Proyecciones Journal of Mathematics
Vol. 31, No 3, pp. 277-297, Sepember 2012.
Universidad Católica del Norte
Antofagasta - Chile
New numerical radius inequalities for certain
operaor matrices
WATHEQ BANI-DOMI
Yarmouk University, Jordan
and
FERAS ALI BANI-AHMAD
Hashemite University, Jordan
Received : June 2012. Accepted : July 2012
Abstract
In this paper we prove some upper and lower bounds for the numerical radius of the off-diagonal part of 3 × 3 operator matrices and
some bounds for the numerical radius inequalities of the general 3 × 3
operator matrix.
Mathematics Subject Classification (2010).
47A63, 47B15.
47A12, 47A30,
Keywords. Numerical radius, Operator norm, Operator matrix, Inequality, Off-diagonal part.
278
Watheq Bani-Domi and Feras Ali Bani-Ahmad
1. Introduction
Let H be a complex Hilbert space with inner product h., .i and let B (H)
be the space of all bounded linear operators on H. For A ∈ B (H) , let
ω (A) and kAk denote the numerical radius and the usual operator norm,
respectively. Recall that
ω (A) = sup {|λ| : λ ∈ W (A)} ,
where W (A) is the numerical range of A which is a subset of the
complex numbers, and
kAk = sup {kAxk : kxk = 1} .
It is well-known that ω (.) defines a norm on B (H) , which is equivalent
to the usual operator norm kAk . In fact, for A ∈ B (H) ,we have
1
kAk ≤ ω (A) ≤ kAk .
2
(1.1)
These inequalities are sharp. The first inequality becomes an equality
if A2 = 0, and the second inequality becomes an equality if A is normal.
One of the important properties of ω (.) is that it is weakly unitarily
invariant, that is, for A ∈ B (H) , we have
ω (U AU ∗ ) = ω (A) ,
(1.2)
for every unitary U ∈ B (H) .
This improvement of the seconed inequality in (1.1) has been given in
[6]. It says that for A ∈ B (H) , we have
µ
(1.3)
¶
° °1
1
° °
ω (A) ≤
kAk + °A2 ° 2 ,
2
consequently, if A2 = 0, then
New Numerical Radius Inequalities for Certain Operator Matrices 279
ω (A) =
1
kAk .
2
(1.4)
The equality (1.4) follows from the inequality (1.3) and the first inequality in (1.1).
A fundamental inequality for the numerical radius is the power inequality, which says that for A ∈ B (H) , we have
ω (An ) ≤ (ω (A))n ,
(1.5)
for n = 1, 2, 3, ... (see, e.g., [4, p. 118]).
Recent numerical radius equalities and inequalities for operator matrices
can be found in [1, 2], and [5].
In this paper, we give some new numerical radius inequalities for certain
3 × 3 operator matrices. In section 2, we establish upper and lower bounds
for the numerical radii of the off-diagonal parts of 3 × 3 operator matrices.
In section 3, we establish upper and lower bounds for the numerical radii
of general 3 × 3 operator matrices.
2. Numerical radius inequalities for the operator matrix
⎡
⎤
0 0 A
⎢
⎥
⎣ 0 B 0 ⎦.
C 0 0
Our goal in this section
for the numerical radius
⎡ is to give bounds
⎤
0
0 A13
⎢
⎥
of the off-diagonal part ⎣ 0 A22 0 ⎦ of a 3 × 3 operator matrix
0
A31 0
⎡
⎤
A11 A12 A13
⎢
⎥
⎣ A21 A22 A23 ⎦ defined on H ⊕ H ⊕ H. To achieve our goal, we need
A31 A32 A33
two basic lemmas. Part (a) of the first lemma is well-known, and it can be
found in [3]. Part (b) is also known (see, e.g., [1]) and it follows by applying
280
Watheq Bani-Domi and Feras Ali Bani-Ahmad
⎡
⎤
A B C
⎢
⎥
the identity (1.2) to the operator matrix ⎣ C A B ⎦ and the unitary opB C A
⎡
⎤
I
I
I
⎢
⎥
erator √13 ⎣ I αI α2 I ⎦ which is defined on H ⊕ H ⊕ H, where 1, α, α2
I α2 I αI
are the cubic roots of unity.
Lemma ⎛
1.⎡ Let
A
⎜⎢
(a) ω ⎝⎣ 0
0
⎛⎡
A
⎜⎢
(b) ω ⎝⎣ C
B
¡
A, B, C ⎤∈⎞B (H) . Then
0 0
⎥⎟
B 0 ⎦⎠ = max (ω (A) , ω (B) , ω (C)) .
0 C
⎤⎞
B C
⎥⎟
A B ⎦⎠ =
C A
¡
¢
¡
max ω (A + B + C) , ω A + αB + α2 C , ω A + α2 B + αC
Lemma 2. Let A, B, C ∈ B (H) . Then
(a)
⎛⎡
⎤⎞
⎛⎡
⎤⎞
⎛⎡
⎤⎞
⎛⎡
⎤⎞
⎛⎡
⎤⎞
0 0 A
⎜⎢
⎥⎟
ω ⎝⎣ 0 B 0 ⎦⎠
C 0 0
⎛⎡
¢¢
.
⎤⎞
0 0 C
B 0 0
B 0 0
⎜⎢
⎥⎟
⎜⎢
⎥⎟
⎜⎢
⎥⎟
=ω ⎝⎣ 0 B 0 ⎦⎠ = ω ⎝⎣ 0 0 C ⎦⎠ = ω ⎝⎣ 0 0 A ⎦⎠
A 0 0
0 A 0
0 C 0
0 A 0
0 C 0
⎜⎢
⎥⎟
⎜⎢
⎥⎟
=ω ⎝⎣ C 0 0 ⎦⎠ = ω ⎝⎣ A 0 0 ⎦⎠
0 0 B
0 0 B
⎛⎡
⎤⎞
⎛⎡
⎤⎞
0
0 α2 C
0
0 αC
⎜⎢
⎥⎟
⎜⎢
⎥⎟
0 ⎦⎠ = ω ⎝⎣ 0
B 0 ⎦⎠
=⎝⎣ 0 B
0
αA 0
0
α2 A 0
(b)
⎛⎡
⎤⎞
0 0 A
⎜⎢
⎥⎟
ω ⎝⎣ 0 A 0 ⎦⎠ = ω (A) .
