平方等於零和平方等於自己的矩陣在數值半徑方面的不等式關係

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平方等於零和平方等於自己的矩陣在數值半徑方面的不等式關係

Numerical Radius Inequalities of Square-zero and Idempotent Matrices

研究生：黃金瑩

指導教授：吳培元 教授

by

CHIN-YING HUANG

Thesis Advisor: PEI YUAN WU

Institute of Applied Mathematics, National Chiao Tung University

Hsinchu, Taiwan, Republic of China

Abstract

For arbitrary n-by-n commuting complex matrices A and it is known that the inequality

, B

( )

w AB ≤ A w B( ) is in general false where denotes the

numerical radius of a matrix, but this still holds for special classes of ( ) w ⋅

A and .

In our Chapter 2 below, we prove that the inequality holds if

B A is square-zero or idempotent.

在研究的這一段時間裡，經過吳培元老師細心的指導、耐心地教

導我這個學生，不時的叮嚀我那裡該要加強，並提醒我哪裡應該要怎

麼樣作會更好，讓我了解到自己在知識與學習的能力方面的不足，要

更加倍的用功、努力。在老師的指導下，完成了對於可交換的特殊矩

陣(square-zero and idempotent matrices)在 numerical radius inequalities

上的結果。雖然只是一個小小的研究，但是老師卻不斷地教導我、提

醒我，因為我知道這是老師希望我能表現出我更應該有的水準。我很

感謝吳培元老師的敦敦教誨，我會銘記在心。感謝老師，辛苦了。

Contents

Abstract………2 Acknowledgment……….3 Table of Contents Chapter 1. Introduction………5 Chapter 2. Numerical Radius Inequalities of Square-zero and Idempotent Matrices………..6

Section 2.1. Numerical Range and Numerical Radius……….6

Section 2.2. Square-Zero Matrices……….11

Section 2.3. Idempotent Matrices………...13

Chapter 1 Introduction

The theory of quadratic forms and their applications appear in many parts of mathematics and sciences. Also, we often have the opportunity to encounter such concepts and applications in linear algebra. This subject and its extensions to infinite dimensions comprise the theory of the numerical range. In fact, a lot of recent researches have been focused on the numerical range and numerical radius in finite dimensions. In Chapter 2, we will present some results concerning the numerical radiusw AB)( when complex matrices A and B commute, namely,

.

AB=BA

We now introduce some of the notations to be used in the following chapter. We use A, B, C, to denote complex square matrices. denotes the algebra of all n-by-n complex matrices.

( )

n

M

A = max

{

Ax :x∈ n, x = denotes the matrix 1

}

norm of an n-by-n matrix A . The numerical range of A is defined by

( )

W A =

{

Ax x, :x∈ n, x =1

}

and the numerical radius is ( )w A = max

{

z z: ∈W A( ) .

}

Here x y, denotes the inner product

1 , n j j j x y x y = =

∑

of

vectors in . Recall that the inner

product is conjugate dual, 1 2

( , , , _n) ,T

x= x x … x y=( ,y y_{1 2},…,y_n)T n

,

y x = x y , and is related to the norm by ,

,

x x = x 2. Recall that A∗ is defined by the duality relationship

,

Chapter 2 Numerical Radius Inequalities of Square-zero and

Idempotent Matrices

Quadratic forms and their use in linear algebra are quite well-known. In this section, we consider the numerical range, denoted by W( )⋅ , of n-by-n complex matrices. Furthermore, we study some basic properties of the numerical range and the numerical radius. We also indicate how one may compute the numerical range

for matrices ( )

W A A and give some examples.

Section 2.1. Numerical Range and Numerical Radius

In this section, we first define the numerical range of a matrix.

Definition 1.1. The numerical range of A in M_n( ) is W A( ) =

{

Ax x, :x∈ n, x = . 1

}

Thus W( )⋅ is a function from M_n( ) into subsets of the complex plane.

The following examples give a rough idea of what the shape of the numerical range can be:

Example 1.2. If 1 0 , then is the closed unit interval

0 0

A= ⎢⎡

⎣ ⎦

⎤

⎥ W A( )

[ ]

0,1 .

