In this note, we show that for each minimal norm $N(\cdot)$ on the algebra
$M_n$ of all $n \times n$ complex matrices, there exist norms $\|\cdot\|_1$ and
$\|\cdot\|_2$ on ${\mathbb C}^n$ such that $$N(A)=\max\{\|Ax\|_2: \|x\|_1=1,
x\in {\mathbb C}^n\}$$ for all $A \in M_n$. This may be regarded as an
extension of a known result on characterization of minimal algebra norms.Comment: 4 pages, to appear in Abstract and Applied Analysi

Mirzavaziri, Madjid

Moslehian, Mohammad Sal

English

arXiv

We show that for each minimal norm N(⋅) on the
algebra ℳn of all n×n complex matrices, there exist
norms ‖⋅‖1 and ‖⋅‖2 on ℂn such that N(A)=max{‖Ax‖2:‖x‖1=1, x∈ℂn} for all A∈ℳn. This may be regarded as an extension of a known result on characterization of minimal algebra norms

Madjid Mirzavaziri

Mohammad Sal Moslehian

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Abstract and Applied Analysis

On Minimal Norms on Mn

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Abstract and Applied Analysis
Volume 2007, Article ID 52840, 4 pages
doi:10.1155/2007/52840
Research Article
On Minimal Norms on Mn
Madjid Mirzavaziri and Mohammad Sal Moslehian
Received 18 June 2007; Accepted 19 August 2007
Recommended by Allan C. Peterson
We show that for each minimal norm N(·) on the algebra n of all n× n complex ma-
trices, there exist norms ‖ · ‖1 and ‖ · ‖2 on Cn such that N(A) =max{‖Ax‖2 : ‖x‖1 =
1, x ∈ Cn} for all A ∈n. This may be regarded as an extension of a known result on
characterization of minimal algebra norms.
Copyright © 2007 M. Mirzavaziri and M. S. Moslehian. This is an open access article dis-
tributed under the Creative Commons Attribution License, which permits unrestricted
use, distribution, and reproduction in any medium, provided the original work is prop-
erly cited.
1. Introduction
Let n denote the algebra of all n×n complex matricesAwith entries inC, together with
the usual matrix operations. By an algebra norm (or a matrix norm) we mean a norm
‖·‖ on n such that ‖AB‖ ≤ ‖A‖‖B‖ for all A,B ∈n. It is easy to see that the norm
‖A‖σ =
∑n
i, j=1|αi j| is an algebra norm, but the norm ‖A‖m =max{|ai, j| : 1≤ i, j ≤ n} is
not an algebra norm, (see [1]).
Let ‖·‖1 and ‖·‖2 be two norms on Cn. Then the norm ‖·‖1,2 on n defined by
‖A‖1,2 :=max{‖Ax‖2 : ‖x‖1 = 1} is called the generalized induced (or g-ind) norm con-
structed via ‖·‖1 and ‖·‖2. If ‖·‖1 = ‖·‖2, then ‖·‖1,1 is called an induced norm.
It is known that ‖A‖C=max{
∑n
i=1|αi, j| :≤ j ≤n},‖A‖R=max{
∑n
j=1|αi, j| : 1≤ i≤n}
and the spectral norm ‖A‖S=max{
√
λ : λ is an eigenvalue of A∗A} are induced by 1,∞,
and 2, respectively, (cf. [2]). Recall that the p-norm (1≤ p ≤∞) on Cn is defined by
p(x)= p
( n∑
i=1
xiei
)
=
⎧
⎪⎪⎪⎨
⎪⎪⎪⎩
( n∑
i=1
|xi|p
)1/p
, 1≤ p <∞,
max
{∣
∣x1
∣
∣, . . . ,
∣
∣xn
∣
∣
}
, p =∞.
(1.1)
2 Abstract and Applied Analysis
It is known that the algebra norm ‖A‖ =max{‖A‖C,‖A‖R} is not induced, and it is not
hard to show that it is not g-ind too (cf. Corollary 3.2.6 of [3]).
A norm N(·) on n is called minimal if for any norm |‖·‖| on n satisfying |‖·‖| ≤
N(·), we have |‖·‖| =N(·). It is known [3, Theorem 3.2.3] that an algebra norm is an in-
duced norm if and only if it is a minimal element in the set of all algebra norms. Note that
a generalized induced norm may not be minimal. For instance, put ‖·‖α = ∞(·),‖·‖β =
22(·), and ‖·‖γ = 2(·). Then ‖·‖γ,β ≤ ‖·‖α,β but ‖·‖γ,β =‖·‖α,β.
In [1], the authors investigate generalized induced norms. In particular, they examine
the problem that “for any norm ‖·‖ on n, are there two norms ‖·‖1 and ‖·‖2 onCn such
that ‖A‖ =max{‖Ax‖2 : ‖x‖1 = 1} for all A ∈n?” In this short note, we utilize some
ideas of [1] to study the minimal norms on n. More precisely, we show that for each