A 0 0
New Numerical Radius Inequalities for Certain Operator Matrices 281
Proof. ⎡To prove part
(a), ⎡let
⎤
0 0 I
0 I 0
⎢
⎥
⎢
U1 = ⎣ 0 I 0 ⎦ , U2 = ⎣ 0 0 I
I 0 0
I 0 0
⎡
⎤
⎡
0 I
0
0
⎢
⎥
⎢
U5 = ⎣ αI 0 0 ⎦ , U6 = ⎣ 0
0 0 α2 I
α2 I
⎤
⎡
⎤
⎡
0 I 0
I 0
⎥
⎢
⎥
⎢
⎦ , U3 = ⎣ I 0 0 ⎦ , U4 = ⎣ 0 0
0 0 I
0 I
⎤
⎡
0 I
0 0 αI
⎥
⎢
αI 0 ⎦ , and U7 = ⎣ 0 I 0
0 0
α2 I 0 0
Then U1 , U2 , U3 , U4 , U5 , U6 , and U7 are unitary operator matrices, where
I is the identity operator
in B (H)
⎡
⎤ .
0 0 A
⎢
⎥
Consider X = ⎣ 0 B 0 ⎦ .
C 0 0
Now, it is easy to prove the following identities
U1 XU1∗
⎡
⎡
⎤
⎡
⎤
0 0 A
B 0 0
⎢
⎥
⎢
⎥
= ⎣ 0 B 0 ⎦ , U2 XU2∗ = ⎣ 0 0 C ⎦ ,
C 0 0
0 A 0
⎤
⎡
⎤
⎡
⎤
B 0 0
0 A 0
⎢
⎥
⎢
⎥
U3 XU3∗ = ⎣ 0 0 A ⎦ , U4 XU4∗ = ⎣ C 0 0 ⎦ ,
0 C 0
0 0 B
⎡
⎤
0 C 0
0 0 α2 C
⎢
⎥
⎢
⎥
0 ⎦,
U∗2 XU2 = ⎣ A 0 0 ⎦ , U7 XU7∗ = ⎣ 0 B
0 0 B
αA 0
0
⎡
⎤
⎡
⎤
0
0 αC
B 0
0
⎢
⎥
⎢
⎥
U6 XU6∗ = ⎣ 0
B 0 ⎦ , U5 XU5∗ = ⎣ 0
0 α2 A ⎦ .
2
0
0 αC
0
α A 0
Hence, from the property (1.2), we obtain
the√required results.
⎡
⎤
I
2I
I
√ ⎥
⎢ √
Now, to prove part (b), take U = 12 ⎣ 2I
0
− 2I ⎦ . Then U
√
I
I
− 2I
is unitary matrix. Thus,
⎡
⎤
⎡
⎤
0 0 A
A 0
0
⎢
⎥
⎢
⎥
U ⎣ 0 A 0 ⎦ U ∗ = ⎣ 0 −A 0 ⎦ .
A 0 0
0
0 A
⎤
0
⎥
I ⎦,
0
⎤
⎥
⎦.
282
Watheq Bani-Domi and Feras Ali Bani-Ahmad
Consequently,
⎛⎡
0 0
⎜⎢
⎝⎣ 0 A
A 0
=
⎛ ⎡
0 0
⎜ ⎢
ω ⎝U ⎣ 0 A
A 0
⎤⎞
A
⎥⎟
0 ⎦⎠
0
⎤
⎞
A
⎥
⎟
0 ⎦ U ∗⎠
0
2
=
⎤⎞
A 0
0
⎜⎢
⎥⎟
ω ⎝⎣ 0 −A 0 ⎦⎠
0
0 A
⎛⎡
=
ω (A)
(by Lemma 1 (a)).
Our first result in this section can be stated as follows.
Theorem 3. Let A, B, C ∈ B (H). Then
⎛⎡
2n
≤
1
2
Proof.
⎤⎞
0 0 A
⎜⎢
⎥⎟
n
n
2n
max (ω((AC) ), ω (B ) , ω((CA) )) ≤ ω ⎝⎣ 0 B 0 ⎦⎠
C 0 0
q
(kAk + kCk) + ω (B) , f orn=1,2,3,.... (2.1)
To prove the first inequality in (2.1), let
⎡
⎤
0 0 A
⎢
⎥
X = ⎣ 0 B 0 ⎦ . Then
C 0 0
X 2n
⎡
⎤
(AC)n
0
A
⎢
⎥
=⎣
0
0
B 2n
⎦,
n
C
0
(CA)
New Numerical Radius Inequalities for Certain Operator Matrices 283
for n = 1,¡2, 3, ..., and so
¡
¢
¢
max ω((AC)n ), ω B 2n , ω((CA)n )
¡
= ω X 2n
¢
(by Lemma 1 (a))
≤ ω 2n (X) (by the inequality (1.5)) 2
⎛⎡
⎤⎞
0 0 A
⎜⎢
⎥⎟
= ω 2n ⎝⎣ 0 B 0 ⎦⎠ .
C 0 0
Thus,
⎛⎡
2n
for n = 1, 2, 3, .... This completes
(2.1).
⎡
⎤2
⎡
0 0 A
0
⎢
⎥
⎢
Now, since ⎣ 0 0 0 ⎦ = ⎣ 0
0 0 0
C
by the identity (1.4) that
⎛⎡
⎤⎞
0 0 A
⎜⎢
⎥⎟
max (ω((AC)n ), ω (B 2n ) , ω((CA)n )) ≤ ω ⎝⎣ 0 B 0 ⎦⎠ ,
C 0 0
q
⎤⎞
⎛⎡
the proof of the first inequality in
⎤2
⎡
⎤
0 0
0 0 0
⎥
⎢
⎥
0 0 ⎦ = ⎣ 0 0 0 ⎦ , it follows
0 0
0 0 0
⎤⎞
⎛⎡
⎤⎞
0 0 A
0 0 A
0 0 0
⎜⎢
⎥⎟
⎜⎢
⎥⎟
⎜⎢
⎥⎟
ω ⎝⎣ 0 B 0 ⎦⎠ ≤ ω ⎝⎣ 0 0 0 ⎦⎠ + ω ⎝⎣ 0 B 0 ⎦⎠
C 0 0
0 0 0
0 0 0
⎛⎡
⎤⎞
0 0 0
⎜⎢
⎥⎟
+ω ⎝⎣ 0 0 0 ⎦⎠
C 0 0
=
=
°⎡
°⎡
⎤°
° 0 0 A °
° 0
°
°
1 °⎢
1°
°⎢
⎥°
°⎣ 0 0 0 ⎦° + ω (B) + °⎣ 0
°
°
2°
2°
° C
0 0 0 °
1
(kAk + kCk) + ω (B) .