Example 1.3. If 0 2 , then is the closed unit disc

0 0 A= ⎢⎡ ⎣ ⎦ ⎤ ⎥ W A( )

{

z∈ :

}

1 z ≤ .

The most fundamental properties of the numerical range W A( ) for A in ( )

n

M are its compactness and convexity.

Theorem 1.4. For all A in M_n( ),W A( )is a compact subset in the complex plane.

Proof. The set W A( ) is the range of the continuous function x→ Ax x,

over the domain

{

x∈ n: x =1

}

, the unit sphere of the Euclidean space , which is a compact set. Since the continuous image of a compact set is compact, it follows that is compact.

n

( ) W A

Theorem 1.5. For any A in M_n( ),W A( )is a convex subset of the complex plane.

This is the classical Toeplitz-Hausdorff theorem (cf. [ 3, Section 1.3] ).

Proposition 1.6. Let A and B be matrices in M_n( ). If U is a unitary matrix in M_n( ) and c is a scalar, then the following hold:

(1) W A cI( + _n)=W A( )+c, (2) W cA( )=cW A( ),

(3) W A B( + )⊆W A( )+W B( ), )

Proof. (1) We have ( _n) W A cI+ =

{

(

A+cI_n

)

x x, : x =1

}

=

{

Ax x, +c: x = 1

}

=

{

Ax x, : x =1

}

+ c = W A( )+c. (2) We have ( ) W cA =

{

(cA x x) , : x =1

}

=

{

c Ax x, : x =1

}

= c

{

Ax x, : x =1

}

= cW A( ). (3) We have ( ) W A B+ =

{

(A+B x x) , : x =1

}

=

{

Ax x, + Bx x, : x = 1

}

⊆

{

Ax x, : x =1

}

+

{

By y, : y =1

}

=W A( )+W B( ). (4) If x∈ n and x =1, we have (U AU x x∗ ) , = Ay y, ∈W A( ) ,

where y=Uxand y = Ux = x = . 1 It implies that W U AU( ∗ )⊆ W A( ). The reverse containment is obtained similarly.

Next, we want to measure the size of W A( ). This can be done by considering the radius of the smallest circular disc centered at the origin that contains . Hence we have the following definition.

( ) W A

Definition 1.7. The numerical radius of A in M_n( ) is

w A( ) =max

{

z z: ∈W A( ) .

}

In the following theorem, we show some basic properties of the numerical radius, one of which says that the numerical radius provides a norm equivalent to the matrix norm.

Theorem 1.8. Let A and B be matrices in M_n( ) and be a scalar. Then c the following hold:

(1)w A( )=0 if and only if A=0, (2)w A B( + )≤w A( )+w B( ), (3) (w cA)=c w A( ),

(4) A / 2 ≤ ( )w A ≤ A .

Proof. (2) Assume that z∈W A B( + ). Since W A B( + )⊆W A( )+W B)( , there exist u∈W A( ) and v∈W B( ) such that z= +u v. Also,

z = +u v ≤ u + v ≤w A( )+w B( ). Thus w A B( + )≤w A( )+w B( ).

(3) Ifc=0,then w cA( )= =0 c w A( ). )

Hence we may assume that Let

. Since , there exists

0.

c≠ ( )

z∈W cA W(

cA

)=cW(

A

u∈W( )

A

such that z=cu. Also, z = cu = c u ≤ c w A . Thus (( ) w c

A

) ≤ c w A( ). On the other hand, using this we

havew A( ) w cA c

⎛ ⎞

= ⎜_⎝ ⎟_⎠≤ 1 w cA( )

c . Hence w cA( )= c w A( ).

(4) We have, for any x =1, by the Schwarz inequality

_{A x x, ≤ A x x = A x ≤ A} .

Thus max

{

_{Ax x, : x} =₁

}

≤ _A , that is,w A( ) ≤ A .

To prove the other inequality, we use the polar identity, which may be verified by a direct computation: 4 Ax y, = A x( + y x), +y − A x( − y x), − y +i A x( +iy x), +iy −i A x( −iy x), −iy . Hence 4 _{Ax y, w A( )⎢}⎡ _{x y} 2 _{x y} 2 _{x iy} 2 _{x iy} 2⎤_⎥ ⎣ ⎦ ≤ + + − + + + − = 2 2 4 ( )w A ⎡_⎣ x + y ⎤_⎦.