minimal norm N(·) on the algebra n of all n× n complex matrices, there exist norms
‖·‖1 and ‖·‖2 on Cn such that N(A) =max{‖Ax‖2 : ‖x‖1 = 1, x ∈ Cn} for all A ∈n.
In particular, if N(·) is an algebra norm, then ‖·‖1 = ‖·‖2. This may be regarded as an
extension of the above known result on characterization of minimal algebra norms.
2. Main result
For x ∈ Cn and 1≤ j ≤ n, let Cx, j ∈n be defined by the operator Cx, j(y)= yjx. Hence
Cx, j is the n× n matrix with x in the j column and 0 elsewhere. Define Cx ∈ n by
Cx =
∑n
j=1Cx, j . Hence Cx is the n×n matrix whose all columns are x.
If ‖·‖1,2 is a generalized induced norm on n obtained via ‖·‖1 and ‖·‖2 then
‖Cx‖1,2 = α‖x‖2, where α=max{|
∑n
j=1yj| : ‖(y1, . . . , yj , . . . , yn)‖1 = 1}.
To achieve our goal, we need the following lemmas.
Lemma 2.1 [1, Theorem 2.7]. Let ‖·‖1 and ‖·‖2 be two norms on Cn. Then ‖·‖1,2 is an
algebra norm on n if and only if ‖·‖1 ≤ ‖·‖2.
Lemma 2.2 [1, Corollary 2.5]. ‖·‖1,2 = ‖·‖3,4 if and only if there exists γ > 0 such that
‖·‖1 = γ‖·‖3 and ‖·‖2 = γ‖·‖4.
Theorem 2.3. Let N(·) be a minimal norm on n, then N(·)= ‖·‖1,2 for some ‖·‖1 and
‖·‖2 on Cn. Moreover, if N(·) is an algebra norm, then ‖·‖1 = ‖·‖2.
Proof. For x ∈ Cn, set
‖x‖1 =max
{
N
(
CAx
)
:N(A)= 1, A∈n
}
,
‖x‖2 =N
(
Cx
)
.
(2.1)
We will show that ‖·‖1 and ‖·‖2 are norms on Cn.
To see that ‖·‖1 is a norm, let x ∈ Cn. Then ‖x‖1 = 0 if and only if N(CAx)= 0 for all
matrix A with N(A) = 1, and this holds if and only if Ax = 0 for all A, or equivalently
x = 0.
M. Mirzavaziri and M. S. Moslehian 3
For α∈ Cn and x, y ∈ Cn, we have
‖αx‖1 =max
{
N
(
CA(αx)
)
:N(A)= 1, A∈n
}
=max{N(αCAx
)
:N(A)= 1, A∈n
}
=max{|α|N(CAx
)
:N(A)= 1, A∈n
}
= |α|max{N(CAx
)
:N(A)= 1, A∈n
}
= |α|‖x‖1,
‖x+ y‖1 =max
{
N
(
CA(x+y)
)
:N(A)= 1, A∈n
}
=max{N(CAx +CAy
)
:N(A)= 1, A∈n
}
≤ max{N(CAx
)
:N(A)= 1, A∈n
}
+ max
{
N
(
CAy
)
:N(A)= 1, A∈n
}
= ‖x‖1 +‖y‖1.
(2.2)
To see that ‖·‖2 is a norm, let x ∈ Cn. Then ‖x‖2 = 0 if and only if Cx = 0 and this holds
if and only if x = 0.
For α∈ Cn and x, y ∈ Cn, we have
‖αx‖2 =N
(
Cαx
)=N(αCx
)= |α|N(Cx
)= |α|‖x‖2,
‖x+ y‖2 =N
(
Cx+y
)=N(Cx +Cy
)≤N(Cx
)
+N
(
Cy
)= ‖x‖2 +‖y‖2.
(2.3)
Now let A∈n\{0}. Then N(A/N(A))= 1 so that
∥
∥
∥
∥
A
N(A)
(x)
∥
∥
∥
∥
2
=N(C(A/N(A))(x)
)≤ ‖x‖1, (2.4)
whence
‖Ax‖2 ≤N(A)‖x‖1. (2.5)
Therefore ‖A‖1,2 ≤ N(A). Since N(·) is a minimal norm, we conclude that ‖A‖1,2 =
N(A).
If N(A) is an algebra norm, then Lemma 2.1 implies that ‖·‖1 ≤ ‖·‖2.
Next, let A ∈ n. It follows from ‖Ax‖1 ≤ ‖A‖11‖x‖1 ≤ ‖A‖1,1‖x‖2, (x ∈ Cn) that
‖A‖2,1 ≤ ‖A‖1,1. In a similar fashion, one can get
‖·‖2,1 ≤ ‖·‖k,k ≤ ‖·‖1,2 (k = 1,2). (2.6)
By the minimality of ‖·‖1,2, we deduce that ‖·‖1,2 = ‖·‖1,1. It then follows from
Lemma 2.2 that ‖·‖1 = ‖·‖2. 
4 Abstract and Applied Analysis
References
[1] S. Hejazian, M. Mirzavaziri, and M. S. Moslehian, “Generalized induced norms,” Czechoslovak
Mathematical Journal, vol. 57, no. 1, pp. 127–133, 2007.
[2] R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge University Press, Cam-
bridge, UK, 1994.
[3] G. R. Belitskiı˘ and Yu. I. Lyubich, Matrix Norms and Their Applications, vol. 36 of Operator
Theory: Advances and Applications, Birkha¨user, Basel, Switzerland, 1988.
Madjid Mirzavaziri: Department of Mathematics, Ferdowsi University, P.O. Box 1159,
Mashhad 91775, Iran; Banach Mathematical Research Group (BMRG), Mashhad, Iran
Email address: madjid@mirzavaziri.com
Mohammad Sal Moslehian: Department of Mathematics, Ferdowsi University, P.O. Box 1159,
Mashhad 91775, Iran; Centre of Excellence in Analysis on Algebraic Structures (CEAAS),
Ferdowsi University, Iran
Email address: moslehian@ferdowsi.um.ac.ir
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