2
This proves the second inequality in (2.1).
⎛⎡
⎤°
0 0 °
°
⎥°
0 0 ⎦°
°
0 0 °
⎤⎞
0 0 A
⎜⎢
⎥⎟
Now, we give some inequalities that involve ω ⎝⎣ 0 B 0 ⎦⎠ .
C 0 0
284
Watheq Bani-Domi and Feras Ali Bani-Ahmad
Theorem 4. Let A, B, C ∈ B (H) . Then
⎛⎡
⎤⎞
0 0 A
³
³
´
1
⎜⎢
⎥⎟
ω ⎝⎣ 0 B 0 ⎦⎠ ≥ max ω (A + B + C) , ω αA + B + α2 C ,
3
C 0 0
¡
¢
⎛⎡
⎤⎞
¡
¢
ω α2 A + B + αC
(2.2)
and
0 0 A
³
´
1³
⎜⎢
⎥⎟
ω (A + B + C) + ω αA + B + α2 C +
ω ⎝⎣ 0 B 0 ⎦⎠ ≤
3
C 0 0
ω α2 A + B + αC
(2.3)
Proof.
First, we prove the inequality (2.2). We have
⎛⎡
⎤⎞
A+B+C
0
0
⎜⎢
⎥⎟
2
0
ω ⎝⎣
0
αA + B + α C
⎦⎠
0
0
α2 A + B + αC
⎛⎡
⎤⎞
⎛⎡
⎤
⎛⎡
⎤⎞
⎛⎡
⎤⎞
⎛⎡
⎤⎞
⎛⎡
⎤⎞
⎛⎡
⎤⎞
⎛⎡
⎤⎞
B A C
⎜⎢
⎥⎟
= ω ⎝⎣ C B A ⎦⎠ (by Lemma 1 (b))
A C B
⎡
⎤
⎡
⎤⎞
B 0 0
0 A 0
0 0 C
⎜⎢
⎥ ⎢
⎥ ⎢
⎥⎟
= ω ⎝⎣ 0 0 A ⎦ + ⎣ C 0 0 ⎦ + ⎣ 0 B 0 ⎦⎠
0 C 0
0 0 B
A 0 0
B 0 0
0 A 0
0 0 C
⎜⎢
⎥⎟
⎜⎢
⎥⎟
⎜⎢
⎥⎟
≤ ω ⎝⎣ 0 0 A ⎦⎠ + ω ⎝⎣ C 0 0 ⎦⎠ + ω ⎝⎣ 0 B 0 ⎦⎠
0 C 0
0 0 B
A 0 0
0 0 A
0 0 A
0 0 A
⎜⎢
⎥⎟
⎜⎢
⎥⎟
⎜⎢
⎥⎟
0
B
0
0
B
0
= ω ⎝⎣
⎦⎠ + ω ⎝⎣
⎦⎠ + ω ⎝⎣ 0 B 0 ⎦⎠
C 0 0
C 0 0
C 0 0
New Numerical Radius Inequalities for Certain Operator Matrices 285
(by Lemma 2 (a))
⎛⎡
⎤⎞
0 0 A
⎜⎢
⎥⎟
=3ω ⎝⎣ 0 B 0 ⎦⎠ ,
C 0 0
and so
¡
1
3 max ω (A + B
¡
¢
¡
+ C) , ω αA + B + α2 C , ω α2 A + B + αC
⎛⎡
⎤⎞
¢¢
0 0 A
⎜⎢
⎥⎟
≤ ω ⎝⎣ 0 B 0 ⎦⎠ .
C 0 0
This completes the proof of the inequality (2.2).
⎡
⎤
I
I
I
⎥
1 ⎢
√
Now, to prove the inequality (2.3), let U = 3 ⎣ I αI α2 I ⎦ and
I α2 I αI
⎡
⎤
0 0 A
⎢
⎥
X = ⎣ 0 B 0 ⎦ . Then U is unitary.
