paper [5], for a given norm ‖ · ‖n on Cn, we define the

functional ‖ ·‖h,n : En → [0,∞) by

‖x‖h,n := sup
‖λ‖n≤1

∥∥∥∥∥
n∑
j=1

λjxj

∥∥∥∥∥, (1.7)
where x = (x1, . . . ,xn) ∈ En and λ = (λ1, . . . ,λn) ∈ Cn.

It is easy to see that [5]:

(i) ‖x‖h,n ≥ 0 for any x ∈ En,
(ii) ‖x + y‖h,n ≤‖x‖h,n + ‖y‖h,n for any x,y ∈ En,
(iii) ‖αx‖h,n = |α|‖x‖h,n for each α ∈ C and x ∈ En,

and therefore ‖·‖h,n is a semi-norm on En. This will be called the hypo-semi-
norm generated by the norm ‖ ·‖n on En.

We observe that ‖x‖h,n = 0 if and only if
∑n

j=1 λjxj = 0 for any

(λ1, . . . ,λn) ∈ B(‖·‖n). If there exists λ01, . . . ,λ
0
n 6= 0 such that (λ01, 0, . . . , 0),

(0,λ02, . . . , 0), . . . , (0, 0, . . . ,λ
0
n) ∈ B(‖ · ‖n) then the semi-norm generated by

‖ ·‖n is a norm on En.
If p ∈ [1,∞] and we consider the p-norms ‖·‖n,p on Cn, then we can define

the following hypo-q-norms on En:

‖x‖h,n,q := sup
‖λ‖n,p≤1

∥∥∥∥∥
n∑
j=1

λjxj

∥∥∥∥∥, (1.8)
with q ∈ [1,∞]. If p = 1, then q = ∞; if p = ∞, then q = 1; if p ∈ (1,∞),
then 1

p
+ 1

q
= 1.

For p = 2, we have the hypo-Euclidean norm on En, i.e.,

‖x‖h,n,e := sup
‖λ‖n,2≤1

∥∥∥∥∥
n∑
j=1

λjxj

∥∥∥∥∥. (1.9)



204 s.s. dragomir

If we consider now E = B(H) endowed with the operator norm ‖·‖, then
we can obtain the following hypo-q-norms on B(n)(H)

‖(T1, . . . ,Tn)‖h,n,q := sup
‖λ‖n,p≤1

∥∥∥∥∥
n∑
j=1

λjTj

∥∥∥∥∥ where p,q ∈ [1,∞], (1.10)
with the convention that if p = 1, q = ∞, if p = ∞, q = 1 and if p > 1, then
1
p

+ 1
q

= 1.
For p = 2 we obtain the hypo-Euclidian norm ‖(·, . . . , ·)‖n,e defined

in (1.2).
If we consider now E = B(H) endowed with the operator numerical radius

w(·), which is a norm on B(H), then we can obtain the following hypo-q-
numerical radius of (T1, . . . ,Tn) ∈ B(n)(H) defined by

wh,n,q(T1, . . . ,Tn) := sup
‖λ‖n,p≤1

w

(
n∑
j=1

λjTj

)
with p,q ∈ [1,∞], (1.11)

with the convention that if p = 1, q = ∞, if p = ∞, q = 1 and if p > 1, then
1
p

+ 1
q

= 1.
For p = 2 we obtain the hypo-Euclidian norm

wh,n,e(T1, . . . ,Tn) := sup
‖λ‖n,2≤1

w

(
n∑
j=1

λjTj

)
(1.12)

and will show further that it coincides with the Euclidean operator radius of
an n-tuple of operators (T1, . . . ,Tn) defined in (1.4).

Using the fundamental inequality between the operator norm and numer-
ical radius w(T) ≤‖T‖≤ 2w(T) for T ∈ B(H) we have

w

(
n∑
j=1

λjTj

)
≤

∥∥∥∥∥
n∑
j=1

λjTj

∥∥∥∥∥ ≤ 2w
(

n∑
j=1

λjTj

)

for any (T1, . . . ,Tn) ∈ B(n)(H) and any λ = (λ1, . . . ,λn) ∈ Cn. By taking the
supremum over λ with ‖λ‖n,p ≤ 1 we get

wh,n,q(T1, . . . ,Tn) ≤‖(T1, . . . ,Tn)‖h,n,q ≤ 2wh,n,q(T1, . . . ,Tn) (1.13)

with the convention that if p = 1, q = ∞, if p = ∞, q = 1 and if p > 1, then
1
p

+ 1
q

= 1.



hypo-q-norms on cartesian products 205

For p = q = 2 we recapture the inequality (1.5).
In 2012, [8] (see also [9, 10]) the author have introduced the concept of

s-q-numerical radius of an n-tuple of operators (T1, . . . ,Tn) for q ≥ 1 as

ws,q(T1, . . . ,Tn) := sup
‖x‖=1

(
n∑
j=1

∣∣〈Tjx,x〉∣∣q
)1/q

(1.14)

and established various inequalities of interest. For more recent results see
also [12, 14].

In the same paper [8] we also introduced the concept of s-q-norm of an
n-tuple of operators (T1, . . . ,Tn) for q ≥ 1 as

‖(T1, . . . ,Tn)‖s,q := sup
‖x‖=‖y‖=1

(
n∑
j=1

|〈Tjx,y〉|q
)1/q

. (1.15)

In [8], [9] and [10], by utilising Kato’s inequality [11]

|〈Tx,y〉|2 ≤
〈
|T|2αx,x

〉〈
|T∗|2(1−α)y,y

〉
(1.16)

for any x,y ∈ H, α ∈ [0, 1], where “absolute value” operator of A is defined
by ‖A‖ :=

√
A∗A, the authors have obtained several inequalities for the s-q-

numerical radius and s-q-norm.
In this paper we investigate the connections between these norms and

establish some fundamental inequalities of interest in multivariate operator
theory.

2. Representation results

We start with the following lemma:

Lemma 1. Let β = (β1, . . . ,βn) ∈ Cn.

(i) If p,q > 1 and 1
p

+ 1
q

= 1, then

sup
‖α‖n,p≤1

∣∣∣∣∣
n∑
j=1

αjβj

∣∣∣∣∣ = ‖β‖n,q . (2.1)
In particular,

sup
‖α‖n,2≤1

∣∣∣∣∣
n∑
j=1

αjβj

∣∣∣∣∣ = ‖β‖n,2. (2.2)



206 s.s. dragomir

(ii) We have

sup
‖α‖n,∞≤1

∣∣∣∣∣
n∑
j=1

αjβj

∣∣∣∣∣ = ‖β‖n,1 and sup‖α‖n,1≤1
∣∣∣∣∣
n∑
j=1

αjβj

∣∣∣∣∣ = ‖β‖n,∞. (2.3)
Proof. (i) Using Hölder’s discrete inequality for p,q > 1 and 1

p
+ 1

q
= 1

we have ∣∣∣∣∣
n∑
j=1

αjβj

∣∣∣∣∣ ≤
(

n∑
j=1

|αj|p
)1/p( n∑

j=1

|βj|q
)1/q

,

which implies that

sup
‖α‖n,p≤1

∣∣∣∣∣
n∑
j=1

αjβj

∣∣∣∣∣ ≤‖β‖n,q (2.4)
where α = (α1, . . . ,αn) and β = (β1, . . . ,βn) are n-tuples of complex numbers.

For (β1, . . . ,βn) 6= 0, consider α = (α1, . . . ,αn) with

αj :=
βj|βj|q−2(∑n
k=1 |βk|q

)1/p
for those j for which βj 6= 0 and αj = 0, for the rest.

