Spectral Radius, Numerical Radius and Unitarily Invariant

Spectral Radius, Numerical Radius and Unitarily Invariant Norm

Inequalities in Hilbert Space

By

Doaa Mahmoud Al-Saafin

Supervisor

Dr. Aliaa Abdel-Jawad Burqan

This Thesis was Submitted in Partial Fulfillment of the Requirements for the

Master’s Degree of Science in Mathematics

Faculty of Graduate Studies

Zarqa University

June, 2016

iii

الإهـــــــــــــداء

مــــــحيالر ن ــــمح الر الله م س ب

ف نفس حب المعرفة والإمان سل الطرق وغر ن أضاء إلى م

ــــأب

زهرة الحاة ونورها المكان .. وغمة الحنان .. مة إلى خ

ـــأم

.. خطوة بخطوة إلى شقق روح .. سندي بعد الله ورفق درب

وسفـأخ

ملاك ف الحاة وقدوت .. لى القلب الطاهر ..إ

أخت فاطمة

.هذا الجهد المتواضع عائلت الحببة .. أهدية أفراد وإلى بق

iv

ACKNOWLEDGEMENT

I would thank first Allah for helping me and inspiring me with patience and

endurance to be able to do this thesis.

I would also like to express my deepest gratitude to my supervisor, Assistant

Professor Aliaa Burqan, for her continuous support, guidance and encouragement to

achieve more than I thought possible.

Finally, my deepest gratitude go to my beloved family and friends. This dissertation

would not have been achieved without their encouragement and support day and night

to complete this thesis.

Doaa Al-Saafin

v

TABLE OF CONTENTS

Dedication………………………………………………………………………… iii

Acknowledgment…………………………………………………………………. iv

Abstract…………………………………………………………………………… vi

Introduction……………………………………………………………………… 1

Chapter One: Fundamentals of Matrix Analysis

Basic Results in matrix Theory…………………………………………………… 4

Positive Semidefinite Matrices……………………………………………………. 12

Unitarily Invariant Norms………………………………………………………… 16

Spectral Radius and Numerical radius……………………………………………. 21

Chapter Two: Numerical and Spectral Radius Inequalities of Matrices

Recent Numerical Radius Inequalities……………………………………………. 28

Cartesian Decomposition and Numerical Radius Inequalities …………………… 50

Spectral Radius Inequalities………………………………………………………. 56

Chapter Three: Two by Two Block Matrix Inequalities

Numerical Radius Inequalities for General Block Matrices……………….. 60

Inequalities for the off-Diagonal Part of Block Matrices………………... 70

On Unitarily Invariant Norm Inequalities and Hermitian Block Matrices……….. 80

References………………………………………………………………………… 96

Abstract in Arabic………………………………………………………………… 100

vi

Spectral Radius, Numerical Radius and Unitarily Invariant Norm

Inequalities in Hilbert Space

By

Doaa Mahmoud Al-Saafin

Supervisor

Dr. Aliaa Abdel-Jawad Burqan

ABSTRACT

In this thesis, we present several inequalities for spectral radius,

numerical radius and unitarily invariant norm for square matrices.

Related inequalities for spectral radius and numerical radius of two by

two block matrices are also given.

1

INTRODUCTION

The study of matrix theory has become more and more popular in the last few

decades. Researchers are attracted to this subject because of its connections with many

other pure and applied areas. In particular, the eigenvalues are crucial in solving system

of differential equations, analyzing population growth models and calculating powers of

matrices. It is not always easy to calculate the eigenvalues. However, in many scientific

problems it is enough to know that the eigenvalues lie in some specified regions. Such

information is provided in this thesis by comparing between spectral radius, numerical

radius and unitarily invariant norm.

Several inequalities involving spectral radius, numerical radius and matrix norm can

be found in many books on inequalities, like Bhatia (1997). Some investigations on

norm and numerical radius inequalities involving the Cartesian decomposition were

obtained by El-Haddad and Kittaneh (2007). In 2011 and 2012, Hirzallah, Kittaneh and

Shebrawi gave several inequalities for the numerical radius of two by two block

matrices. An estimate for the numeral radius of the matrix was given by Kittaneh

(2003), Yamazaki (2007) improved this result by using Aluthge transform. In 2013,

Abu-Omar and Kittaneh studied similar topics and gave inequalities that involve the

generalized Aluthge transform. Bahatia and Kittaneh (1990) and Zhan (2000) proved

important inequalities for the singular value of matrices. Tao (2006) employed these

inequalities to establish different equivalent inequalities for singular values of matrices.