C 0 0
Consequently,
ω (X) = ω (UXU ∗ ) (by the identity (1.2))
=
⎛⎡
⎛
=
⎤⎞
A+B+C
αA + α2 B + C
α2 A + αB + C
1 ⎜⎢
⎥⎟
2
2
2
ω ⎝⎣ A + αB + α C αA + B + α C α A + α2 B + α2 C ⎦⎠
3
α2 A + B + αC
A + α2 B + αC αA + αB + αC
⎜
⎜
⎜
⎜
⎜
⎜
1 ⎜
⎜
ω⎜
3 ⎜
⎜
⎜
⎜
⎜
⎜
⎝
⎡
⎤
0
0
α2 A + αB + C
⎢
⎥
0
0
αA + B + α2 C
⎣
⎦
2 B + αC
A
+
α
0
0
⎡
⎤
0
αA + α2 B + C
0
⎢
⎥
+ ⎣ A + αB + α2 C
0
0
⎦
2
A
+
B
+
αC
0
0
α
⎡
⎤
A+B+C
0
0
⎢
⎥
0
0
α2 A + α2 B + α2 C ⎦
+⎣
0
αA + αB + αC
0
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
286
Watheq Bani-Domi and Feras Ali Bani-Ahmad
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
≤ 13 ⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
= 13 ⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
⎤⎞ ⎞
⎛⎡
0
0
α2 A + αB + C
⎜⎢
⎥⎟
2
0
0
αA + B + α C
ω ⎝⎣
⎦⎠
2
0
0
A + α B + αC
⎛⎡
⎤⎞
0
αA + α2 B + C
0
⎜⎢
⎥⎟
+ω ⎝⎣ A + αB + α2 C
0
0
⎦⎠
2 A + B + αC
0
0
α
⎛⎡
⎤⎞
A+B+C
0
0
⎜⎢
⎥⎟
+ω ⎝⎣
0
0
α2 A + α2 B + α2 C ⎦⎠
0
αA + αB + αC
0
⎛⎡
¡
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
¢ ⎤⎞
⎞
0
0
α αA + B + α2 C
⎜⎢
⎥⎟ ⎟
2
0
αA + B + α C
0
ω ⎝⎣
⎦⎠ ⎟
¡
¢
⎟
2
2
⎟
0
0
α αA + B + α C
⎛⎡
⎤⎞ ⎟
¡
¢
⎟
2
2
0
0
α α A + B + αC
⎟
⎟
⎜⎢
⎥
⎟
2 A + B + αC
+ω ⎝⎣ ¡
0
α
0
⎟
⎦
⎠
¢
⎟
2 A + B + αC
⎟
α
α
0
0
⎟
⎛⎡
⎤⎞
⎟
2
0
0
α (A + B + C)
⎟
⎟
⎜⎢
⎥⎟
+ω ⎝⎣
0
A+B+C
0
⎠
⎦⎠
α (A + B + C)
0
0
⎛ ⎛⎡
⎤⎞ ⎞
0
0
αA + B + α2 C
⎜ ⎜⎢
⎥⎟ ⎟
0
0
αA + B + α2 C
⎜ ω ⎝⎣
⎦⎠ ⎟
⎟
⎜
2C
⎟
⎜
αA
+
B
+
α
0
0
⎜
⎛⎡
⎤⎞ ⎟
⎟
⎜
2
0
0
α A + B + αC
⎟
⎜
⎜⎢
⎥⎟ ⎟
1⎜
2
= 3 ⎜ +ω ⎝⎣
0
0
α A + B + αC
⎦⎠ ⎟
⎟
⎜
2
⎟
⎜
0
0
α A + B + αC
⎟
⎜
⎛⎡
⎤⎞
⎟
⎜
0
0
A+B+C
⎟
⎜
⎟
⎜
⎜⎢
⎥⎟
0
A+B+C
0
⎠
⎝ +ω ⎝⎣
⎦⎠
A+B+C
0
0
(by Lemma 2 (a))
=
´
³
´
´
1³ ³
ω αA + B + α2 C + ω α2 A + B + αC + ω (A + B + C)
3
(by Lemma 2 (b)).
2
New Numerical Radius Inequalities for Certain Operator Matrices 287
Remark 5. If A = B = C, then the inequalities in (2.2) and (2.3) becomes
equalities.
In the following two ⎡results we give
⎤ further upper and lower bounds for
0 0 A
⎢
⎥
the numerical radius of ⎣ 0 B 0 ⎦ . In these results, we use the observaC 0 0
⎡
⎤2
⎡
⎤
X
X
X
0 0 0
⎢
⎥
⎢
⎥
tion that for X ∈ B (H) , we have ⎣ α2 X α2 X α2 X ⎦ = ⎣ 0 0 0 ⎦ .
αX αX αX
0 0 0
So by the identity (1.4) we have
⎛⎡
° ⎡
°
°
° ⎢
° ⎢
⎢
1°
°1 ⎢
= ° ⎢
⎢
2°
°3 ⎢
° ⎣
°
°
⎤⎞
⎜⎢
⎜⎢
⎜⎢
⎜⎢
ω ⎜⎢
⎜⎢
⎜⎢
⎝⎣
X
X
⎥⎟
X
⎥⎟
⎟
2
2
α X α X ⎥
⎥⎟
⎟
⎥
⎥⎟
α2 X
⎥⎟
αX αX ⎦⎠
αX
°⎡
°
°
°⎢
°⎢
⎢
1°
°⎢
= °⎢
⎢
2°
°⎢
°⎣
°
°
°
X
X
°
°
⎥
X
⎥°
⎥
α2 X α2 X ⎥°
°
⎥°
⎥°
α2 X
⎥°
αX αX ⎦°
°
°
αX
⎤°
⎤⎡
⎤⎡
I
I
X
X
I
I
⎢
⎢
⎥
⎥
I
⎥⎢ X
⎥⎢ I
⎢ 2
⎥⎢
2
I
α2 I ⎥
αI
⎥⎢ α X α X ⎥⎢ I
⎥⎢ 2
⎥⎢ 2
⎥⎢ α X
⎥⎢ α I
αI
⎥⎢
⎥⎢
α2 I
I
αI ⎦ ⎣ αX αX ⎦ ⎣ I
2
αI
αX
α I
⎤°
°
°
⎥°
⎥°
⎥°
⎥°
⎥°
⎥°
⎥°
⎦°
°
°
288
Watheq Bani-Domi and Feras Ali Bani-Ahmad
°⎡
°
°
°⎢
°⎢
⎢
1°
°⎢
= °⎢
⎢
6°
°⎢
°⎣
°
°
⎤°
0
0 °
°
⎥°
0
⎥°
°
3
0
0 ⎥
⎥°
⎥° = kXk
⎥°
0
2
⎥°
9X 0 ⎦°
°
°
0
Theorem 6. Let A, B, C ∈ B (H) . Then
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎝
³ ´
ω (X) ≤
°
¡
° °
°¢
min kA + B + Ck , °α2 A + B + αC ° , °αA + B + α2 C °
¡¡
¢
¡
¢ ¢
¡¡
¢
¡
¢ ¢
ω ¡¡ 1 + 2α2¢ A + ¡2 + α2 ¢ B ¢ + ω ¡¡2 + α2 ¢ A + ¡1 + 2α2 ¢ B ¢ ,
⎜ ω 1 + 2α2 C + 2 + α2 B + ω 2 + α2 C + 1 + 2α2 B ,
³ ´
⎜
¡
¡
¢ ¢
¡
¡
¢ ¢
+ 13 min ⎜
⎝ ω (2 + α) C + 2 + α2 B + ω (1 + 2α) C + 1 + 2α2 B ,
¡¡
¢
¡
¢
¢
¡¡
¢
¡
¢ ¢
ω 2α + α2 C + 2 + α2 B + ω α + 2α2 C + 1 + 2α2 B
⎡
⎤
0 0 A
⎢
⎥
where X = ⎣ 0 B 0 ⎦ .