For x = y =1, we have 4 Ax y, ≤8 (w A) , which implies that A ≤ 2 ( )w A .

Thus A / 2 ≤ ( )w A ≤ A .

(1) By (4), we have w A( ) =0 if and only if A =0. Thus w A( )=0 if and

Theorem 1.9. If 1 2 3 4 , A A A A A ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦ then W A( 1),W A( 4)⊆ ( )W A and w A( 1), 4 ( ) w A ≤ ( )w A .

This follows easily from the definitions of the numerical range and numerical radius.

In the following section, it is known that in general for A commuting with

,

B w AB ( ) ≤ w A B( ) is false. It was resolved by a counterexample in [4]. But for special matrices, this inequality may still hold. This is the case when A is a normal matrix or a 2-by-2 matrix commuting with B (cf. [1, pp. 38 39] and [2]). In the following, we prove that the inequality holds if

−

A is square-zero or idempotent.

Section 2.2. Square-Zero Matrices

In this section, we first define a square-zero matrix.

Definition 2.1. If a matrix A in Mn( ) satisfies 2

0

A = , then A is called a square-zero matrix.

The next lemma tells us that a square-zero matrix can be unitarily equivalent to a canonical form:

Lemma 2.2. Let A be a square-zero matrix. Then A is unitarily equivalent to , where with 1 0 0 0 A ⎡ ⎢ ⎣ ⎦ ⎤ ⎥ 1 ⎤⎥ 0 ' 0 0 A A = ⎢⎡

⎣ ⎦ A positive definite. Moreover, '' A is unique up to unitary equivalence.

This result can be found in [5, pp. 1140−1142].

The next theorem says that for commuting A and B in M_n( ), inequalities

( )

w AB ≤ A w B( ) and w AB( ) ≤ w A B( ) hold if A is a square-zero

matrix.

Theorem 2.3. If A is a square-zero matrix in M_n( ) and B is an n-by-n matrix commuting with A , then w AB ≤( ) A w B( ) and (w AB) ≤w A B( ) .

Proof. By Lemma 2.2, we have that A is unitarily equivalent to ,

where and 1 0 0 0 A ⎡ ⎤ ⎢ ⎥ ⎣ ⎦ 1 0 ' 0 0 A A = ⎢⎡ ⎤ ⎣ ⎦⎥ 1 2 ' m a a A a ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ , Assume that . Since 1 2 m 0. a ≥a ≥ ≥a ≥ 1 2 3 4 B B B B B ⎡ ⎤ = ⎢ ⎣ ⎦⎥ AB =BA, we have 1 1 2 1 2 1 , 3 4 3 4 0 0 0 0 0 0 B B B B A A B B B B ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ = ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦⎣ ⎦ ⎣ ⎦⎣ ⎦ that is,

1 1 1 2 1 1 3 1 0 0 0 0 B A A B A B B A ⎡ ⎤ ⎡ ⎤ = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦.

Thus A B_{1 1}=B A_{1 1}, A B_{1 2} =0,and B A_{3 1} = So we have 0. 1 1 0

0 0 A B AB= ⎢⎡ ⎤_⎥ ⎣ ⎦ and 1 2 3 4 , B B B B B ⎡ = ⎢ ⎣ ⎦ ⎤

⎥ where A B_{1 2} =0 and B A_{3 1} = It implies that 0.

( )

w AB = w A B( _{1 1}) = 1 _{1 1}

2 A B (cf. [5, Theorem 2.1] sinceA B1 1is square-zero)

≤ 1 _{1 1} 2 A B ≤ A w B1 ( ) = A w B( ). Similarly, we have ( ) w AB = ( _{1 1}) 1 _{1 1} 2 A B ≤ 1 1 1 2 A B ≤ ( )w A B . w A B =

This completes the proof.

Section 2.3. Idempotent Matrices

We first define an idempotent matrix.

Definition 3.1. If a matrix A in M_n( ) satisfies A2 = , then A is called A an idempotent matrix.

Lemma 3.2. Let A be an idempotent matrix in M_n( ). Then A is unitarily equivalent to 0 ' 0 0 I A I ⎡ ⊕ ⊕ ⎢ ⎣ ⎦ ⎤ ⎥, where 1 2 ' m a a A a ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ , a1≥a2 ≥a3 ≥ ≥a_m> . 0

This result can be found in [5, pp. 1140−1142].