On the other hand, Abu-Omar and Kittaneh (2015), applied spectral radius and norm

inequalities to two by two block matrices to give simple proofs and refinements of some

norm inequalities. Bourin and Lee (2012) proved a remarkable decomposition lemma

that plays a major role in several recent inequalities for positive semidefinite two by two

2

block matrices. Burqan (2013) proved a unitarily invariant norm inequality for positive

semidefinite two by two block matrices which gives a relation between the real part of

the off-diagonal blocks and the sum of the diagonal blocks.

This thesis is divided into three chapters.

Chapter One, which consists of four sections, highlights basic definitions and

properties for the square matrices in Hilbert space that are useful through the thesis.

Section (1.1) and section (1.2) present the most important properties of unitary,

Hermitian, normal and positive matrices. Also, the spectral mapping theorem, Schur's

unitary triagularization theorem and other famous decomposition results for matrices

such as the singular value decomposition and polar decomposition are presented.

Section (1.3) deals with unitarily invariant norms and various classes of norms such

as the Hilbert-Schmidt norm, spectral norm and Ky Fan norms.

Section (1.4) introduces the concepts of spectral radius and the numerical radius.

Also, some of the well-known facts about them are presented.

Chapter Two, which consists of three sections, concerns with numerical radius and

spectral radius inequalities.

Section (2.1) presents some improvements of basic numerical radius inequalities and

gives generalizations for these improvements.

Section (2.2) presents several interesting inequalities and identities for the numerical

radius which involve the Cartesian decomposition of matrices.

Section (2.3) deals with the spectral radius inequalities. several interesting

inequalities were proved by Abu-Omar and Kittaneh (2013).

3

Chapter Three, which consists of three sections, concerns with two by two block

matrices.

Section (3.1) presents recent inequalities for the numerical radius of general two by

two block matrices.

Section (3.2) deals with the off-diagonal part of two by two block matrices. we

present several numerical radius inequalities for this off-diagonal.

Section (3.3) discusses several inequalities for singular values of matrices and

employ them to prove several equivalent theorems. After that, we deal with special case

of positive semidefinite two by two block matrices. At the end of this section, we

establish new estimates for the spectral norm and numerical radius of the off-diagonal

part of positive semidefinite two by two block matrices.

4

Chapter One

Fundamentals of Matrix Analysis

This chapter contains a brief review of the basic concepts and results which are

important in this thesis. In section (1.1), we present some basic results in matrix theory.

In section (1.2), we consider the class of positive semidefinite matrices. This class,

which is included in the class of Hermitian matrices, arises naturally in many

applications. In section (1.3), we review the basic concepts and results taught in

unitarily invariant matrix norms. In section (1.4), we introduce the concepts of spectral

radius and the numerical radius and give the basic results concerning them that will be

used later.

1.1. Basic Results in Matrix Theory:

We will denote the algebra of all complex matrices by . It should be

mentioned that most results which hold for matrices can be generalized for operators

acting on Hilbert spaces.

Definition 1.1.1:

Let Then a complex number λ is called an eigenvalue of if there exists a

nonzero vector such that The vector x is called an eigenvector of

corresponding to λ.

Remark:

If with eigenvalues then ∑ ∏

where tr and det are the trace and determinant functions, respectively.

5

Definition 1.1.2:

If , then is called the characteristic equation of . The

polynomial is called the characteristic polynomial of . The set of

all λ that are eigenvalues of is called the spectrum of and it is denoted by ( ).

For , B the product matrices and need not be equal. However, we

have the following theorem.

Theorem 1.1.1.(Horn and Johnson, 1985):

Let . Then

.

Theorem 1.1.2:

Let such that . Then

and

Theorem 1.1.3.(The Spectral Mapping Theorem):

Let Then for every polynomial p,

( { }

6

Definition 1.1.3:

Let [ ] Then the adjoint of , denoted by is the matrix given by

[ ]

Remark:

For and , we have

1)

2)

3)

4)

5) det

6)

7) { } .

8) is invertible if and only if is invertible and .

9) If [ ] then ∑ | |

Thus and if and

only if

Definition 1.1.4:

For

the Euclidean inner product of x

and y is defined as ⟨ ⟩ ∑

Remark:

For all and , we have

1) ⟨ ⟩ ⟨ ⟩

7

2) ⟨ ⟩ ⟨ ⟩ and ⟨ ⟩ ⟨ ⟩

3) ⟨ ⟩ ⟨ ⟩ ⟨ ⟩

Theorem 1.1.4.(Cauchy-Schwarz Inequality):

Let Then

|⟨ ⟩| ⟨ ⟩ ⁄ ⟨ ⟩ ⁄

with equality if and only if x and y are linearly dependent.

From which it follows that the equation

‖ ‖ ⟨ ⟩ ⁄ , for every ,

defines a norm on called the Euclidean norm.

Definition 1.1.5:

Let . Then is said to be unitary matrix if

In the following theorem, we list some of the basic conditions for a matrix to be

unitary.