C 0 0
1
2⎛
⎡
⎤
⎡
⎤
I
I
I
0 0 A
⎢
⎥
⎢
⎥
1
2
Proof. Let U = √3 ⎣ I αI α I ⎦ and X = ⎣ 0 B 0 ⎦ .
I α2 I αI
C 0 0
Then U is unitary. It follows that
ω (X) = ω (U XU ∗ )
⎛⎡
(by the identity (1.2))
⎤⎞
A+B+C
αA + α2 B + C
α2 A + αB + C
1 ⎜⎢
⎥⎟
= ω ⎝⎣ A + αB + α2 C αA + B + α2 C α2 A + α2 B + α2 C ⎦⎠
3
α2 A + B + αC
A + α2 B + αC αA + αB + αC
0
⎞
⎞ ⎟
⎟
⎟
⎟ ⎟,
⎟ ⎟
⎟ ⎟
⎠ ⎠
New Numerical Radius Inequalities for Certain Operator Matrices 289
⎡
⎛
⎤
⎞
A+B+C
A+B+C
A+B+C
⎢ 2
⎥
⎣ α (A + B + C) α2 (A + B + C) α2 (A + B + C) ⎦ +
α (A + B + C) α (A + B + C) α (A + B + C)
⎜
⎜
⎜
⎜
⎜
⎜
⎜ ⎡
⎜
0¡
¢
¢
1 ⎜
⎜ ⎢ ¡
= ω ⎜ ⎣ 1 − α2 A + α − α2 B
3 ⎜
⎜
0
⎜
⎜
⎜
⎡
⎜
0
⎜
⎜
⎢
⎝ +⎣
0¡
¢
2
(1 − α) A + α − α B
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎤
¡ 2
¢
⎟
(α − 1) A + α − 1 B
0
⎟
⎥ ⎟
0
0
⎟
⎦
¡ 2
¢
⎟
⎟
α − α A + (1 − α) B
0
⎟
⎟
⎤ ⎟
¡ 2
¢
⎟
0¡
α − 1 A + (α − 1) B
⎟
¡
¢
¢
⎟
⎥
α − α2 A + 1 − α2 B
0
⎦ ⎠
0
0
⎛⎡
⎛
⎤⎞
A+B+C
A+B+C
A+B+C
⎜⎢ 2
⎥⎟
2
2
ω ⎝⎣ α (A + B + C) α (A + B + C) α (A + B + C) ⎦⎠ +
α (A + B + C) α (A + B + C) α (A + B + C)
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎛⎡
⎤⎞
¡ 2
¢
1⎜
0
(α
−
1)
A
+
α
−
1
B
0
⎜
¢
¡
¢
≤ ⎜
⎜⎢ ¡
⎥⎟
ω ⎝⎣ 1 − α2 A + α − α2 B
0
0
⎦⎠
3⎜
¡ 2
¢
⎜
⎜
α
−
α
A
+
(1
−
α)
B
0
0
⎜
⎛⎡
⎤⎞
¡ 2
¢
⎜
0
0
α
−
1
A
+
(α
−
1)
B
⎜
¡
¢
¡
¢
⎜
⎜⎢
⎥⎟
⎝ +ω ⎝⎣
0¡
α − α2 A + 1 − α2 B
0
⎦⎠
¢
(1 − α) A + α2 − α B
1
=
3
Ã
0
3
2
0
kA
+ B + ¡¡
Ck
¡¡
¢
¡
¢ ¢
¢
¡
¢ ¢
+ω 1 + 2α2 A + 2 + α2 B + ω 2 + α2 A + 1 + 2α2 B
(2.6)
(by the identity (2.4) and Lemma 2 (a) and (b)).
!
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
290
Watheq Bani-Domi and Feras Ali Bani-Ahmad
In a similar way, we can prove the following
⎛⎡
⎤⎞
0 0 C
⎜⎢
⎥⎟
ω (X) = ω ⎝⎣ 0 B 0 ⎦⎠
A 0 0
≤
1
3
Ã
3
2
(2.7)
¡¡
¢
¡
¢
2 C + 2 + α2 B
kA + B + ¡¡
Ck + ω ¢ 1 + 2α
¡
¢ ¢
2
+ω 2 + α C + 1 + 2α2 B
⎛⎡
≤
1
3
Ã
3
2
⎤⎞
0
0 αC
⎜⎢
⎥⎟
B 0 ⎦⎠
ω (X) = ω ⎝⎣ 0
0
α2 A 0
° 2
°
¡
¡
¢ ¢ !
°α A + B + αC ° + ω (2 + α) C + 2 + α2 B
¡
¡
¢ ¢
2
+ω (1 + 2α) C + 1 + 2α
(2.8)
B
⎛⎡
≤
1
3
Ã
(2.9)
⎛
⎜
⎜
⎜
≤⎜
⎜
⎜
⎝
3
2
¢ !
⎤⎞
0 0 α2 C
⎜⎢
⎥⎟
0 ⎦⎠
ω (X) = ω ⎝⎣ 0 B
αA 0
0
°
°
¡¡
¢
¡
¢ ¢ !
°αA + B + α2 C ° + ω 2α + α2 C + 2 + α2 B
¡¡
¢
¡
¢ ¢
2
2
+ω
α + 2α
C + 1 + 2α
B
Now, the result follows from the inequalities (2.6), (2.7), (2,8), and
(2.9). Thus,
ω (X)
³ ´
¡
°
° °
°¢
min kA + B + Ck , °α2 A + B + αC ° , °αA + B + α2 C °
¡¡
¢
¡
¢ ¢
¡¡
¢
¡
¢ ¢
ω ¡¡ 1 + 2α2¢ A + ¡2 + α2 ¢ B ¢ + ω ¡¡2 + α2 ¢ A + ¡1 + 2α2 ¢ B ¢ ,
⎜ ω 1 + 2α2 C + 2 + α2 B + ω 2 + α2 C + 1 + 2α2 B ,
³ ´
⎜
¡
¡
¢ ¢
¡
¡
¢ ¢
+ 13 min ⎜
⎝ ω (2 + α) C + 2 + α2 B + ω (1 + 2α) C + 1 + 2α2 B ,
¡¡
¢
¡
¢
¢
¡¡
¢
¡
¢ ¢
ω 2α + α2 C + 2 + α2 B + ω α + 2α2 C + 1 + 2α2 B
1
2⎛
⎞
⎞ ⎟
⎟
⎟
⎟ ⎟ 2
⎟ ⎟
⎟ ⎟
⎠ ⎠
New Numerical Radius Inequalities for Certain Operator Matrices 291
Remark 7. If A = B = C, then the inequalities in Theorem 6 becomes
equalities.