Recall that matrices A and B are said to doubly commute if AB=BA and AB∗=B A∗ .

Theorem 3.3. If matrices A and B in M_n( ) doubly commute, then

( )

w AB ≤ A w B( ) and w AB( ) ≤ w A B( ) .

This result can be found in [1, pp. 38−39].

Lemma 3.4. Let a₁≥a_j ≥0and let K be a compact subset of n with

3 2

2 3

( , i , i , , in)

n

x y eθ y eθ … y eθ ∈K for all ( ,x y y₂, ₃,…,y_n)∈K and θ_j∈ , Then 2, 3, j= , .n … max 2 2 : ( , , , ) n j j n j x a y x y y K = ⎫ ⎧⎪ ⎪ ⎬ + ∈ ⎨ ⎪ ⎪ ⎩

∑

… ⎭ 1 2 2 : ( , , , ) n j n j x a y x y y = ⎧⎪ ₊ ⎨ ⎪⎩

∑

… ≤ max

}

K ∈ .

Proof. If a₁=0, then this lemma obviously holds. Next, we assume that

1 0

a > and claim that for someθ₂,…,θ_n∈ ,

(1) 2 n j j j

x

a y

=

+

∑

≤ 1 2 j n i j j

x a

y e

θ =

+

∑

. Since 2 2 2 2 2 R e(xa y ) = R e(xa y e−iθ )for someθ₂∈ , so x+a y_{2 2}2 = x2+ a y_{2 2}2+ 2Re(xa y_{2 2}) ≤ x2 + a y_{2 2}2+ 2 Re(xa y2 2) = 2 2 2 2 2 i

x

+

a y e

θ

+

2Re 2 2 2 (xa y e−iθ ) ≤ 2 2 2 1 2 i

x

+

a y e

θ

+

2Re 2 1 2 (xa y e−iθ ) = 2 2 1 2 i x + a y eθ , namely, x+a y_{2 2} ≤ 2 1 2 i

x

+

a y e

θ for someθ₂∈ .

Now, assume that for n=k, (1) is true, that is,

(2) 2 k j j j

x

a y

=

+

∑

_{≤ 1} 2 j k i j j

x a

y e

θ =

+

∑

for some θ₂,…,θ_k in . Then forn= +k 1,

1 2

(

)

k j j j

x

a y

+ =

+

∑

_≤ 1 1 1 2

(

)

k k i j j k j

x

a y

a y e

θ+ + =

+

∑

+

_{for some} 1 k θ ₊ ∈ = 1 1 1 2

(

k

)

k i k j j

x a y e

θ+

a y

+ =

+

+

∑

_j ≤ 1 1 1 1 2

(

k

)

j k i i k j j

x a y e

θ+

a

y e

θ + =

+

+

∑

_{(by (2))}

= 1 1 2 j k i j j

x a

y e

θ + =

+

∑

_.

Hence, by induction, we see that (1) is true, that is,

2 n j j j

x

a y

=

+

∑

≤ 1 2 j n i j j

x

a y e

θ =

+

∑

≤ max 1 2 2 : ( , , , ) n j n j x a y x y y K = ⎫ ⎧⎪ _{+ ∈} ⎪ ⎨ ⎬ ⎪ ⎪ ⎩

∑

⎭ … . So max ₂ 2 : ( , , , ) n j j n j x a y x y y K = ⎫ ⎧⎪ _{+ ∈} ⎪ ⎨ ⎬ ⎪ ⎪ ⎩

∑

… ⎭ ≤ max 1 2 2 : ( , , , ) n j n j x a y x y y K = ⎫ ⎧⎪ _{+ ∈} ⎪ ⎨ ⎬ ⎪ ⎪ ⎩

∑

… ⎭ .

This completes the proof.

By the preceding lemma, we can prove the following theorem.

Theorem 3.5. If A is an idempotent matrix in M_n( ) and B is an n-by-n matrix commuting with A , then w AB( ) ≤ A w B( ) and (w AB) ≤ w A B( ) .