Theorem 1.1.5:

If the following are equivalent:

a) is unitary.

b) is invertible and

c)

d) is unitary.

8

e) The columns (rows) of form an orthonormal set.

f) ‖ ‖ ‖ ‖ ‖ ‖ for all

g) ⟨ ⟩ ⟨ ⟩ for all

Remark:

For any unitary matrix , we have

1) Every eigenvalue of has modulus one.

2) | | .

3) If is unitary, then is unitary.

Definition 1.1.6:

Two matrices are said to be unitary equivalent if there is a unitary matrix

such that If is unitary equivalent to a diagonal matrix, is

said to be unitary diagonalizable.

Theorem 1.1.6.(Schur's Unitary Triagularization Theorem):

Let with { } Then there is a unitary matrix such

that , where [ ] is an upper triangular matrix with diagonal

entries ,

Definition 1.1.7:

A matrix is called Hermitian (or self-adjoint) if . It is called skew-

Hermitian if

9

Remark:

1) The sum of two Hermitian matrices is Hermitian.

2) The product of two Hermitian matrices is Hermitian if and only if the matrices

commute.

3) If , then and are Hermitian, but is skew-Hermitian.

4) If is Hermitian, then the main diagonal entries of are all real and if is skew-

Hermitian, then the main diagonal entries of are all pure imaginary.

5) If is Hermitian, then the eigenvalues of are all real and if is skew-Hermitian,

then the eigenvalues of are all pure imaginary.

Matrix normality is one of the most interesting topics in linear algebra and matrix

theory, since normal matrices have not only simple structures under unitary equivalence

but also many applications.

Definition 1.1.8:

Let Then is called normal if

Remark:

1) It is obvious that Hermitian, skew-Hermitian and unitary matrices are normal

matrices.

2) The sum and product of two commuting normal matrices are normal.

Next, we present the most fundamental facts about normal matrices.

10

Theorem 1.1.7:

If [ ] the following statements are equivalent:

a) is normal.

b) is unitary diagonalizable.

c) ‖ ‖ ‖ ‖ for all

d) ∑ | |

∑ | |

, where are the eigenvalues of

The equivalent of (a) and (b) in the previous theorem is called the spectral theorem

for normal matrices.

Theorem 1.1.8.(Cartesian Decomposition):

Let Then there exist Hermitian matrices B and C such that

Necessarily,

and

The matrices B and C are called the

real part and the imaginary part of , and denoted by Re and Im , respectively.

It is easy to verify that and is normal if and only if Re

and Im commute.

For all the expression of the inner product of and as

⟨ ⟩

‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖

is called the polarization identity.

11

The following formula is a generalized of the identity .

Theorem 1.1.9.(Generalized Polarization Identity):

Let and Then

⟨ ⟩

⟨ ⟩

⟨ ⟩

⟨ ⟩

⟨ ⟩

12

1.2. Positive Semidefinite Matrices:

Definition 1.2.1:

A Hermitian matrix is said to be positive semidefinite, written as , if

⟨ ⟩

is further called positive definite, written as , if

⟨ ⟩

Remark:

1) The sum of any two positive definite (semidefinite) matrices of the same size is

positive definite (semidefinite).

2) The product of any two positive definite (semidefinite) matrices is positive definite

(semidefinite) if and only if the two matrices commute.

3) Each eigenvalue of a positive definite (semidefinite) matrix is positive

(nonnegative) real number.

4) The Hermitian matrix is positive definite (semidefinite) if and only if all

eigenvalues of are positive (nonnegative) real numbers.

5) The trace and determinant of positive definite (semidefinite) matrices are positive

(nonnegative) real numbers.

6) If is positive semidefinite and then is positive semidefinite,

and if is positive definite matrix, then is positive definite if and only if is

invertible.

13

Theorem 1.2.1:

Let be a positive semidefinite (definite) matrix and let be given

integer. Then there exists a unique positive semidefinite (definite) matrix such

that , written as ⁄ or √

Remark:

1) Let and be two Hermitian matrices of the same size. If , we write

or

2) If are positive semidefinite matrices, then ⁄ ⁄ .

3) If are positive semidefinite matrices, then the eigenvalues of are all

nonnegative.

Theorem 1.2.2.(Weyl's Monotonicity Principle Theorem).(Zhang, 1999):

Let be two Hermitian matrices. Then

Theorem 1.2.3:

Let Then is positive semidefinite if and only if for some

In the positive definite case B is taken to be invertible.