We observe that∣∣∣∣∣
n∑
j=1

αjβj

∣∣∣∣∣ =
∣∣∣∣∣
n∑
j=1

βj |βj|q−2(∑n
k=1 |βk|

q
)1/pβj

∣∣∣∣∣ =
∑n

j=1 |βj|
q(∑n

k=1 |βk|
q
)1/p

=

(
n∑
j=1

|βj|q
)1/q

= ‖β‖n,q

and

‖α‖pn,p =
n∑
j=1

|αj|p =
n∑
j=1

∣∣∣βj |βj|q−2 ∣∣∣p(∑n
k=1 |βk|q

) = n∑
j=1

(
|βj|q−1

)p
(∑n

k=1 |βk|
q
)

=

n∑
j=1

|βj|qp−p(∑n
k=1 |βk|

q
) = n∑

j=1

|βj|q(∑n
k=1 |βk|

q
) = 1.

Therefore, by (2.4) we have the representation (2.1).



hypo-q-norms on cartesian products 207

(ii) Using the properties of the modulus, we have∣∣∣∣∣
n∑
j=1

αjβj

∣∣∣∣∣ ≤ maxj∈{1,...,n} |αj|
n∑
j=1

|βj| ,

which implies that

sup
‖α‖n,∞≤1

∣∣∣∣∣
n∑
j=1

αjβj

∣∣∣∣∣ ≤‖β‖n,1, (2.5)
where α = (α1, . . . ,αn) and β = (β1, . . . ,βn).

For (β1, . . . ,βn) 6= 0, consider α = (α1, . . . ,αn) with αj :=
βj
|βj|

for those j

for which βj 6= 0 and αj = 0, for the rest.
We have ∣∣∣∣∣

n∑
j=1

αjβj

∣∣∣∣∣ =
∣∣∣∣∣
n∑
j=1

βj
|βj|

βj

∣∣∣∣∣ =
n∑
j=1

|βj| = ‖β‖n,1

and

‖α‖n,∞ = max
j∈{1,...,n}

|αj| = max
j∈{1,...,n}

∣∣∣∣ βj|βj|
∣∣∣∣ = 1

and by (2.5) we get the first representation in (2.3).
Moreover, we have∣∣∣∣∣

n∑
j=1

αjβj

∣∣∣∣∣ ≤
n∑
j=1

|αj| max
j∈{1,...,n}

|βj| ,

which implies that

sup
‖α‖n,1≤1

∣∣∣∣∣
n∑
j=1

αjβj

∣∣∣∣∣ ≤‖β‖n,∞, (2.6)
where α = (α1, . . . ,αn) and β = (β1, . . . ,βn).

For (β1, . . . ,βn) 6= 0, let j0 ∈{1, . . . ,n} such that

‖β‖∞ = max
j∈{1,...,n}

|βj| = |βj0| .

Consider α = (α1, . . . ,αn) with αj0 =
βj0
|βj0|

and αj = 0 for j 6= j0. For this
choice we get

n∑
j=1

|αj| =
∣∣βj0∣∣
|βj0|

= 1 and

∣∣∣∣∣
n∑
j=1

αjβj

∣∣∣∣∣ =
∣∣∣∣ βj0|βj0|βj0

∣∣∣∣ = |βj0| = ‖β‖n,∞ ,



208 s.s. dragomir

therefore by (2.6) we obtain the second representation in (4).

Theorem 2. Let (T1, . . . ,Tn) ∈ B(n)(H) and x,y ∈ H, then for p,q > 1
and 1

p
+ 1

q
= 1 we have

sup
‖α‖n,p≤1

∣∣∣∣∣
〈(

n∑
j=1

αjTj

)
x,y

〉∣∣∣∣∣ =
(

n∑
j=1

|〈Tjx,y〉|q
)1/q

(2.7)

and in particular

sup
‖α‖n,2≤1

∣∣∣∣∣
〈(

n∑
j=1

αjTj

)
x,y

〉∣∣∣∣∣ =
(

n∑
j=1

|〈Tjx,y〉|2
)1/2

. (2.8)

We also have

sup
‖α‖n,∞≤1

∣∣∣∣∣
〈(

n∑
j=1

αjTj

)
x,y

〉∣∣∣∣∣ =
n∑
j=1

|〈Tjx,y〉| (2.9)

and

sup
‖α‖n,1≤1

∣∣∣∣∣
〈(

n∑
j=1

αjTj

)
x,y

〉∣∣∣∣∣ = maxj∈{1,...,n}{|〈Tjx,y〉|} . (2.10)
Proof. If we take β = (〈T1x,y〉 , . . . ,〈Tnx,y〉) ∈ Cn in (2.1), then we get(
n∑
j=1

|〈Tjx,y〉|q
)1/q

= ‖β‖n,q = sup
‖α‖p≤1

∣∣∣∣∣
n∑
j=1

αjβj

∣∣∣∣∣
= sup
‖α‖n,p≤1

∣∣∣∣∣
n∑
j=1

αj 〈Tjx,y〉

∣∣∣∣∣ = sup‖α‖n,p≤1
∣∣∣∣∣
〈

n∑
j=1

αjTjx,y

〉∣∣∣∣∣,
which proves (2.7).

The equalities (2.9) and (2.10) follow by (2.3).

Corollary 3. Let (T1, . . . ,Tn) ∈ B(n)(H) and x ∈ H, then for p,q > 1
and 1

p
+ 1

q
= 1 we have

sup
‖α‖n,p≤1

∣∣∣∣∣
〈(

n∑
j=1

αjTj

)
x,x

〉∣∣∣∣∣ =
(

n∑
j=1

|〈Tjx,x〉|q
)1/q

(2.11)



hypo-q-norms on cartesian products 209

and, in particular

sup
‖α‖n,2≤1

∣∣∣∣∣
〈(

n∑
j=1

αjTj

)
x,x

〉∣∣∣∣∣ =
(

n∑
j=1

|〈Tjx,x〉|2
)1/2

. (2.12)

We also have

sup
‖α‖n,∞≤1

∣∣∣∣∣
〈(

n∑
j=1

αjTj

)
x,x

〉∣∣∣∣∣ =
n∑
j=1

|〈Tjx,x〉| (2.13)

and

sup
‖α‖n,1≤1

∣∣∣∣∣
〈(

n∑
j=1

αjTj

)
x,x

〉∣∣∣∣∣ = maxj∈{1,...,n}{|〈Tjx,x〉|} . (2.14)
Corollary 4. Let (T1, . . . ,Tn) ∈ B(n)(H) and x ∈ H, then for p,q > 1

and 1
p

+ 1
q

= 1 we have

sup
‖α‖n,p≤1

∥∥∥∥∥
n∑
j=1

αjTjx

∥∥∥∥∥ = sup‖y‖=1
(

n∑
j=1

|〈Tjx,y〉|q
)1/q

(2.15)

and in particular

sup
‖α‖n,2≤1

∥∥∥∥∥
n∑
j=1

αjTjx

∥∥∥∥∥ = sup‖y‖=1
(

n∑
j=1

|〈Tjx,y〉|2
)1/2

. (2.16)

We also have

sup
‖α‖n,∞≤1

∥∥∥∥∥
n∑
j=1

αjTjx

∥∥∥∥∥ = sup‖y‖=1
n∑
j=1

|〈Tjx,y〉| (2.17)

and

sup
‖α‖n,1≤1

∥∥∥∥∥
n∑
j=1

αjTjx

∥∥∥∥∥ = maxj∈{1,...,n}{‖Tjx‖}. (2.18)
Proof. By the properties of inner product, we have for any u ∈ H, u 6= 0

that
‖u‖ = sup

‖y‖=1
|〈u,y〉|.



210 s.s. dragomir

Let x ∈ H, then by taking the supremum over ‖y‖ = 1 in (2.7) we get for
p,q > 1 with 1

p
+ 1

q
= 1 that

sup
‖y‖=1

(
n∑
j=1

|〈Tjx,y〉|q
)1/q

= sup
‖y‖=1


 sup
‖α‖n,p≤1

∣∣∣∣∣
〈(

n∑
j=1

αjTj

)
x,y

〉∣∣∣∣∣



= sup
‖α‖n,p≤1


 sup
‖y‖=1

∣∣∣∣∣
〈(

n∑
j=1

αjTj

)
x,y

〉∣∣∣∣∣



= sup
‖α‖n,p≤1

∥∥∥∥∥
(

n∑
j=1

αjTj

)
x

∥∥∥∥∥,
which proves the equality (2.15).