Proof. By Lemma 3.2, we have A2 = if and only if A is unitarily A equivalent to ' 0 0 I A ⎡ ⎤ ⎢ ⎣ ⎦⎥ , where 1 2 ' , m a a A a ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ 1 0 m k m a a × ⎡⎡ ⎤⎤ ⎢⎢ ⎥⎥ ⎢⎢ ⎥⎥ ⎢⎢_⎣ ⎥_⎦⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ or

1 0 , m m k a a _× ⎡⎡ ⎤ ⎤ ⎢⎢ ⎥ ⎥ ⎢⎢ ⎥ ⎥ ⎢⎢_⎣ ⎥_⎦ ⎥ ⎣ ⎦ 1 2 m 0 a ≥a ≥ ≥a ≥ and k >m.

Case 1: Let and

1 ' m a A a ⎡ ⎤ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ 1 2 3 4 B B B B B ⎡ ⎤ = ⎢ ⎥

⎣ ⎦. Then AB=BA, that is,

1 2 3 4 ' 0 0 B B I A B B ⎡ ⎤ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ = 1 2 3 4 B B B B ⎡ ⎤ ⎢ ⎥ ⎣ ⎦ ' 0 0 I A ⎡ ⎤ ⎢ ⎥ ⎣ ⎦ implies that 1 ' 3 2 ' 0 0 4 B +A B B +A B ⎡ ⎤ ⎢ ⎥ ⎣ ⎦ = 1 1 3 3 ' ' B B A B B A ⎡ ⎤ ⎢ ⎥ ⎣ ⎦. Thus B₃ =0, B₂+A B' ₄ =B A₁ 'or B₂ =B A₁ '−A B' ₄. So 1 1 4 4 ' ' 0 B B A A B B B − ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦ and 1 1 ' 0 0 B B A AB= ⎢⎡ ⎤_⎥ ⎣ ⎦.

Now if a₁ =0, then we have A'=0 and w A( )= A =1. So it

follows that ( ) w AB = w B( ₁) ≤ ( )w B (by Theorem 1.9) ( ) A w B = . Similarly, we have (w AB) = w B( ₁) ≤ ( )w B ≤ B = w A B( ) .

Next, we consider a₁ >0 and the special case when A' = a I₁ . Then

1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 B B 1 I a I B a B I a I B B ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ₌⎡ ⎤₌ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎣ ⎦

and 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 I a I B B a B B I a I B B ∗ ∗ ∗ ∗ ∗ ∗ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ = = ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎣ ⎦. Hence 1 0 0 I a I A≡ ⎢⎡ ⎣ ⎦ ⎤ ⎥ and 1 1 0 0 B B B ⎡ ⎤ ≡ ⎢ ⎥

⎣ ⎦ doubly commute. So we have that

w AB ( ) ≤ A w B( ) = 2 1

1 a+ w B (by Theorem 3.3) ( ₁)

≤ A w B( ) (by Theorem 1.9).

Next, we check w AB( ) ≤ w AB) , that is,( 1 1 '

0 0 B B A w⎛_⎜⎡_⎢ ⎤_⎥⎞_⎟ ⎣ ⎦ ⎝ ⎠ ≤ . So 1 1 1 0 0 B a B w⎛_⎜⎡_⎢ ⎤_⎥⎞_⎟ ⎣ ⎦ ⎝ ⎠ 1 1 ' 0 0 B B A w⎛⎜_⎜⎡_⎢ ⎤_⎥⎞⎟_⎟ ⎣ ⎦ ⎝ ⎠ = max 1 1 ' , 0 0 B B A x y ⎧ ⎡ ⎤ ⎡ ⎤ ⎪ ⎨ ⎢_⎣ ⎥ ⎢ ⎥_{⎦ ⎣ ⎦} ⎪⎩ 2 2 : 1 x x y y ⎫ ⎡ ⎤ ₊ ⎪ = ⎬ ⎢ ⎥ ⎣ ⎦ _⎪⎭ = max

{

B x x₁ , + B A y x₁ ' , : x 2+ y 2 =1

}

= max 1 1 1 , : ( , , m j j j m j 2 , ), B x x a y z y y y y = ⎧⎪ _{+ =} ⎨ ⎪⎩

∑

… B x₁∗ =( ,z z_{1 2},…,z_m), x 2+ y 2 =1

}

.