The absolute value of a matrix is defined as the square root of the positive

semidefinite matrix and denoted by | |. That is,

| | ⁄

14

Theorem 1.2.4:

Let be Hermitian, and let be any vector. Then

|⟨ ⟩| ⟨| | ⟩

Theorem 1.2.5:

Let be positive semidefinite, and let be any unit vector. Then

(a) ⟨ ⟩ ⟨ ⟩

(b) ⟨ ⟩ ⟨ ⟩

The eigenvalues of | | are called the singular values of . We will always

enumerate them in decreasing order and use for them the notation

The singular value decomposition is one of the most important factorization of

complex matrices which depends on the singular values.

Theorem 1.2.6.(Singular Value Decomposition):

If then

for some unitary matrices , and the matrix

( )

15

Theorem 1.2.7.(Polar Decomposition):

If then there exists a unitary matrix such that

where is positive semidefinite and it is uniquely determined as | |. If is

invertible, then is uniquely determined as

16

1.3. Unitarily Invariant Norms:

If one has several matrices in , what might it mean to say that some are "small" or

that others are "large"? One way to answer this question is to study norms of matrices.

A matrix norm is a number defined in terms of the entries of the matrix. The norm is

a useful quantity which can give important information about a matrix.

Definition 1.3.1:

A function is called a matrix norm if for all and all it

satisfies the following axioms:

1) , and if and only if .

2) | | .

3)

4)

Notice that the properties (1) (3) are identical to the axiom for a vector norm. A

vector norm on matrices, that is a function satisfies (1) (3) and not necessarily (4), is

often called a generalized matrix norm.

Example 1.3.1:

If [ ] then

1. The Frobenius (Hilbert Schmidt) norm of is given by

‖ ‖ (∑| |

)

*∑

+

17

2. The spectral (operator) norm of is given by

‖ ‖

‖ ‖

‖ ‖

‖ ‖ ‖ ‖

where is the largest singular value of

Theorem 1.3.1:

Let . Then

‖ ‖ ‖ ‖ √ ‖ ‖

Definition 1.3.2:

A matrix norm is called unitarily invariant norm, if

whenever and are unitary matrices, and it is denoted by ‖ ‖

The following are the most familiar unitarily invariant norms.

1. The Schatten p-norms are defined as

‖ ‖ [∑( )

]

⁄

[ | | ] ⁄

Notice that the Frobenius and the spectral norms are important special cases of the

Schatten p-norms, corresponding to the values and , respectively.

2. The Ky Fan k-norms are defined as

‖ ‖ ∑

18

It is clear that the norm ‖ ‖ is the same as ‖ ‖ and the norm ‖ ‖ is the same

as ‖ ‖

Remark:

For any unitarily invariant norm ‖ ‖ and for any we have

‖ ‖ ‖ ‖ ‖| |‖

Theorem 1.3.2.(Fan Dominace Theorem):

Let . Then

‖ ‖ ‖ ‖

for all unitarily invariant norms on if and only if

‖ ‖ ‖ ‖ , .

By using the Ky Fan k-norms formula we can rewrite Theorem (1.3.2) as:

For

‖ ‖ ‖ ‖

for all unitarily invariant norms if and only if

∑ ∑

This is known as the Fan Dominace property.

19

Remark:

If are positive semidefinite such that , then

‖ ‖ ‖ ‖

for every unitarily invariant norm, and

Theorem 1.3.3:

Let be Hermitian matrices. Then

‖ ‖ ‖ ‖

for every unitarily invariant norm.

Theorem 1.3.4:

Let be positive semidefinite matrices. Then

‖ ‖ ‖ ‖

Theorem 1.3.5.(Bhatia, 1997):


‖*

+‖ ‖*

+‖


We end this section with a matrix versions of the arithmetic-geometric mean.

20

Theorem 1.3.6.(Bhatia and Kittaneh, 1990):


‖ ‖

‖ ‖


The following is a generalization of Theorem (1.3.6).

Theorem 1.3.7.(Bhatia and Davis, 1993):

Let such that are positive semidefinite matrices. Then

‖ ‖

‖ ‖


21

1.4. Spectral Radius and Numerical Radius:

Definition 1.4.1:

The spectral radius of a matrix is defined as

{| | }

It is well known that

for every matrix norm Moreover, if is normal, then

‖ ‖

Let and n is positive integer. It follows readily from the spectral

mapping theorem and Theorem (1.1.1) that

| |

and

Theorem 1.4.1.(Spectral Radius Formula):

Let Then

⁄

22

Theorem 1.4.2:

Let such that . Then

and

The spectral radius is not a norm, this can be easily seen by considering the

matrix *

+ and noting that

Definition 1.4.2:

The numerical range of a matrix is the subset of the complex numbers ,

given by

{⟨ ⟩ ‖ ‖ }

Note that: ⟨ ⟩

Let and let Then the following are immediate:

{ }

and

23

A very important property of the numerical range of a matrix is that it includes the

spectrum of the matrix as in the following theorem.