The other equalities can be proved in a similar way by using Theorem 2,
however the details are omitted.

We can state and prove our main result.

Theorem 5. Let (T1, . . . ,Tn) ∈ B(n)(H).

(i) For q ≥ 1 we have the representation for the hypo-q-norm

‖(T1, . . . ,Tn)‖h,n,q = sup
‖x‖=‖y‖=1

(
n∑
j=1

|〈Tjx,y〉|q
)1/q

= ‖(T1, . . . ,Tn)‖s,q

(2.19)

and in particular

‖(T1, . . . ,Tn)‖n,e = sup
‖x‖=‖y‖=1

(
n∑
j=1

|〈Tjx,y〉|2
)1/2

. (2.20)

We also have

‖(T1, . . . ,Tn)‖h,n,∞ = max
j∈{1,...,n}

{
‖Tj‖

}
. (2.21)



hypo-q-norms on cartesian products 211

(ii) For q ≥ 1 we have the representation for the hypo--numerical radius

wh,n,q(T1, . . . ,Tn) = sup
‖x‖=1

(
n∑
j=1

|〈Tjx,x〉|q
)1/q

= ws,q(T1, . . . ,Tn)

(2.22)

and in particular

wn,e (T1, . . . ,Tn) := sup
‖x‖=1

(
n∑
j=1

|〈Tjx,x〉|2
)1/2

. (2.23)

We also have

wh,n,∞ (T1, . . . ,Tn) = max
j∈{1,...,n}

{
w(Tj)

}
. (2.24)

Proof. (i) By using the equality (2.15) we have for (T1, . . . ,Tn) ∈ B(n)(H)
that

sup
‖x‖=‖y‖=1

(
n∑
j=1

|〈Tjx,y〉|q
)1/q

= sup
‖x‖=1


 sup
‖y‖=1

(
n∑
j=1

|〈Tjx,y〉|q
)1/q

= sup
‖x‖=1


 sup
‖α‖n,p≤1

∥∥∥∥∥
n∑
j=1

αjTjx

∥∥∥∥∥



= sup
‖α‖n,p≤1


 sup
‖x‖=1

∥∥∥∥∥
n∑
j=1

αjTjx

∥∥∥∥∥



= sup
‖α‖n,p≤1

∥∥∥∥∥
n∑
j=1

αjTj

∥∥∥∥∥
= ‖(T1, . . . ,Tn)‖h,n,q,

which proves (2.19). The rest is obvious.



212 s.s. dragomir

(ii) By using the equality (2.11) we have for (T1, . . . ,Tn) ∈ B(n)(H) that

sup
‖x‖=1

(
n∑
j=1

|〈Tjx,x〉|q
)1/q

= sup
‖x‖=1


 sup
‖α‖n,p≤1

∣∣∣∣∣
〈(

n∑
j=1

αjTj

)
x,x

〉∣∣∣∣∣



= sup
‖α‖n,p≤1


 sup
‖x‖=1

∣∣∣∣∣
〈(

n∑
j=1

αjTj

)
x,x

〉∣∣∣∣∣



= sup
‖α‖n,p≤1

w

(
n∑
j=1

αjTj

)
= wh,n,q(T1, . . . ,Tn),

which proves (2.22). The rest is obvious.

Remark 6. The case q = 2 was obtained in a different manner in [5] by
utilising the rotation-invariant normalised positive Borel measure on the unit
sphere.

We can consider on B(n)(H) the following usual operator and numerical
radius q-norms, for q ≥ 1

‖(T1, . . . ,Tn)‖n,q :=

(
n∑
j=1

‖Tj‖q
)1/q

,

wn,q(T1, . . . ,Tn) :=

(
n∑
j=1

wq(Tj)

)1/q
,

where (T1, . . . ,Tn) ∈ B(n)(H). For q = ∞ we put

‖(T1, . . . ,Tn)‖n,∞ := max
j∈{1,...,n}

{
‖Tj‖

}
,

wn,∞(T1, . . . ,Tn) := max
j∈{1,...,n}

{
w(Tj)

}
.

Corollary 7. With the assumptions of Theorem 5 we have for q ≥ 1
that

1

n1/q
‖(T1, . . . ,Tn)‖n,q ≤‖(T1, . . . ,Tn)‖h,n,q ≤‖(T1, . . . ,Tn)‖n,q (2.25)



hypo-q-norms on cartesian products 213

and

1

n1/q
wn,q(T1, . . . ,Tn) ≤ wh,n,q(T1, . . . ,Tn) ≤ wn,q(T1, . . . ,Tn) (2.26)

for any (T1, . . . ,Tn) ∈ B(n)(H).
In particular, we have [5]

1
√
n
‖(T1, . . . ,Tn)‖n,2 ≤‖(T1, . . . ,Tn)‖h,n,e ≤‖(T1, . . . ,Tn)‖n,2 (2.27)

and

1
√
n
wn,2(T1, . . . ,Tn) ≤ wh,n,e(T1, . . . ,Tn) ≤ wn,2(T1, . . . ,Tn) (2.28)

for any (T1, . . . ,Tn) ∈ B(n)(H).

Proof. Let (T1, . . . ,Tn) ∈ B(n)(H) and x,y ∈ H with ‖x‖ = ‖y‖ = 1.
Then by Schwarz’s inequality we have(

n∑
j=1

|〈Tjx,y〉|q
)1/q

≤

(
n∑
j=1

‖Tjx‖q ‖y‖q
)1/q

=

(
n∑
j=1

‖Tjx‖q
)1/q

.

By the operator norm inequality we also have(
n∑
j=1

‖Tjx‖q
)1/q

≤

(
n∑
j=1

‖Tj‖q ‖x‖q
)1/q

= ‖(T1, . . . ,Tn)‖n,q.

Therefore (
n∑
j=1

|〈Tjx,y〉|q
)1/q

≤‖(T1, . . . ,Tn)‖n,q

and by taking the supremum over ‖x‖ = ‖y‖ = 1 we get the second inequality
in (2.25).

By the properties of complex numbers, we have

max
j∈{1,...,n}

{
|〈Tjx,y〉|

}
≤

(
n∑
j=1

|〈Tjx,y〉|q
)1/q

x,y ∈ H with ‖x‖ = ‖y‖ = 1.



214 s.s. dragomir

By taking the supremum over ‖x‖ = ‖y‖ = 1 we get

sup
‖x‖=‖y‖=1

(
max

j∈{1,...,n}

{
|〈Tjx,y〉|

})
≤‖(T1, . . . ,Tn)‖h,n,q (2.29)

and since

sup
‖x‖=‖y‖=1

(
max

j∈{1,...,n}

{
|〈Tjx,y〉|

})
= max

j∈{1,...,n}

{
sup

‖x‖=‖y‖=1
|〈Tjx,y〉|

}

= max
j∈{1,...,n}

{
‖Tj‖

}
= ‖(T1, . . . ,Tnt)‖n,∞,

then by (2.29) we get

‖(T1, . . . ,Tn)‖n,∞ ≤‖(T1, . . . ,Tn)‖h,n,q (2.30)

for any (T1, . . . ,Tn) ∈ B(n)(H).
Since

‖(T1, . . . ,Tn)‖n,q : =

(
n∑
j=1

‖Tj‖q
)1/q

≤
(
n‖(T1, . . . ,Tn)‖qn,∞

)1/q
= n1/q‖(T1, . . . ,Tn)‖n,∞,

(2.31)

then by (2.30) and (2.31) we get

1

n1/q
‖(T1, . . . ,Tn)‖n,q ≤‖(T1, . . . ,Tn)‖h,n,q

for any (T1, . . . ,Tn) ∈ B(n)(H).
The inequality (2.26) follows in a similar way and we omit the details.