Thus, by Lemma 3.4, we have

1 1 ' 0 0 B B A w⎛⎜_⎜⎡_⎢ ⎤_⎥⎞⎟_⎟ ⎣ ⎦ ⎝ ⎠ = max _{1 1 1} 1 _{1 2} 2 , : ( m j j j m j , , ), B x x a y z a y z y y y y = ⎧⎪ _{+ + =} ⎨ ⎪⎩

∑

… B x₁ ( ,z z_{1 2}, ,zm), ∗ _{= … x} 2+ _y 2 =₁

}

≤ max _{1 1 1} 1 _{1 1 2} 2 , : ( , m j j m j , , ), B x x a y z a y z y y y y = ⎧⎪ _{+ + =} ⎨ ⎪⎩

∑

… 1 ( ,1 2, , m), B x∗ = z z … z x 2+ y 2 =1

}

= max _{1 1 1 2} 1 , : ( , , m j j m j , ), B x x a y z y y y y = ⎧⎪ _{+ =} ⎨ ⎪⎩

∑

…

B x₁ ( ,z z_{1 2}, ,zm), ∗ _{= … x} 2+ _y 2 =₁

}

= 1 1 1 . 0 0 B a B w⎛_⎜⎡_⎢ ⎤_⎥⎞_⎟ ⎣ ⎦ ⎝ ⎠ Finally, ( ) w AB = 1 1 ' 0 0 B B A w⎛_⎜⎡_⎢ ⎤_⎥⎞_⎟ ⎣ ⎦ ⎝ ⎠ ≤ 1 1 1 0 0 B a B w⎛⎜⎡_⎢ ⎤_⎥⎞⎟ ⎣ ⎦ ⎝ ⎠ = w AB

( )

≤ A w B( ).

Moreover, since A and B doubly commute, so

( )

w AB ≤ ( )w A B = w A B ( ) ₁ ≤ ( )w A B ( by Theorem 3.3).

Also, we have A = 1 a+ 12 = A and ( )w A =w A( )(cf. [5, Theorem 2.1]). It

follows that w AB( ) ≤ w AB ( ) ≤ w A B( ) . Case 2: LetA'= 1 0 m _{m k} a a _× ⎡⎡ ⎤ ⎤ ⎢⎢ ⎥ ⎥ ⎢⎢ ⎥ ⎥ ⎢⎢_⎣ ⎥_⎦ ⎥ ⎣ ⎦ and 1 2 3 4 B B B B B ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦. Then AB=BA, that is, 1 2 3 4 ' 0 0 m k m k k I A B B B B × × ⎡ ⎤ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ = 1 2 3 4 B B B B ⎡ ⎤ ⎢ ⎥ ⎣ ⎦ ' 0 0 m k m k k I A × × ⎡ ⎤ ⎢ ⎥ ⎣ ⎦ implies that 1 ' 3 2 ' 0k m 0k k 4 B A B B A B × × + + ⎡ ⎤ ⎢ ⎥ ⎣ ⎦ = 1 1 3 3 ' ' B B A B B A ⎡ ⎤ ⎢ ⎥ ⎣ ⎦. Thus B₃=0_{k m×} , B₂ +A B' ₄ =B A₁ 'or B₂ =B A₁ '−A B' ₄. So 1 1 4 4 ' ' 0_{k m} B B A A B B B × − ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦ and 1 1 ' 0_{k m} 0_{k k} B B A AB × × ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦.

Now if a₁ =0, then we have A'=0 and w A( )= A =1. So ( ) w AB = w B( ₁) ≤ ( )w B (by Theorem 1.9) ( ) A w B = . Similarly, we have (w AB) = w B( ₁) ≤ ( )w B ≤ B = w A B( ) . Next, we assume that a₁>0. Then

1 1 ' 0k m 0k k B B A w × × ⎛⎡ ⎤⎞ ⎥ ⎜⎢ ⎟ ⎜_{⎣ ⎦}⎟ ⎝ ⎠ = max 1 1 ' , 0k m 0k k B B A x y × × ⎧ ⎡ ⎤ ⎡ ⎤ ⎪ ⎨ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ⎪⎩ 2 2 : 1 x x y y ⎫ ⎡ ⎤ _{+ = ⎬}⎪ ⎢ ⎥ ⎣ ⎦ _⎪⎭ = max