Theorem 1.4.3.(Horn and Johnson, 1991):

Let Then

Theorem 1.4.4.(Gustafson and Rao, 1997):

If such that is positive definite and , then

.

The following example can be found in Halmos (1982).

Example 1.4.1:

1. (*

+) [ ]

2. (*

+) { | | ⁄ }

Definition 1.4.3:

The numerical radius of a matrix is given by

| |

It is easy to verify that defined a vector norm on This norm is weakly

unitarily invariant (i.e., for any matrix and any unitary

matrix ) and satisfies for any matrix We will denote

| | by for any

24

Theorem 1.4.5:

The numerical radius norm and the matrix norm ‖ ‖ on are equivalent. In

fact,

‖ ‖ ‖ ‖

Theorem 1.4.6:

Let Then

‖ ‖

Moreover, if is normal, then

‖ ‖

A matrix is called nilpotent if for some positive integer The

smallest such is sometimes called the power of nilpotency of

Theorem 1.4.7:

Let be a nilpotent matrix. Then

‖ ‖ (

)

where is the power of nilpotency of .

It follows from Theorem and Theorem that both inequalities in

Theorem are sharp. The first inequality becomes an equality if . The

second inequality becomes an equality if is normal.

25

Remark:

Numerical radius is not submultiplicative. But for any we have

In particular, if are commute, then

and if are normal, then

Theorem 1.4.8:

Let Then

for every positive integer

The following useful theorem provides alternative way to compute the numerical

radius of a matrix, and will be used frequently throughout this thesis.

Theorem 1.4.9:

Let Then

‖ ( )‖

‖ ( )‖

26

Theorem 1.4.10:

Let [ ] The following statements hold:

a) ([ ]) ([| |])

b) If for all then

([ ])

([ ])

Definition 1.4.4:

Let | | be the polar decomposition of The Aluthge transform of

is defined by

| | | |

Aluthge transform was first defined by Aluthge (1990). The following are among the

well-known relations:

1) ( )

2) ‖ ‖ ‖ ‖

3) ( )

4) ( )

Definition 1.4.5:

For any two matrices and in we let denote the direct sum of and ,

that is matrix *

+

27

Remark:

Let Then

1) ‖ ‖ ‖*

+‖

2) ‖ ‖ ‖ ‖

3) ‖ ‖ ‖ ‖ ‖ ‖

4)

5) (*

+) √

6) ( )

The material in this chapter can be found in almost every book on matrix analysis.

Here we mention the books of , and Aluthge .

28

Chapter Two

Numerical Radius and Spectral Radius Inequalities of Matrices

This chapter is devoted to the recent numerical and spectral radius inequalities. In

section (2.1), we present some improvements of basic numerical radius inequalities,

then we give generalizations of these improvements. In section (2.2), we present some

generalizations and results of numerical radius inequalities that are concerning the

Cartesian decomposition. In section (2.3), we present several spectral radius inequalities

for sum, product and power of matrices in

2.1. Recent Numerical Radius Inequalities:

It has been mentioned earlier that if then

‖ ‖ ‖ ‖

The inequalities (2.1.1) have been improved by many mathematicians. In this section

we will present recent improvements of these inequalities.

Theorem 2.1.1.(Kittaneh, 2003):

Let Then

‖ ‖

‖ ‖ ⁄

To prove Theorem (2.1.1), Kittaneh used the following useful lemmas. The first

lemma, which contains a mixed Schwarz inequality, can be found in Halmos (1982).

29

Lemma 2.1.1:

If , then

|⟨ ⟩| ⟨| | ⟩ ⟨| | ⟩

for all

The second lemma contains a special case of more general norm inequality. see

Furuta (1989).

Lemma 2.1.2:

If are positive semidefinite matrices, then

‖

‖ ‖ ‖

The third lemma contains a norm inequality for sums of positive semidefinite

matrices that is sharper than the triangle inequality. See Kittaneh (2002).

Lemma 2.1.3:

If are positive semidefinite matrices, then

‖ ‖

‖ ‖ ‖ ‖

√ ‖ ‖ ‖ ‖ ‖

‖

Proof of Theorem (2.1.1):

By Lemma (2.1.1) and by the arithmetic-geometric mean inequality, we have for

every

|⟨ ⟩| ⟨| | ⟩ ⟨| | ⟩

⟨| | ⟩ ⟨| | ⟩

30

And so

|⟨ ⟩|

⟨ | | | | ⟩

By taking the maximum on both sides in the above inequality over with

‖ ‖ and observing that | | | | is positive semidefinite matrix, we get

‖| | | |‖

Applying Lemma (2.1.2) and Lemma (2.1.3) to the positive semidefinite matrices | |

and | |, and using the fact that ‖| |‖ ‖| |‖ ‖ ‖ and ‖| || |‖ ‖ ‖ we have

‖| | | |‖ ‖ ‖ ‖ ‖

and so

‖| | | |‖

‖ ‖

‖ ‖

as required.