Corollary 8. With the assumptions of Theorem 5 we have for r ≥ q ≥ 1
that

‖(T1, . . . ,Tn)‖h,n,r ≤‖(T1, . . . ,Tn)‖h,n,q ≤ n
r−q
rq ‖(T1, . . . ,Tn)‖h,n,r (2.32)

and [14]

wh,n,r(T1, . . . ,Tn) ≤ wh,n,q(T1, . . . ,Tn) ≤ n
r−q
rq wh,n,r(T1, . . . ,Tn) (2.33)

for any (T1, . . . ,Tn) ∈ B(n)(H).



hypo-q-norms on cartesian products 215

Proof. We use the following elementary inequalities for the nonnegative
numbers aj, j = 1, . . . ,n and r ≥ q > 0 (see for instance [14])(

n∑
j=1

arj

)1/r
≤

(
n∑
j=1

a
q
j

)1/q
≤ n

r−q
rq

(
n∑
j=1

arj

)1/r
. (2.34)

Let (T1, . . . ,Tn) ∈ B(n)(H) and x,y ∈ H with ‖x‖ = ‖y‖ = 1. Then by (2.34)
we get(

n∑
j=1

|〈Tjx,y〉|r
)1/r

≤

(
n∑
j=1

|〈Tjx,y〉|q
)1/q

≤ n
r−q
rq

(
n∑
j=1

|〈Tjx,y〉|r
)1/r

.

By taking the supremum over ‖x‖ = ‖y‖ = 1 we get (2.32).
The inequality (2.33) follows in a similar way and we omit the details.

Remark 9. For q ≥ 2 we have by (2.32) and (2.33)

‖(T1, . . . ,Tn)‖h,n,q ≤‖(T1, . . . ,Tn)‖h,n,e ≤ n
q−2
2q ‖(T1, . . . ,Tn)‖h,n,q (2.35)

and

wh,n,q(T1, . . . ,Tn) ≤ wh,n,e(T1, . . . ,Tn) ≤ n
q−2
2q wh,n,q(T1, . . . ,Tn) (2.36)

and for 1 ≤ q ≤ 2 we have

‖(T1, . . . ,Tn)‖h,n,e ≤‖(T1, . . . ,Tn)‖h,n,q ≤ n
2−q
2q ‖(T1, . . . ,Tn)‖h,n,e (2.37)

and

wh,n,e(T1, . . . ,Tn) ≤ wh,n,e(T1, . . . ,Tn) ≤ n
2−q
2q wh,n,e(T1, . . . ,Tn) (2.38)

for any (T1, . . . ,Tn) ∈ B(n)(H).
Also, if we take q = 1 and r ≥ 1 in (2.32) and (2.33), then we get

‖(T1, . . . ,Tn)‖h,n,r ≤‖(T1, . . . ,Tn)‖h,n,1 ≤ n
r−1
r ‖(T1, . . . ,Tn)‖h,n,r (2.39)

and

wh,n,r(T1, . . . ,Tn) ≤ wh,n,1(T1, . . . ,Tn) ≤ n
r−1
r wh,n,r(T1, . . . ,Tn) (2.40)

for any (T1, . . . ,Tn) ∈ B(n)(H).



216 s.s. dragomir

In particular, for r = 2 we get

‖(T1, . . . ,Tn)‖h,n,e ≤‖(T1, . . . ,Tn)‖h,n,1 ≤
√
n‖(T1, . . . ,Tn)‖h,n,e (2.41)

and

wn,e(T1, . . . ,Tn) ≤ wh,n,1(T1, . . . ,Tn) ≤
√
nwn,e(T1, . . . ,Tn) (2.42)

for any (T1, . . . ,Tn) ∈ B(n)(H).

We have:

Proposition 10. For any (T1, . . . ,Tn) ∈ B(n)(H) and p,q > 1 with
1
p

+ 1
q

= 1, then we have

‖(T1, . . . ,Tn)‖h,n,q ≥
1

n1/p

∥∥∥∥∥
n∑
j=1

Tj

∥∥∥∥∥ (2.43)
and

wh,n,q(T1, . . . ,Tn) ≥
1

n1/p
w

(
n∑
j=1

Tj

)
. (2.44)

Proof. Let λj =
1

n1/p
for j ∈ {1, . . . ,n}, then

∑n
j=1 |λj|

p = 1. Therefore
by (1.8) we get

‖(T1, . . . ,Tn)‖h,n,q = sup
‖λ‖n,p≤1

∥∥∥∥∥
n∑
j=1

λjTj

∥∥∥∥∥ ≥
∥∥∥∥∥

n∑
j=1

1

n1/p
Tj

∥∥∥∥∥ = 1n1/p
∥∥∥∥∥

n∑
j=1

Tj

∥∥∥∥∥.
The inequality (2.44) follows in a similar way.

We can also introduce the following norms for (T1, . . . ,Tn) ∈ B(n)(H),

‖(T1, . . . ,Tn)‖s,n,p := sup
‖x‖=1

(
n∑
j=1

‖Tjx‖p
)1/p

where p ≥ 1 and

‖(T1, . . . ,Tn)‖s,n,∞ := sup
‖x‖=1

(
max

j∈{1,...,n}
‖Tjx‖

)
= max

j∈{1,...,n}
{‖Tj‖}.

The triangle inequality ‖·‖s,n,q follows by Minkowski inequality, while the
other properties of the norm are obvious.



hypo-q-norms on cartesian products 217

Proposition 11. Let (T1, . . . ,Tn) ∈ B(n)(H).

(i) We have for p ≥ 1, that

‖(T1, . . . ,Tn)‖h,n,p ≤‖(T1, . . . ,Tn)‖s,n,p

≤‖(T1, . . . ,Tn)‖n,p,
(2.45)

(ii) For p ≥ 2 we also have

‖(T1, . . . ,Tn)‖s,n,p =
[
wh,n,p/2

(
|T1|2 , . . . , |Tn|2

)]1/2
, (2.46)

where the absolute value |T| is defined by |T| := (T∗T)1/2.

Proof. (i) We have for p ≥ 2 and x,y ∈ H with ‖x‖ = ‖y‖ = 1, that

|〈Tjx,y〉|p ≤‖Tjx‖p‖y‖p

= ‖Tjx‖p ≤‖Tj‖p‖x‖p = ‖Tj‖p

for j ∈{1, . . . ,n}.
This implies

n∑
j=1

|〈Tjx,y〉|p ≤
n∑
j=1

‖Tjx‖p ≤
n∑
j=1

‖Tj‖p ,

namely

(
n∑
j=1

|〈Tjx,y〉|p
)1/p

≤

(
n∑
j=1

‖Tjx‖p
)1/p

≤

(
n∑
j=1

‖Tj‖p
)1/p

, (2.47)

for any x,y ∈ H with ‖x‖ = ‖y‖ = 1.
Taking the supremum over ‖x‖ = ‖y‖ = 1 in (2.47), we get the desired

result (2.45).



218 s.s. dragomir

(ii) We have

‖(T1, . . . ,Tn)‖s,n,p

= sup
‖x‖=1

(
n∑
j=1

‖Tjx‖p
)1/p

= sup
‖x‖=1

(
n∑
j=1

(
‖Tjx‖2

)p/2 )1/p

= sup
‖x‖=1

(
n∑
j=1

〈Tjx,Tjx〉p/2
)1/p

= sup
‖x‖=1


 n∑
j=1

〈
T∗j Tjx,x

〉p/21/p

= sup
‖x‖=1

(
n∑
j=1

〈
|Tj|2x,x

〉p/2 )1/p
=


 sup
‖x‖=1

(
n∑
j=1

〈
|Tj|2x,x

〉p/2 )1/(p/2)1/2

=
[
wh,n,p/2

(
|T1|2, . . . , |Tn|2

)]1/2
,

which proves the equality (2.46).