{

B x x₁ , + B A y x₁ ' , : x 2+ y 2 = 1

}

= max _{1 1 1} 1 _{1 2} 2 , : ( , m j j j k j , , ), B x x a y z a y z y y y y = ⎧⎪ _{+ + =} ⎨ ⎪⎩

∑

… B x₁∗ =( ,z z_{1 2},…,z_m), x 2+ y 2 =1

}

≤ max _{1 1 1} 1 _{1 1 2} 2 , : ( , m j , , ), j k j B x x a y z a y z y y y y = ⎧⎪ _{+ + =} ⎨ ⎪⎩

∑

… 1 ( ,1 2, , m), B x∗ = z z … z x 2+ y 2 =1

}

(by Lemma 3.4) = _{1 1 1 2} 1 max , : ( , , , ), m j j k j B x x a y z y y y y = ⎧⎪ _{+ =} ⎨ ⎪⎩

∑

… B x₁ ( ,z z_{1 2}, ,zm), ∗ _{= … x} 2+ _y 2 =₁

}

= 1 1 1 0 ( ) 0 0 m m k m k m k k B a B I w × − × × ⎛⎡ ⎡_⎣ ⎤_⎦⎤⎞ ⎜⎢ ⎥⎟ ⎜_{⎢⎣ ⎥⎦}⎟ ⎝ ⎠ . Let 1 0 ( and 0 0 m m m k m k m k k I a I A × − × × ⎡ ⎡_⎣ ⎤_⎦⎤ ≡ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ) 1 1 0 0 0 0 0 m k k m B B B × × ⎡ ⎤ ⎢ ⎥ ≡_⎢ ⎡ ⎤_⎥ ⎢ ⎥ ⎢ _{⎣ ⎦}⎥ ⎣ ⎦ . Then AB=B A

( ) w AB ≤ A w B( ) = 2 1 1 a+ w B (by Theorem 3.3) ( ) = 1+a w B₁2 ( ₁) ≤ A w B( ) (by Theorem 1.9). Finally, ( ) w AB = 1 1 ' 0k m 0k k B B A w × × ⎛⎡ ⎤⎞ ⎜_{⎢ ⎥}⎟ ⎣ ⎦ ⎝ ⎠ ≤ 1 1 1 0 ( ) ( ) 0 0 m m k m k m k k B a B I w × − w × × ⎛⎡ ⎡_⎣ ⎤_⎦⎤⎞ ⎜⎢ ⎥⎟ = ⎜_{⎢⎣ ⎥⎦}⎟ ⎝ ⎠ AB ≤ A w B( ).

Moreover, since A and B doubly commute, so

( ) w AB ≤ w A B ( ) = w A B ( ) ₁ ≤ ( )w A B . Also, we have 2 1 1 A = +a = A and ( )w A =w A( )(cf. [5, Theorem 2.1]). It follows that w AB( ) ≤ w AB ( ) ≤ w A B( ) . Case 3: Let 1 ' 0 m k m a A a × ⎡⎡ ⎤⎤ ⎢⎢ ⎥⎥ ⎢⎢ ⎥⎥ = ⎢⎢_⎣ ⎥_⎦⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ and 1 2 3 4 B B B B B ⎡ ⎤ = ⎢ ⎥

⎣ ⎦. Then AB=BA, that is,

1 2 3 4 ' 0 0 k m k m m I A B B B B × × ⎡ ⎤ ⎡ ⎤ ⎥ ⎦ ⎢ ⎥ ⎢ ⎣ ⎦ ⎣ 1 2 3 4 = B B B B ⎡ ⎤ ⎢ ⎥ ⎣ ⎦ ' 0 0 k m k m m I A × × ⎡ ⎤ ⎢ ⎥ ⎣ ⎦ implies that 1 ' 3 2 ' 0_{m k} 0_{m m} 4 B A B B A B × × + + ⎡ ⎤ ⎢ ⎥ ⎣ ⎦ = 1 1 3 3 ' ' B B A B B A ⎡ ⎤ ⎢ ⎥ ⎣ ⎦. Thus B₃=0_{m k×} , B₂ +A B' ₄ =B A₁ 'or B₂ =B A₁ '−A B' ₄. So

1 1 4 4 ' ' 0m k B B A A B B B × − ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦ and 1 1 ' 0m k 0m m B B A AB × × ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦.