Since ‖ ‖ ‖ ‖ for every the inequality (2.1.2) is a refinement of the

second inequality in (2.1.1).

Theorem 2.1.2.(Kittaneh, 2005):

Let Then

‖ ‖

‖ ‖

31

Proof:

Let be the Cartesian decomposition of , and let be any vector in

Then by the convexity of the function we have

|⟨ ⟩| ⟨ ⟩ ⟨ ⟩

|⟨ ⟩| |⟨ ⟩|

|⟨ ⟩|

By taking the maximum on both sides in the above inequality over ‖ ‖

we get

‖ ‖

‖ ‖

Thus,

‖ ‖

‖ ‖

‖ ‖

‖ ‖

‖ ‖

and so

‖ ‖

which proves the first inequality in (2.1.4).

32

To prove the second inequality in (2.1.4) let be any unit vector. Then by

Cauchy-Schwarz inequality, we have

|⟨ ⟩| ⟨ ⟩ ⟨ ⟩

‖ ‖ ‖ ‖

⟨ ⟩ ⟨ ⟩

⟨ ⟩

Thus,

|⟨ ⟩| ⟨ ⟩


‖ ‖ we get

‖ ‖

‖ ‖

which proves the second inequality in (2.1.4) and completes the proof.

To see that the inequalities (2.1.4) improve the inequalities (2.1.1), consider the

chain of inequalities

‖ ‖

‖ ‖

‖ ‖ ‖ ‖

In order to prove the inequality (2.1.6), we need the following lemma which can be

found in Kittaneh (2004).

33

Lemma 2.1.4:

For any we have

‖ ‖ ( ) ‖ ‖

Now, the first inequality in (2.1.6) is an immediate consequence of Lemma (2.1.4)

while the last follows by the triangle inequality and the fact that

‖ ‖ ‖ ‖ ‖ ‖

By using Aluthge transform and the generalized polarization identity, Yamazaki

(2007) improved the second inequality in (2.1.1) as follows:

Theorem 2.1.3.(Yamazaki, 2007):

If then

‖ ‖

( )

Proof:

Let | | be the polar decomposition of and let . Then by the

generalized polarization identity, we have for any unit vector ,

⟨ ⟩ ⟨ | | ⟩

⟨| |( ) ( ) ⟩

⟨| |( ) ( ) ⟩

⟨| |( ) ( ) ⟩

⟨| |( ) ( ) ⟩

Now, since | | is positive semidefinite, all inner products of the terminal side are

positive.

34

Thus,

⟨ ⟩

⟨| |( ) ( ) ⟩

⟨| |( ) ( ) ⟩

⟨| |( ) ( ) ⟩

⟨( )| |( ) ⟩

‖( )| |( )‖

‖( )| |

(( )| |

)

‖

‖(( )| |

)

( )| | ‖

‖| |

( )( )| |

‖

‖( | |

| |

) ( | |

| |

)‖

‖| | | |

| |

| |

| |

| |‖

‖ | | ‖

‖| | ( )‖

‖ ‖

‖ ( )‖

‖ ‖

( )

35

Thus,

⟨ ⟩

‖ ‖

( )

Now, since

|⟨ ⟩|

( ⟨ ⟩)

⟨ ⟩

we get

‖ ‖

( )

as required.

Remark:

If then

( ) ‖ ‖ ‖| | | |

‖

‖(| | | |

) (| |

| |

)

‖

((| |

| |

) (| |

| |

)

)

(| |

| | | |

)

| | | |

( | | | |)

36

Now, since

( | | | |) ‖ | | | |‖

‖ ‖

we have

( ) ‖ ‖

Thus, the inequality (2.1.7) is sharper than the inequality (2.1.2).

By Theorem (2.1.3) and Theorem (1.4.5), we have the following corollary.

Corollary 2.1.1:

Let If then

‖ ‖

Another improvement of the inequalities (2.1.1) is due to Abu-Omar and Kittaneh

(2015),(c) as follows:.

Theorem 2.1.4.(Abu-Omar and Kittaneh, 2015):

Let Then

√‖ ‖

√‖ ‖

37

Proof:

Let be a unit vector and let be a real number such that

⟨ ⟩ |⟨ ⟩|

We have

‖ ( )‖

‖ ‖

‖( )

‖

√‖ ‖

√|⟨ ⟩|

√|⟨ ⟩ ⟨ ⟩|

√|⟨ ⟩ ⟨ ⟩ |

√|⟨ ⟩ |⟨ ⟩||

√|⟨ ⟩ |

Thus,

‖ ‖

√|⟨ ⟩ |

√‖ ‖

38


To prove the second inequality in (2.1.9), we have

‖ ( )‖

‖ ‖

‖( ) ‖

‖ ( )‖

√ ‖ ‖

‖ ‖

√‖ ‖

which proves the second inequality in (2.1.9) and completes the proof.