3. Some reverse inequalities

Recall the following reverse of Cauchy-Buniakowski-Schwarz inequality [2]
(see also [3, Theorem 5.14]):

Lemma 12. Let a,A ∈ R and z = (z1, . . . ,zn), y = (y1, . . . ,yn) be two
sequences of real numbers with the property that:

ayj ≤ zj ≤ Ayj for each j ∈{1, . . . ,n}. (3.1)

Then for any w = (w1, . . . ,wn) a sequence of positive real numbers, one has
the inequality

0 ≤
n∑
j=1

wjz
2
j

n∑
j=1

wjy
2
j −

(
n∑
j=1

wjzjyj

)2
≤

1

4
(A−a)2

(
n∑
j=1

wjy
2
j

)2
. (3.2)

The constant 1
4

is sharp in (3.2).

O. Shisha and B. Mond obtained in 1967 (see [15]) the following counter-
parts of (CBS )–inequality (see also [3, Theorem 5.20 & 5.21]):



hypo-q-norms on cartesian products 219

Lemma 13. Assume that a = (a1, . . . ,an) and b = (b1, . . . ,bn) are such
that there exists a,A,b,B with the property that:

0 ≤ a ≤ aj ≤ A and 0 < b ≤ bj ≤ B for any j ∈{1, . . . ,n}, (3.3)

then we have the inequality

n∑
j=1

a2j

n∑
j=1

b2j −

(
n∑
j=1

ajbj

)2
≤

(√
A

b
−
√
a

B

)2 n∑
j=1

ajbj

n∑
j=1

b2j. (3.4)

and

Lemma 14. Assume that a, b are nonnegative sequences and there exists
γ, Γ with the property that

0 ≤ γ ≤
aj
bj
≤ Γ < ∞ for any j ∈{1, . . . ,n}. (3.5)

Then we have the inequality

0 ≤

(
n∑
j=1

a2j

n∑
j=1

b2j

)1
2

−
n∑
j=1

ajbj ≤
(Γ −γ)2

4 (γ + Γ)

n∑
j=1

b2j. (3.6)

We have:

Theorem 15. Let (T1, . . . ,Tn) ∈ B(n)(H).

(i) We have

0 ≤‖(T1, . . . ,Tn)‖2h,n,e −
1

n
‖(T1, . . . ,Tn)‖2h,n,1

≤
1

4
n‖(T1, . . . ,Tn)‖2n,∞

(3.7)

and

0 ≤ w2n,e(T1, . . . ,Tn) −
1

n
w2h,n,1(T1, . . . ,Tn)

≤
1

4
n‖(T1, . . . ,Tn)‖2n,∞.

(3.8)



220 s.s. dragomir

(ii) We have

0 ≤‖(T1, . . . ,Tn)‖2h,n,e −
1

n
‖(T1, . . . ,Tn)‖2h,n,1

≤‖(T1, . . . ,Tn)‖n,∞‖(T1, . . . ,Tn)‖h,n,1
(3.9)

and

0 ≤ w2n,e(T1, . . . ,Tn) −
1

n
w2h,n,1(T1, . . . ,Tn)

≤‖(T1, . . . ,Tn)‖n,∞wh,n,1(T1, . . . ,Tn).
(3.10)

(iii) We have

0 ≤‖(T1, . . . ,Tn)‖h,n,e −
1
√
n
‖(T1, . . . ,Tn)‖h,n,1

≤
1

4

√
n‖(T1, . . . ,Tn)‖n,∞

(3.11)

and

0 ≤ wn,e(T1, . . . ,Tn) −
1
√
n
wh,n,1(T1, . . . ,Tn)

≤
1

4

√
n‖(T1, . . . ,Tn)‖n,∞.

(3.12)

Proof. (i) Let (T1, . . . ,Tn) ∈ B(n)(H) and put

R = max
j∈{1,...,n}

{
‖Tj‖

}
= ‖(T1, . . . ,Tn)‖n,∞.

If x,y ∈ H, with ‖x‖ = ‖y‖ = 1 then |〈Tjx,y〉| ≤ ‖Tjx‖ ≤ ‖Tj‖ ≤ R for any
j ∈{1, . . . ,n}.

If we write the inequality (3.2) for zj = |〈Tjx,y〉|, wj = yj = 1, A = R and
a = 0, we get

0 ≤ n
n∑
j=1

|〈Tjx,y〉|2 −

(
n∑
j=1

|〈Tjx,y〉|

)2
≤

1

4
n2R2

for any x,y ∈ H, with ‖x‖ = ‖y‖ = 1.



hypo-q-norms on cartesian products 221

This implies that

n∑
j=1

|〈Tjx,y〉|2 ≤
1

n

(
n∑
j=1

|〈Tjx,y〉|

)2
+

1

4
nR2 (3.13)

for any x,y ∈ H, with ‖x‖ = ‖y‖ = 1 and, in particular

n∑
j=1

|〈Tjx,x〉|2 ≤
1

n

(
n∑
j=1

|〈Tjx,x〉|

)2
+

1

4
nR2 (3.14)

for any x ∈ H, with ‖x‖ = 1.
Taking the supremum over ‖x‖ = ‖y‖ = 1 in (3.13) and ‖x‖ = 1 in (3.14),

then we get (3.7) and (3.8).
(ii) Let (T1, . . . ,Tn) ∈ B(n)(H). If we write the inequality (3.4) for aj =

|〈Tjx,y〉|, bj = 1, b = B = 1, a = 0 and A = R, then we get

0 ≤ n
n∑
j=1

|〈Tjx,y〉|2 −

(
n∑
j=1

|〈Tjx,y〉|

)2
≤ nR

n∑
j=1

|〈Tjx,y〉| ,

for any x,y ∈ H, with ‖x‖ = ‖y‖ = 1.
This implies that

n∑
j=1

|〈Tjx,y〉|2 ≤
1

n

(
n∑
j=1

|〈Tjx,y〉|

)2
+ R

n∑
j=1

|〈Tjx,y〉| , (3.15)

for any x,y ∈ H, with ‖x‖ = ‖y‖ = 1 and, in particular

n∑
j=1

|〈Tjx,x〉|2 ≤
1

n

(
n∑
j=1

|〈Tjx,x〉|

)2
+ R

n∑
j=1

|〈Tjx,x〉| , (3.16)

for any x ∈ H with ‖x‖ = 1.
Taking the supremum over ‖x‖ = ‖y‖ = 1 in (3.15) and ‖x‖ = 1 in (3.16),

then we get (3.9) and (3.10).
(iii) If we write the inequality (3.6) for aj = |〈Tjx,y〉|, bj = 1, b = B = 1,

γ = 0 and Γ = R we have

0 ≤

(
n

n∑
j=1

|〈Tjx,y〉|2
)1

2

−
n∑
j=1

|〈Tjx,y〉| ≤
1

4
nR,



222 s.s. dragomir

for any x,y ∈ H, with ‖x‖ = ‖y‖ = 1.
This implies that(

n∑
j=1

|〈Tjx,y〉|2
)1

2

≤
1
√
n

n∑
j=1

|〈Tjx,y〉| +
1

4

√
nR, (3.17)

for any x,y ∈ H, with ‖x‖ = ‖y‖ = 1 and, in particular(
n∑
j=1

|〈Tjx,x〉|2
)1

2

≤
1
√
n

n∑
j=1

|〈Tjx,x〉| +
1

4

√
nR, (3.18)

for any x ∈ H with ‖x‖ = 1.
Taking the supremum over ‖x‖ = ‖y‖ = 1 in (3.17) and ‖x‖ = 1 in (3.18),

then we get (3.11) and (3.12).