3. Upper and lower bounds for the numerical radius of the
general 3 × 3 operator matrix.
We start our results by the following lemma which satisfies certain pinching
inequalities (see, e.g., [3]).
Lemma 1.
⎛⎡Let Aij ∈ B(H), for
⎤⎞all i, j⎛=
⎡ 1, 2, 3. Then
⎤⎞
A11 0
0
A11 A12 A13
⎜⎢
⎥⎟
⎜⎢
⎥⎟
(a) ω ⎝⎣ 0 A22 0 ⎦⎠ ≤ ω ⎝⎣ A21 A22 A23 ⎦⎠ ,
A31 A32 A33
0
0 A33
⎛⎡
⎤⎞
⎛⎡
⎤⎞
0 0 A13
A11 A12 A13
⎜⎢
⎥⎟
⎜⎢
⎥⎟
(b) ω ⎝⎣ 0 0 0 ⎦⎠ ≤ ω ⎝⎣ A21 A22 A23 ⎦⎠ ,
0 0
A
A32 A33
A
⎛⎡ 31
⎤⎞
⎛⎡ 31
⎤⎞
0 0
0
A11 A12 A13
⎜⎢
⎥⎟
⎜⎢
⎥⎟
(c) ω ⎝⎣ 0 0 A23 ⎦⎠ ≤ ω ⎝⎣ A21 A22 A23 ⎦⎠ ,
0 A32 0
A31 A32 A33
⎛⎡
⎤⎞
⎛⎡
⎤⎞
0 A12 0
A11 A12 A13
⎜⎢
⎥⎟
⎜⎢
⎥⎟
(d) ω ⎝⎣ A21 0 0 ⎦⎠ ≤ ω ⎝⎣ A21 A22 A23 ⎦⎠ .
A31 A32 A33
0
0 0
Proof.
U1
U4
Let
⎡
⎤
⎡
⎤
⎡
⎤
I 0 0
I 0 0
−I 0 0
⎢
⎥
⎢
⎥
⎢
⎥
= ⎣ 0 −I 0 ⎦ , U2 = ⎣ 0 I 0 ⎦ , U3 = ⎣ 0 I 0 ⎦ ,
0 0 I
0 0 −I
0 0 I
⎡
⎤
⎡
⎤
I 0 0
A11 A12 A13
⎢
⎥
⎢
⎥
= ⎣ 0 I 0 ⎦ , and X = ⎣ A21 A22 A23 ⎦ .
A31 A32 A33
0 0 I
Then, to prove part (c) for example, it is ⎡easy to prove that ⎤
0
0
0
⎢
⎥
U1 XU1∗ + U2 XU2∗ − U3 XU3∗ − U4 XU4∗ = ⎣ 0
0
−4A23 ⎦ ,
0 −4A32
0
292
Watheq Bani-Domi and Feras Ali Bani-Ahmad
and from the fact that the numerical radius is a norm, which is weakly
unitarily invariant, we have
⎛⎡
⎤⎞
⎛⎡
⎤⎞
0 0
0
A11 A12 A13
⎜⎢
⎥⎟
⎜⎢
⎥⎟
ω ⎝⎣ 0 0 A23 ⎦⎠ ≤ ω ⎝⎣ A21 A22 A23 ⎦⎠ .
0 A32 0
A31 A32 A33
2
Based on the Lemmas 1 and 8, we have our first result in this section.
Theorem 2. Let A, B, C ∈ B (H) . Then
⎛⎡
⎤⎞
A αB α2 C
⎜⎢
⎥⎟
max (ω (A) , ω (B) , ω (C)) ≤ ω ⎝⎣ B αC α2 A ⎦⎠ ≤ ω (A)+ω (B)+ω (C) .
C αA α2 B
Proof.
For the second inequality, we have
⎛⎡
⎛ ⎡
⎤⎞
A αB α2 C
⎜⎢
⎥⎟
ω ⎝⎣ B αC α2 A ⎦⎠
2
C αA α B
⎤
⎡
A 0
0
⎜ ⎢
⎥ ⎢
2
0 α A ⎦+⎣
⎜ ⎣ 0
⎜
⎜
0 αA ⎡0
= ω⎜
⎜
0
0
⎜
⎜
⎢
+ ⎣ 0 αC
⎝
C 0
⎛
⎛⎡
⎤⎞
⎤ ⎞
0 αB
0
⎥ ⎟
B 0
0 ⎦ ⎟
⎟
⎟
0
0⎤ α2 B
⎟
⎟
2
α C
⎟
⎟
⎥
0 ⎦
⎠
0
⎛⎡
⎤⎞ ⎞
A 0
0
0 αB
0
⎜ ⎜⎢
⎥⎟
⎜⎢
⎥⎟
2
0 α A ⎦⎠ + ω ⎝⎣ B 0
0 ⎦⎠
⎜ ω ⎝⎣ 0
⎜
⎜
0 αA ⎛
0⎡
0 ⎤⎞0 α2 B
≤⎜
⎜
0
0 α2 C
⎜
⎜
⎜⎢
⎥⎟
+ω ⎝⎣ 0 αC
0 ⎦⎠
⎝
C 0
0
= ω (A) + ω (B) + ω (C) . (by Lemma 2 (a) and (b))
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
The first inequality follows from Lemma 1 (a), Theorem 4, and Lemma
8.