The following theorems are generalizations of the first inequality in (2.1.3) and the

second inequality in (2.1.4).

Theorem 2.1.5.(El-Haddad and Kittaneh, 2007):

Let Then

‖| | | | ‖


Let Then

‖ | | | | ‖

39

In prooving their generalizations, El-Haddad and Kittaneh (2007) used the following

lemmas. The first lemma is an application of Jensen's inequality, and can be found in

Hardy, Littlewood and Pólya (1988).

Lemma (2.1.5):

For ,

a) [ ]

b)

The second lemma is known as the generalized mixed Schwarz's inequality, and can

be found in Kittaneh (1988).

Lemma (2.1.6):

Let Then

|⟨ ⟩| ⟨| | ⟩⟨| | ⟩


For every unit vector we have

|⟨ ⟩| ⟨| | ⟩ ⟨| | ⟩

( )

(⟨| | ⟩ ⟨| | ⟩

)

( )

(⟨| | ⟩ ⟨| | ⟩

)

( )

40

Thus,

|⟨ ⟩|

⟨(| | | | ) ⟩

taking the maximum on both sides in the above inequality over ‖ ‖

produces

‖| | | | ‖

as required.


We have

|⟨ ⟩| ⟨| | ⟩ ⟨| | ⟩ ( )

⟨| | ⟩ ⟨| | ⟩ ( )

⟨| | ⟩ ⟨| | ⟩ ( )

⟨| | ⟩ ⟨| | ⟩ ( )

Thus,

|⟨ ⟩| ⟨ | | | | ⟩

By taking the maximum on both sides in the above inequality over ‖ ‖ , we

get

‖ | | | | ‖

as required.

41

Dragomir (2008,2009), established other inequalities related to the spectral norm and

the numerical radius as follows:

Theorem 2.1.7.(Dragomir, 2008):

Let Then

‖ ‖

Dragomir used the following useful lemma in proving Theorem which can

be considered as a refinement of Cauchy-Schwarz inequality.

Lemma 2.1.8:

For such that ‖ ‖

‖ ‖‖ ‖ |⟨ ⟩ ⟨ ⟩⟨ ⟩| |⟨ ⟩⟨ ⟩| |⟨ ⟩|

Proof:

By the first inequality in (2.1.11), we deduce

‖ ‖‖ ‖ |⟨ ⟩| |⟨ ⟩⟨ ⟩|

Let be any unit vector and put in the inequality

(2.1.12).

Thus,

‖ ‖‖ ‖ |⟨ ⟩| |⟨ ⟩|

By taking the maximum on both sides in the inequality (2.1.13) over with

‖ ‖ we have

42

‖ ‖

as required.

Theorem 2.1.8.(Dragomir, 2009):

Let Then

‖ ‖

for all .

Proof:

Let be any vector in By the Schwarz's inequality, we have

|⟨ ⟩| |⟨ ⟩| ‖ ‖ ‖ ‖

⟨ ⟩ ⟨ ⟩

By the arithmetic-geometric mean inequality and the convexity of

we have

⟨ ⟩ ⟨ ⟩

⟨ ⟩ ⟨ ⟩

(⟨ ⟩ ⟨ ⟩

)

(⟨ ⟩ ⟨ ⟩

)

(⟨ ⟩

)

43

Thus,

|⟨ ⟩|

⟨ ⟩

Note that, is Hermitian. So by taking the maximum on both sides in

the above inequality over ‖ ‖ we deduce the desired inequality.

Sattari, Moslehian and Yamazaki (2015) generalized inequality (2.1.10) as follows:

Theorem 2.1.9.(Sattari, Moslehian and Yamazaki, 2015):

Let Then

‖ ‖

for all

Proof:

By applying Lemma (2.1.5) (a) on the inequality (2.1.13), we get

|⟨ ⟩|

‖ ‖‖ ‖ |⟨ ⟩| (

‖ ‖ ‖ ‖ |⟨ ⟩|

)

Hence,

|⟨ ⟩|

‖ ‖ ‖ ‖ |⟨ ⟩|

Taking the maximum on both sides in the above inequality over with ‖ ‖ ,

we obtain the desired inequality.

44

The following is another upper bound for given by Sattari, Moslehian and

Yamazaki (2015).