Before we proceed with establishing some reverse inequalities for the hypo-
Euclidean numerical radius, we recall some reverse results of the Cauchy-
Bunyakovsky-Schwarz inequality for complex numbers as follows:

If γ, Γ ∈ C and αj ∈ C, j ∈{1, . . . ,n} with the property that

0 ≤ Re [(Γ −αj) (αj −γ)] (3.19)

= (Re Γ − Re αj) (Re αj − Re γ) + (Im Γ − Im αj) (Im αj − Im γ)

or, equivalently, ∣∣∣∣αj − γ + Γ2
∣∣∣∣ ≤ 12 |Γ −γ| (3.20)

for each j ∈{1, . . . ,n}, then (see for instance [4, p. 9])

n
n∑
j=1

|αj|2 −

∣∣∣∣∣
n∑
j=1

αj

∣∣∣∣∣
2

≤
1

4
n2 |Γ −γ|2 . (3.21)

In addition, if Re
(
Γγ̄
)
> 0, then (see for example [4, p. 26]):

n

n∑
j=1

|αj|2 ≤
1

4

{
Re
[(

Γ + γ
)∑n

j=1 αj

]}2
Re

(Γγ)

≤
1

4

|Γ + γ|2

Re
(Γγ)

∣∣∣∣∣
n∑
j=1

αj

∣∣∣∣∣
2

.

(3.22)



hypo-q-norms on cartesian products 223

Also, if Γ 6= −γ, then (see for instance [4, p. 32]):

(
n

n∑
j=1

|αj|2
)1

2

−

∣∣∣∣∣
n∑
j=1

αj

∣∣∣∣∣ ≤ 14n|Γ −γ|
2

|Γ + γ|
. (3.23)

Finally, from [7] we can also state that

n
n∑
j=1

|αj|2 −

∣∣∣∣∣
n∑
j=1

αj

∣∣∣∣∣
2

≤ n
[
|Γ + γ|− 2

√
Re (Γγ)

]∣∣∣∣∣
n∑
j=1

αj

∣∣∣∣∣, (3.24)
provided Re (Γγ) > 0.

We notice that a simple sufficient condition for (3.19) to hold is that

Re Γ ≥ Re αj ≥ Re γ and Im Γ ≥ Im αj ≥ Im γ (3.25)

for each j ∈{1, . . . ,n}.

Theorem 16. Let (T1, . . . ,Tn) ∈ B(n)(H) and γ, Γ ∈ C with Γ 6= γ.
Assume that

w

(
Tj −

γ + Γ

2
I

)
≤

1

2
|Γ −γ| for any j ∈{1, . . . ,n}. (3.26)

(i) We have

w2h,n,e(T1, . . . ,Tn) ≤
1

n
w2

(
n∑
j=1

Tj

)
+

1

4
n|Γ −γ|2. (3.27)

(ii) If Re (Γγ) > 0, then

wh,n,e(T1, . . . ,Tn) ≤
1

2
√
n

|Γ + γ|√
(Γγ)

w

(
n∑
j=1

Tj

)
(3.28)

and

w2h,n,e(T1, . . . ,Tn) (3.29)

≤


1
n
w

(
n∑
j=1

Tj

)
+
[
|Γ + γ|− 2

√
(Γγ)

] ·w
(

n∑
j=1

Tj

)
.



224 s.s. dragomir

(iii) If Γ 6= −γ, then

wh,n,e(T1, . . . ,Tn) ≤
1
√
n


w

(
n∑
j=1

Tj

)
+

1

4

|Γ −γ|2

|Γ + γ|


 . (3.30)

Proof. Let x ∈ H with ‖x‖ = 1 and (T1, . . . ,Tn) ∈ B(n)(H) with the
property (3.26). By taking αj = 〈Tjx,x〉 we have∣∣∣∣αj − γ + Γ2

∣∣∣∣ =
∣∣∣∣〈Tjx,x〉− γ + Γ2 〈x,x〉

∣∣∣∣ =
∣∣∣∣
〈(

Tj −
γ + Γ

2
I

)
x,x

〉∣∣∣∣
≤ sup
‖x‖=1

∣∣∣∣
〈(

Tj −
γ + Γ

2
I

)
x,x

〉∣∣∣∣
= w

(
Tj −

γ + Γ

2

)
≤

1

2
|Γ −γ|

for any j ∈{1, . . . ,n}.
(i) By using the inequality (3.21), we have

n∑
j=1

|〈Tjx,x〉|2 ≤
1

n

∣∣∣∣∣
n∑
j=1

〈Tjx,x〉

∣∣∣∣∣
2

+
1

4
n|Γ −γ|2

=
1

n

∣∣∣∣∣
〈

n∑
j=1

Tjx,x

〉∣∣∣∣∣
2

+
1

4
n |Γ −γ|2

(3.31)

for any x ∈ H with ‖x‖ = 1.
By taking the supremum over ‖x‖ = 1 in (3.31) we get

sup
‖x‖=1

(
n∑
j=1

|〈Tjx,x〉|2
)
≤

1

n
sup
‖x‖=1

∣∣∣∣∣
〈

n∑
j=1

Tjx,x

〉∣∣∣∣∣
2

+
1

4
n|Γ −γ|2

=
1

n
w2

(
n∑
j=1

Tj

)
+

1

4
n|Γ −γ|2,

which proves (3.27).



hypo-q-norms on cartesian products 225

(ii) If Re (Γγ) > 0, then by (3.22) we have for αj = 〈Tjx,x〉, j ∈{1, . . . ,n}
that

n∑
j=1

|〈Tjx,x〉|2 ≤
1

4n

|Γ + γ|2

Re(Γγ)

∣∣∣∣∣
n∑
j=1

〈Tjx,x〉

∣∣∣∣∣
2

=
1

4n

|Γ + γ|2

Re (Γγ)

∣∣∣∣∣
〈

n∑
j=1

Tjx,x

〉∣∣∣∣∣
2

(3.32)

for any x ∈ H with ‖x‖ = 1.
On taking the supremum over ‖x‖ = 1 in (3.32) we get (3.32).
Also, by (3.24) we get

n∑
j=1

|〈Tjx,x〉|2 ≤
1

n

∣∣∣∣∣
n∑
j=1

〈Tjx,x〉

∣∣∣∣∣
2

+
[
|Γ + γ|− 2

√
Re (Γγ)

]∣∣∣∣∣
n∑
j=1

〈Tjx,x〉

∣∣∣∣∣,
for any x ∈ H with ‖x‖ = 1.

By taking the supremum over ‖x‖ = 1 in this inequality, we have

sup
‖x‖=1

n∑
j=1

|〈Tjx,x〉|2

≤ sup
‖x‖=1


1
n

∣∣∣∣∣
n∑
j=1

〈Tjx,x〉

∣∣∣∣∣
2

+
[
|Γ + γ|− 2

√
(Γγ)

]∣∣∣∣∣
n∑
j=1

〈Tjx,x〉

∣∣∣∣∣



≤
1

n
sup
‖x‖=1

∣∣∣∣∣
〈

n∑
j=1

Tjx,x

〉∣∣∣∣∣
2

+
[
|Γ + γ|− 2

√
(Γγ)

]
sup
‖x‖=1

∣∣∣∣∣
〈

n∑
j=1

Tjx,x

〉∣∣∣∣∣
=

1

n
w2

(
n∑
j=1

Tj

)
+
[
|Γ + γ|− 2

√
(Γγ)

]
w

(
n∑
j=1

Tj

)
,

which proves (3.29).



226 s.s. dragomir

(iii) By the inequality (3.23) we have(
n∑
j=1

|〈Tjx,x〉|2
)1

2

≤
1
√
n


∣∣∣∣∣

n∑
j=1

〈Tjx,x〉

∣∣∣∣∣ + 14 |Γ −γ|
2

|Γ + γ|




=
1
√
n


∣∣∣∣∣
〈

n∑
j=1

Tjx,x

〉∣∣∣∣∣ + 14 |Γ −γ|
2

|Γ + γ|




for any x ∈ H with ‖x‖ = 1.
By taking the supremum over ‖x‖= 1 in this inequality, we get (3.30).