New Numerical Radius Inequalities for Certain Operator Matrices 293
⎛⎡
⎛
⎛⎡
⎛
⎛⎡
⎜
⎜⎢
⎜ ω ⎝⎣
⎜
⎜
⎛⎡
≥ max ⎜
⎜
⎜
⎜ ⎜⎢
⎝ ω ⎝⎣
⎜
⎜
⎜
⎜
= max⎜
⎜
⎜
⎜
⎝
≥ max
³
⎤⎞
A αB α2 C
⎜⎢
⎥⎟
ω ⎝⎣ B αC α2 A ⎦⎠
C αA α2 B
0
0
0
0
0
C
0
⎜⎢
ω ⎝⎣ 0
αA
⎛⎡
0
⎜⎢
ω ⎝⎣ 0
C
⎞
⎤⎞
⎛⎡
⎤⎞
⎤⎞
⎛⎡
⎤⎞ ⎞
0
0
0 αB 0
⎥⎟
⎜⎢
⎥⎟
2
0 α A ⎦⎠ , ω ⎝⎣ B 0 0 ⎦⎠ ,
αA
0 ⎤⎞ ⎛⎡ 0
0 0 ⎤⎞
2
0 α C
A 0
0
⎥⎟
⎜⎢
⎥⎟
0
0 ⎦⎠ , ω ⎝⎣ 0 αC
0 ⎦⎠
2
0
0
0 0 α B
0 α2 A
0 0 αB
⎥⎟
⎜⎢
⎥⎟
0
0 ⎦⎠ , ω ⎝⎣ 0 0 0 ⎦⎠ ,
0
0 ⎤⎞
B 0 0
2
0 α C
⎥⎟
0
0 ⎦⎠ , ω (A) , ω (B) , ω (C)
0
0
´
2
2
2
3 ω (A) , 3 ω (B) , 3 ω (C) , (ω (A) , ω (B) , ω (C))
= max(ω (A) , ω (B) , ω (C)) 2
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
At the end of this section, we present a general numerical radius inequalities for 3 × 3 operator matrices. These new inequalities are based
on the pinching inequalities given in Lemma 8, the triangle inequality for
ω (.) , Lemma 1 (a) and Lemma 2 (a), concerning the numerical radii of
the diagonal parts of 3 × 3 operator matrices, and our estimates of the
numerical radii of the off-diagonal parts of these operator matrices given in
Theorem 4.
Theorem 3. Let Aij ∈ B(H), for all i, j = 1, 2, 3. Then
ω ([Aij ])
⎡
¡
¢
¡
¢ ⎤
ω (A11 + A23 + A32 ) + ω ¡α2 A23 + A11 + αA32 ¢+ ω ¡αA23 + A11 + α2 A32 ¢
⎥
1⎢
≤ 3 ⎣ +ω (A12 + A33 + A21 ) + ω ¡α2 A12 + A33 + αA21 ¢ + ω ¡αA12 + A33 + α2 A21 ¢ ⎦
2
2
+ω (A13 + A22 + A31 ) + ω α A13 + A22 + αA31 + ω αA13 + A22 + α A31
(3.1)
and
294
Watheq Bani-Domi and Feras Ali Bani-Ahmad
ω ([Aij ])
⎡
⎤
¡
¢
¡
¢
ω (A23 + A32 ) , ω ¡α2 A23 + αA32 ¢ , ω ¡αA23 + α2 A32 ¢ , 3ω (A11 ) ,
⎢
⎥
≥ 13 max ⎣ ω (A12 + A21 ) , ω ¡α2 A12 + αA21 ¢, ω ¡αA12 + α2 A21 ¢, 3ω (A22 ) , ⎦ .
2
2
ω (A13 + A31 ) , ω α A13 + αA31 , ω αA13 + α A31 , 3ω (A33 )
(3.2)
Proof.
To prove the inequality (3.3), note that Lemma 2 (a) and Theorem 4 imply that
ω ([Aij ])
⎛
⎛⎡
⎤⎞
⎛⎡
⎤⎞ ⎞
⎛
⎛⎡
⎤⎞
⎛⎡
⎤⎞ ⎞
A11 0
0
0 A12 0
⎜ ⎜⎢
⎥⎟
⎜⎢
⎥⎟
0 ⎦⎠
0 A23 ⎦⎠ + ω ⎝⎣ A21 0
⎜ ω ⎝⎣ 0
⎜
⎜
0 A32 ⎛⎡
0
0 ⎤⎞ 0 A33
≤⎜
⎜
0
0 A13
⎜
⎜
⎜⎢
⎥⎟
+ω ⎝⎣ 0 A22 0 ⎦⎠
⎝
A31 0
0
⎡
0
0 A23
0
0 A12
⎜ ⎜⎢
⎥⎟
⎜⎢
⎥⎟
A11 0 ⎦⎠ + ω ⎝⎣ 0 A33 0 ⎦⎠
⎜ ω ⎝⎣ 0
⎜
⎜
0
A32 0 ⎛0⎡
A21⎤⎞ 0
=⎜
⎜
0
0 A13
⎜
⎜
⎜⎢
⎥⎟
+ω ⎝⎣ 0 A22 0 ⎦⎠
⎝
0
A31 0
¡
¢
¡
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
¢ ⎤
ω (A11 + A23 + A32 ) + ω ¡α2 A23 + A11 + αA32 ¢+ ω ¡αA23 + A11 + α2 A32 ¢
⎢
⎥
≤ 13 ⎣ +ω (A12 + A33 + A21 ) + ω ¡α2 A12 + A33 + αA21 ¢ + ω ¡αA12 + A33 + α2 A21 ¢ ⎦
+ω (A13 + A22 + A31 ) + ω α2 A13 + A22 + αA31 + ω αA13 + A22 + α2 A31
Now, it follows from Lemma 8 that
⎛
⎜
⎜
⎜
⎜
ω ([Aij ]) ≥ max ⎜
⎜
⎜
⎜
⎝
⎛⎡
⎤⎞
⎛⎡
0 0
0
0 A12
⎜⎢
⎥⎟
⎜⎢
ω ⎝⎣ 0 0 A23 ⎦⎠ , ω ⎝⎣ A21 0
0 A32 0 ⎤⎞ ⎛⎡ 0
0
⎛⎡
0 0 A13
A11 0
⎜⎢
⎥⎟
⎜⎢
ω ⎝⎣ 0 0 0 ⎦⎠ , ω ⎝⎣ 0 A22
A31 0 0
0
0
⎤⎞
0
⎥⎟
0 ⎦⎠ ,
0 ⎤⎞
0
⎥⎟
0 ⎦⎠
A33
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
New Numerical Radius Inequalities for Certain Operator Matrices 295
⎛
⎜
⎜
⎜
⎜
= max ⎜
⎜
⎜
⎜
⎝
⎛⎡
⎤⎞
⎛⎡
⎤⎞ ⎞
0 0
0
0 A12 0
⎜⎢
⎥⎟
⎜⎢
⎥⎟
ω ⎝⎣ 0 0 A23 ⎦⎠ , ω ⎝⎣ A21 0 0 ⎦⎠ ,
0 A32 0 ⎤⎞
0
0 0
⎛⎡
0 0 A13
⎜⎢
⎥⎟
ω ⎝⎣ 0 0 0 ⎦⎠ , ω (A11 ) , (A22 ) , ω (A33 )
A31 0 0
(by Lemma 1 (a))
⎡
¡
¢
¡
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
⎤
¢
ω (A23 + A32 ) , ω ¡α2 A23 + αA32 ¢ , ω ¡αA23 + α2 A32 ¢ , 3ω (A11 ) ,
⎢
⎥
1
≥ 3 max ⎣ ω (A12 + A21 ) , ω ¡α2 A12 + αA21 ¢, ω ¡αA12 + α2 A21 ¢, 3ω (A22 ) , ⎦ .