Now if a₁ =0, then we have A'=0 and w A( )= A =1. So it

follows that ( ) w AB = w B( ₁) ≤ ( )w B (by Theorem 1.9) ( ) A w B = . Similarly, we have (w AB) = w B( ₁) ≤ ( )w B ≤ B = w A B( ) . Next, we assume that a₁>0. Then

1 1 ' 0m k 0m m B B A w × × ⎛⎡ ⎤⎞ ⎥ ⎜_⎢ ⎟ ⎜_{⎣ ⎦}⎟ ⎝ ⎠ = max 1 1 ' , 0_{m k} 0_{m m} B B A x y × × ⎧ ⎡ ⎤ ⎡ ⎤ ⎪ ⎨ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ⎪⎩ 2 2 : 1 x x y y ⎫ ⎡ ⎤ ₊ ⎪ = ⎬ ⎢ ⎥ ⎣ ⎦ _⎪⎭ = max

{

2 2

}

1 , 1 ' , : B x x + B A y x x + y = 1 = max _{1 1 1} 1 _{1 2} 2 , : ( , m j j j m j , , ), B x x a y z a y z y y y y = ⎧⎪ _{+ + =} ⎨ ⎪⎩

∑

… B x₁∗ =( ,z z_{1 2},…,z_k), x 2+ y 2 =1

}

≤ max _{1 1 1} 1 _{1 1 2} 2 , : ( , m j j m j , , ), B x x a y z a y z y y y y = ⎧⎪ _{+ + =} ⎨ ⎪⎩

∑

… 1 ( ,1 2, , k), B x∗ = z z … z x 2 + y 2 =1

}

(by Lemma 3.4) ≤ max _{1 1 1 2} 1 , : ( , , k j , ), j k j B x x a y z y y y y = ⎧⎪ _{+ =} ⎨ ⎪⎩

∑

… B x₁∗ =( ,z z_{1 2},…,z_k), x 2 + y 2 =1

}

= 1 1 1 0_k 0_k B a B w⎛_⎜⎡_⎢ ⎤_⎥⎞_⎟ ⎣ ⎦ ⎝ ⎠.

As in Case 1 before, we have 1 1 1 0_k 0_k B a B w⎛_⎜⎡_⎢ ⎤_⎥⎞_⎟ ⎣ ⎦ ⎝ ⎠ = w AB ( ) ≤ A w B ( ) ≤ A w B( ) and 1 1 1 0_k 0_k B a B w⎛_⎜⎡_⎢ ⎤_⎥⎞_⎟ ⎣ ⎦ ⎝ ⎠ = w AB ( ) ≤ ( )w A B ≤ w A B( ) , where 1 0 0 k k k k I a I A= ⎢⎡ ⎤_⎥ ⎣ ⎦ and 1 1 0 . 0 k k B B B ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦ So it follows that w AB( ) = 1 1 ' 0m k 0m m B B A w × × ⎛⎡ ⎤⎞ ⎜⎢ ⎥⎟ ⎣ ⎦ ⎝ ⎠ ≤ 1 1 1 0_k 0_k B a B w⎛⎜⎡_⎢ ⎤_⎥⎞⎟ ⎣ ⎦ ⎝ ⎠ ≤ A w B( ), w A B( )

This completes the proof.

In conclusion, we remark that it is unknown that whether the inequalities

( )

w AB ≤ A w B( ) and (w AB) ≤ w A B( ) hold for commuting n-by-n matrices A and B with A satisfying a quadratic polynomial equation.

References

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2. J. A. R. Holbrook, Inequalities of von Neumann type for small matrices, in: Function Space, K. Jarosz, ed. , Marcel Dekker, New York, 1992, pp. 189 193. −

−

3. R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge Univ. Press, Cambridge, 1991.

4. V. Muller, The numerical radius of a commuting product, Michigan Math. J. 35 (1988), 255−260.

5. S.-H. Tso and P. Y. Wu, Matricial ranges of quadratic operators, Rocky Mountain J. Math. 3 (1999), 1139 1142.