Remark 17. By the use of the elementary inequality w (T) ≤ ‖T‖ that
holds for any T ∈ B(H), a sufficient condition for (3.26) to hold is that∥∥∥∥Tj − γ + Γ2

∥∥∥∥ ≤ 12 |Γ −γ| for any j ∈{1, . . . ,n}. (3.33)
4. Inequalities for δp and ϑp norms

For T ∈ B(H) and p ≥ 1 we can consider the functionals

δp(T) := sup
‖x‖=‖y‖=1

(
|〈Tx,y〉|p + |〈T∗x,y〉|p

)1/p
= ‖(T,T∗)‖h,2,p (4.1)

and
ϑp(T) := sup

‖x‖=1

(
‖Tx‖p + ‖T∗x‖p

)1/p
= ‖(T,T∗)‖s,2,p . (4.2)

It is easy to see that both δp and ϑp are norms on B(H). The case p = 2
for the norm δ := δ2 was considered and studied in [5].

Observe that, for any T ∈ B(H) and p ≥ 1, we have

wh,2,p((T,T
∗)) = sup

‖x‖=1

(
|〈Tx,x〉|p + |〈T∗x,x〉|p

)1/p
= sup
‖x‖=1

(
|〈Tx,x〉|p + |〈Tx,x〉|p

)1/p
= 21/p sup

‖x‖=1
|〈Tx,x〉| = 21/pw(T).

(4.3)

Using the inequality (1.13) we have

21/pw(T) ≤ δp(T) ≤ 21+1/pw(T) (4.4)



hypo-q-norms on cartesian products 227

for any T ∈ B(H) and p ≥ 1.
For p = 2, we get

√
2w(T) ≤ δ(T) ≤

√
8w(T) (4.5)

while for p = 1 we get

2w(T) ≤ δ1(T) ≤ 4w(T) (4.6)

for any T ∈ B(H).
We have for any T ∈ B(H) and p ≥ 1 that

‖(T,T∗)‖2,p =
(
‖T‖p + ‖T∗‖p

)1/p
= 21/p‖T‖

and by (2.25) we get
‖T‖≤ δp(T) ≤ 21/p‖T‖ (4.7)

for any T ∈ B(H) and p ≥ 1.
For p = 2, we get

‖T‖≤ δ(T) ≤
√

2‖T‖ (4.8)

while for p = 1 we get
‖T‖≤ δ1 (T) ≤ 2‖T‖ (4.9)

for any T ∈ B(H).
From (2.32) we get for r ≥ q ≥ 1 that

δr(T) ≤ δq(T) ≤ 2
r−q
rq δr(T) (4.10)

for any T ∈ B(H).
For any T ∈ B(H) and p,q > 1 with 1

p
+ 1

q
= 1, then by (2.43) we have

δq(T) ≥
1

21/p
‖T + T∗‖. (4.11)

In particular, for p = q = 2 we get

δ(T) ≥
√

2

2
‖T + T∗‖, (4.12)

for any T ∈ B(H).
By using the inequality (2.45) we get

δp(T) ≤ ϑp(T) ≤ 21/p‖T‖ (4.13)



228 s.s. dragomir

for any T ∈ B(H) and p ≥ 1.
For p = 1 we get

δ1(T) ≤ ϑ1(T) ≤ 2‖T‖ (4.14)

for any T ∈ B(H).
For p ≥ 2, by employing the equality (2.46) we get

ϑp(T) =
[
wh,2,p/2

(
|T|2 , |T∗|2

)]1/2
=
[
22/pw

(
|T |2

)]1/2
= 21/p‖T‖ (4.15)

for any T ∈ B(H).
On utilising (3.7), (3.9) and (3.11) we get

0 ≤ δ2(T) −
1

2
δ21(T) ≤

1

2
‖T‖2, (4.16)

0 ≤ δ2(T) −
1

2
δ21(T) ≤‖T‖δ1(T) (4.17)

and

0 ≤ δ(T) −
1
√

2
δ1(T) ≤

√
2

4
‖T‖ (4.18)

for any T ∈ B(H).
Observe, by (4.3) we have that

wh,2,e((T,T
∗)) =

√
2w(T),

for any T ∈ B(H).
Assume that T ∈ B(H) and γ, Γ ∈ C with Γ 6= γ such that

w

(
T −

γ + Γ

2
I

)
, w

(
T∗ −

γ + Γ

2
I

)
≤

1

2
|Γ −γ|, (4.19)

then by (3.27) we get

w2(T) ≤‖Re (T)‖2 +
1

4
|Γ −γ|2, (4.20)

where Re (T) := T+T
∗

2
.

If Re (Γγ) > 0, then by (3.28) and (3.29)

w(T) ≤
1

2

|Γ + γ|√
Re (Γγ)

‖Re (T)‖ (4.21)



hypo-q-norms on cartesian products 229

and

w2(T) ≤
[
‖Re (T)‖ +

[
|Γ + γ|− 2

√
(Γγ)

]]
‖Re (T)‖. (4.22)

If Γ 6= −γ, then by (3.30) we get

w(T) ≤‖Re (T)‖ +
1

8

|Γ −γ|2

|Γ + γ|
. (4.23)

Due to the fact that w(A) = w(A∗) for any A ∈ B(H), the condition (4.19)
can be simplified as follows.

If m,M are real numbers with M > m and if

w

(
T −

m + M

2
I

)
≤

1

2
(M −m),

then

w2(T) ≤‖Re (T)‖2 +
1

4
(M −m)2. (4.24)

If m > 0, then

w(T) ≤
1

2

m + M
√
mM

‖Re (T)‖ (4.25)

and

w2(T) ≤
[
‖Re (T)‖ +

(√
M −

√
m
)2]
‖Re (T)‖. (4.26)

If M 6= −m, then

w(T) ≤‖Re (T)‖ +
1

8

(M −m)2

m + M
. (4.27)

For other numerical radius and norm inequalities, the interested reader
may also consult [1] and [6] and compare these results. The details are not
provided here.

5. Inequalities for real norms

If X is a complex linear space, then the functional ‖·‖ is a real norm, if
the homogeneity property in the definition of the norms is satisfied only for
real numbers, namely we have

‖αx‖ = |α|‖x‖ for any α ∈ R and x ∈ X.



230 s.s. dragomir

For instance if we consider the complex linear space of complex numbers C
then the functionals

|z|p :=
(
|Re (z)|p + |Im (z)|p

)1/p
, p ≥ 1,

|z|∞ := max{|Re (z)|, |Im (z)|}, p = ∞,

are real norms on C.
For T ∈ B(H) we consider the Cartesian decomposition

T = Re (T) + i Im (T)

where the selfadjoint operators Re (T) and Im (T) are uniquely defined by

Re (T) =
T + T∗

2
and Im (T) =

T −T∗

2i
.

We can introduce the following functionals

‖T‖r,p :=
(
‖Re (T)‖p + ‖Im (T)‖p

)1/p
, p ≥ 1,

and

‖T‖r,∞ := max
{
‖Re (T)‖,‖Im (T)‖

}
, p = ∞,

where ‖·‖ is the usual operator norm on B(H). The definition can be extended
for any other norms on B(H) or its subspaces.

Using the properties of the norm ‖·‖ and the Minkowski’s inequality

(
|a + b|p + |c + d|p

)1/p ≤ ( |a|p + |c|p )1/p + ( |b|p + |d|p )1/p
for p ≥ 1 and a,b,c,d ∈ C, we observe that ‖ · ‖r,p, p ∈ [1,∞] is a real norm
on B(H).