ω (A13 + A31 ) , ω α2 A13 + αA31 , ω αA13 + α2 A31 , 3ω (A33 )
(by Theorem 4)
2
This proves the inequality (3.4)
Theorem 4. Let Aij ∈ B(H), for all i, j = 1, 2, 3. Then
ω ([Aij ])
⎛
⎜
⎜
⎝
≤⎜
(3.3)
Proof.
max (ω (A11 ) , (A22 ) , ω (A33 )) +
⎡
¡
¢
¡
¢ ⎤
ω (A23 + A32 ) + ω ¡α2 A23 + αA32 ¢+ ω ¡αA23 + α2 A32 ¢
⎥
1⎢
2
2
3 ⎣ +ω (A12 + A21 ) + ω ¡α A12 + αA21 ¢ + ω ¡αA12 + α A21 ¢ ⎦
+ω (A13 + A31 ) + ω α2 A13 + αA31 + ω αA13 + α2 A31
To prove the inequality (3.5), note that
ω ([Aij ])
⎛
⎜
⎜
⎜
⎜
≤⎜
⎜
⎜
⎜
⎝
⎛⎡
A11 0
0
⎜⎢
ω ⎝⎣ 0 A22 0
0
0 A33
⎛⎡
0 0
0
⎜⎢
+ω ⎝⎣ 0 0 A23
0 A32 0
⎤⎞
⎛⎡
⎤⎞
⎛⎡
⎥⎟
⎜⎢
⎦⎠ + ω ⎝⎣
⎥⎟
⎜⎢
⎦⎠ + ω ⎝⎣
0
A21
0
0
0
A31
A12 0
0 0
0 0
0 A13
0 0
0 0
⎤⎞ ⎞
⎥⎟ ⎟
⎦⎠ ⎟
⎟
⎟
⎤⎞ ⎟
⎟
⎟
⎥⎟ ⎟
⎦⎠ ⎠
⎞
⎟
⎟
⎟
⎠
296
Watheq Bani-Domi and Feras Ali Bani-Ahmad
⎛
⎜
⎜
⎜
⎜
=⎜
⎜
⎜
⎜
⎝
⎛
⎜
⎜
⎝
≤⎜
⎛⎡
0
⎜⎢
max (ω (A11 ) , (A22 ) , ω (A33 )) + ω ⎝⎣ 0
A21
⎛⎡
⎤⎞
⎛⎡
0 0 A23
0 0
⎜⎢
⎥⎟
⎜⎢
+ω ⎝⎣ 0 0 0 ⎦⎠ + ω ⎝⎣ 0 0
A32 0 0
A31 0
⎤⎞ ⎞
0 A12
⎥⎟
0 0 ⎦⎠
0 0⎤⎞
A13
⎥⎟
0 ⎦⎠
0
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
⎞
max (ω (A11 ) , (A22 ) , ω (A33 )) +
⎡
¡
¢
¡
¢ ⎤ ⎟
ω (A23 + A32 ) + ω ¡α2 A23 + αA32 ¢+ ω ¡αA23 + α2 A32 ¢
⎟
⎥ ⎟. 2
1⎢
2 A + αA
2A
+ω
(A
+
A
)
+
ω
α
+
ω
αA
+
α
12
21
21 ¢
12
21 ¢ ⎦ ⎠
3⎣
¡ 2 12
¡
2
+ω (A13 + A31 ) + ω α A13 + αA31 + ω αA13 + α A31
Remark 5. If Aij = A for all i, j = 1, 2, 3, then the inequality in (3.3)
becomes equality, but the inequality (3.5) does not.
Acknowledgement 6. The authors would like to thank Dr. Sharifa AlSharif from Yarmouk University for her comments and suggestions.
References
[1] Bani-Domi, W., Kittaneh, F.: Norm equalities and inequalities for
operator
matrices. Linear Algebra and its Applications 429, pp. 57-67, (2008).
[2] Bani-Domi, W., Kittaneh, F.: Numerical radius inequalities for operator matrices. Linear Multilinear Algebra 57, pp. 421-427, (2009).
[3] Bhatia, R.: Matrix Analysis. Springer, New York, (1997).
[4] Halmos, P. R.: A Hilbert Space Problem Book, 2nd ed. Springer, New
York, (1982).
[5] Hirzallah, O., Kittaneh, F., Shebrawi, K.: Numerical radius inequalities for commutators of Hilbert space operators. Numer. Funct. Anal.
Optim, pp. 32, pp. 739-749, (2011).
New Numerical Radius Inequalities for Certain Operator Matrices 297
[6] Kittaneh, F.: A numerical radius inequalities and an estimate for the
numerical radius of the Frobenius companion matrix. Studia Math.
158, pp. 11-17, (2003).
Watheq Bani-Domi
Department of Mathematics
Yarmouk University
Irbed, Jordan
[email protected]
and
Feras Ali Bani-Ahmad
Department of Mathematics
Hashemite University
Zarqa, Jordan
[email protected]