For p ≥ 1 and T ∈ B we can introduce the following functionals

ηr,p(T) : = sup
‖x‖=‖y‖=1

(
|Re〈Tx,y〉|p + |Im〈Tx,y〉|p

)1/p
= sup
‖x‖=‖y‖=1

(
|〈Re Tx,y〉|p + |〈Im Tx,y〉|p

)1/p
= ‖(Re T, Im T)‖h,2,p ,



hypo-q-norms on cartesian products 231

θr,p(T) : = sup
‖x‖=1

(
|Re〈Tx,x〉|p + |Im〈Tx,x〉|p

)1/p
= sup
‖x‖=1

(
|〈Re Tx,x〉|p + |〈Im Tx,x〉|p

)1/p
= wh,2,p(Re T, Im T)

and

κr,p(T) := sup
‖x‖=1

(
‖Re Tx‖p + ‖Im Tx‖p

)1/p
= ‖(Re T, Im T)‖s,2,p .

The case p = 2 is of interest since for T ∈ B(H) we have

ηr,2(T) : = sup
‖x‖=‖y‖=1

(
|Re〈Tx,y〉|2 + |Im〈Tx,y〉|2

)1/2
= sup
‖x‖=‖y‖=1

|〈Tx,y〉| = ‖T‖ ,

θr,2(T) : = sup
‖x‖=1

(
|Re〈Tx,x〉|2 + |Im〈Tx,x〉|2

)1/2
= sup
‖x‖=1

|〈Tx,x〉| = w(T)

and

κr,2(T) := sup
‖x‖=1

(
‖Re Tx‖2 + ‖Im Tx‖2

)1/2
= sup
‖x‖=1

(〈
(Re T)2x,x

〉
+
〈

(Im T)2x,x
〉)1/2

= sup
‖x‖=1

(〈[
(Re T)

2
+ (Im T)

2
]
x,x

〉)1/2

=
∥∥∥(Re T)2 + (Im T)2∥∥∥1/2 =

∥∥∥∥∥|T|
2

+ |T∗|2

2

∥∥∥∥∥
1/2

.



232 s.s. dragomir

For p = ∞ we have

ηr,∞(T) : = sup
‖x‖=‖y‖=1

(
max

{
|Re〈Tx,y〉| , |Im〈Tx,y〉|

})

= max

{
sup

‖x‖=‖y‖=1
|〈Re Tx,y〉| , sup

‖x‖=‖y‖=1
|〈Im Tx,y〉|

}

= max
{
‖Re T‖,‖Im T‖

}
,

and in a similar way

θr,∞(T) = κr,∞(T) = max
{
‖Re T‖,‖Im T‖

}
= ‖T‖r,∞ .

The functionals ηr,p, θr,p and κr,p with p ∈ [1,∞] are real norms on B(H).
We have

ηr,p (T) = sup
‖x‖=‖y‖=1

(
|Re〈Tx,y〉|p + |Im〈Tx,y〉|p

)1/p

≤

(
sup

‖x‖=‖y‖=1
|Re〈Tx,y〉|p + sup

‖x‖=‖y‖=1
|Im〈Tx,y〉|p

)1/p
= (‖Re (T)‖p + ‖Im (T)‖p)1/p = ‖T‖r,p

and

‖T‖r,∞ = sup
‖x‖=‖y‖=1

(
max{|Re〈Tx,y〉|, |Im〈Tx,y〉|}

)
≤ sup
‖x‖=‖y‖=1

(
|Re〈Tx,y〉|p + |Im〈Tx,y〉|p

)1/p
= ηr,p(T)

for any p ≥ 1 and T ∈ B(H).
In a similar way we have

‖T‖r,∞ ≤ θr,p(T) ≤‖T‖r,p

and

‖T‖r,∞ ≤ κr,p(T) ≤‖T‖r,p
for any p ≥ 1 and T ∈ B(H).



hypo-q-norms on cartesian products 233

If we write the inequality (1.13) for n = 2, T1 = Re T and T2 = Im T then
we get

θr,p(T) ≤ ηr,p(T) ≤ 2θr,p(T) (5.1)

for any p ≥ 1 and T ∈ B(H).
Using the inequalities (2.25) and (2.26) for n = 2, T1 = Re T and T2 = Im T

then we get
1

21/p
‖T‖r,p ≤ ηr,p(T) ≤‖T‖r,p (5.2)

and
1

21/p
‖T‖r,p ≤ θr,p(T) ≤‖T‖r,p (5.3)

for any p ≥ 1 and T ∈ B(H).
If we use the inequalities (2.32) and (2.33) for n = 2, T1 = Re T and

T2 = Im T then we get for t ≥ p ≥ 1 that

ηr,t(T) ≤ ηr,p(T) ≤ 2
t−p
tp ηr,t(T) (5.4)

and
θr,t(T) ≤ θr,p(T) ≤ 2

t−p
tp θr,t(T) (5.5)

for any T ∈ B(H).
For p = 1 we have the functionals

ηr,1(T) = sup
‖x‖=‖y‖=1

(
|〈Re Tx,y〉| + |〈Im Tx,y〉|

)
= ‖(Re T, Im T)‖h,2,1 ,

θr,1(T) : = sup
‖x‖=1

(
|〈Re Tx,x〉| + |〈Im Tx,x〉|

)
= wh,2,1(Re T, Im T)

and

κr,1(T) : = sup
‖x‖=1

(
‖Re Tx‖ + ‖Im Tx‖

)
= ‖(Re T, Im T)‖s,2,1 .

By utilising the inequalities (3.7), (3.9) and (3.11) for n = 2, T1 = Re T
and T2 = Im T, then

0 ≤‖T‖2 −
1

2
η2r,1(T) ≤

1

2

(
max{‖Re T‖,‖Im T‖}

)2
, (5.6)

0 ≤‖T‖2 −
1

2
η2r,1(T) ≤ max

{
‖Re T‖,‖Im T‖

}
ηr,1(T) (5.7)



234 s.s. dragomir

and

0 ≤‖T‖−
√

2

2
ηr,1(T) ≤

√
2

4
max{‖Re T‖,‖Im T‖} (5.8)

for any T ∈ B(H).
Also, by utilising the inequalities (3.8), (3.10) and (3.12) for n = 2, T1 =

Re T and T2 = Im T, then

0 ≤ w2(T) −
1

2
θ2r,1(T) ≤

1

2

(
max{‖Re T‖ ,‖Im T‖}

)2
, (5.9)

0 ≤ w2(T) −
1

2
θ2r,1(T) ≤ max

{
‖Re T‖,‖Im T‖

}
θr,1(T) (5.10)

and

0 ≤ w(T) −
√

2

2
θr,1(T) ≤

√
2

4
max

{
‖Re T‖,‖Im T‖

}
(5.11)

for any T ∈ B(H).
If m,M are real numbers with M > m and if∥∥∥∥ Re T − m + M2 I

∥∥∥∥,
∥∥∥∥ Im T − m + M2 I

∥∥∥∥ ≤ 12 (M −m), (5.12)
then by (3.27) we get

w2(T) ≤
1

2
‖Re T + Im T‖2 +

1

2
(M −m)2 . (5.13)

If m > 0, then (3.28) and (3.29) we have

w(T) ≤
1

2
√

2

m + M
√
mM

‖Re T + Im T‖ (5.14)

and

w2(T) ≤
[

1

2
‖Re T + Im T‖ +

(√
M −

√
m
)2 ]
‖Re T + Im T‖. (5.15)

If M 6= −m, then by (3.30) we get

w(T) ≤
1
√

2

(
‖Re T + Im T‖ +

1

4

(M −m)2

M + m

)
. (5.16)

Finally, we observe that a simple sufficient condition for (5.12) to hold, is
that

mI ≤ Re T, Im T ≤ MI
in the operator order of B(H).



hypo-q-norms on cartesian products 235

Acknowledgements

The author would like to thank the anonymous referee for valuable
suggestions that have been implemented in the final version of the paper.

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http://rgmia.org/papers/v6e/CSIIPS.pdf
https://www.emis.de/journals/JIPAM/article301.html?sid=301
https://www.emis.de/journals/JIPAM/article854.html?sid=854

	Introduction
	Representation results
	Some reverse inequalities
	Inequalities for p and p norms
	Inequalities for real norms