Theorem 2.1.10.(Sattari, Moslehian and Yamazaki, 2015):

Let Then

‖ ‖

for all

Proof:

For any unit vector we have

⟨ ⟩

‖( ) ‖

‖( ) ‖

( )

‖( ) ‖

‖ ‖

‖ ‖

‖( )

( )‖

‖ ‖

‖ ‖

‖ ( )‖

‖ ‖

45

Since ⟨ ⟩ ⟨ ⟩ the inequality

‖ ‖

|⟨ ⟩|

⟨ ⟩

For since and

are convex and matrix concave functions, respectively, we

have

(

‖

‖

)

‖

‖

‖(

)

‖

‖

‖

as required.

Remark:

By Theorem (2.1.10) and Theorem (2.1.8), we have

‖ ‖

‖ ‖

Hence if both and are normal matrices, then the inequality (2.1.16) is sharper than

the inequality (2.1.14).

46

Kittaneh (2006) established a general spectral radius inequality which gives spectral

radius inequalities for sums and products of matrices. In fact, Kittaneh has shown that if

then

‖ ‖ ‖ ‖

√ ‖ ‖ ‖ ‖ ‖ ‖‖ ‖

Recently, by using the inequality (2.1.17), Abu-Omar and Kittaneh (2015),(a),

improved the triangle inequality of numerical radius as follows:


Let Then

( )

√( )

‖ ‖

Proof:

Let be any real number. Then by letting ( ) ( )

and in the inequality (2.1.17), we have

‖ ( )‖ ( ( ) ( ))

(‖ ( )‖ ‖ ( )‖)

√ ‖ ‖ ‖ ‖ ‖ ‖

47

Thus,

‖ ( )‖ ‖[‖ ( )‖ √‖ ‖

√‖ ‖ ‖ ( )‖]‖

Hence, by the norm monotonicity of matrices with nonnegative entries and then by

Theorem (1.4.9), we have

‖ ( )‖

‖*

‖ ( )‖

√‖ ‖

√‖ ‖

‖ ( )‖+‖

‖*

√‖ ‖

√‖ ‖ +‖

( )

√( )

‖ ‖

Now, the inequality (2.1.18) follows by taking the maximum over all and by

using Theorem (1.4.9).

Hou and Du (1995) established useful estimates for the spectral radius, the numerical

radius and the spectral norm of matrix [ ] with entries In particular,

they proved that

([‖ ‖])

([‖ ‖])

and

‖ ‖ ‖[‖ ‖]‖

48

Abu-Omar and Kittaneh (2015),(b) improved the inequality as follows:


Let [ ] be a matrix with Then

([ ])

where

, ( )

‖ ‖

Proof:

Let [

] be a unit vector in Then

|⟨ ⟩| |∑ ⟨ ⟩

| ∑ |⟨ ⟩|

∑|⟨ ⟩|

∑|⟨ ⟩|

∑ ‖ ‖

∑‖ ‖‖ ‖‖ ‖

∑

‖ ‖‖ ‖

⟨[ ] ⟩

49

where [

‖ ‖

‖ ‖

‖ ‖

]

Now, since is a unit vector in then

|⟨ ⟩| ([ ])

and so

‖ ‖

|⟨ ⟩| ([ ])

as required.

The following corollary is an immediate consequence of Theorem (2.1.12) and

Corollary 2.1.2:

Let Then

(*

+)

( )

√( )

‖ ‖ ‖ ‖

50

2.2. Cartesian Decomposition and Numerical Radius Inequalities:

It well-known that if is the Cartesian decomposition of a matrix

then

| | | |

Thus, the inequalities (2.1.4) can be written as

‖ ‖ ‖ ‖

Or equivalently, as

‖ ‖

‖ ‖

El-Haddad and Kittaneh (2007) gave generalizations of the second inequality of

(2.2.1) and the inequalities (2.2.2) by using the Cartesian decomposition of the matrix as

the following theorems.


Let with the Cartesian decomposition and let Then

‖| | | | ‖

To prove Theorem (2.2.1), we need the following lemma which is an application of

Jensen's inequality, can be found in Hardy, Littlewood and Pólya (1988).

Lemma (2.2.1):

Let . Then

51

Proof:

For any unit vector and for we have

|⟨ ⟩| ⟨ ⟩ ⟨ ⟩

|⟨ ⟩| |⟨ ⟩| ( )

⟨| | ⟩ ⟨| | ⟩ ( )

⟨| | ⟩ ⟨| | ⟩

⟨ | | | | ⟩

Thus, we obtain the inequality

|⟨ ⟩| ⟨ | | | | ⟩


‖ ‖ , we obtain

‖| | | | ‖

For the case we have

‖ ‖ ( )

‖ ‖

‖| | | | ‖ ( )

as required.

52



‖| | | | ‖

Proof:

For any unit vector we have

|⟨ ⟩|

√ (

⟨ ⟩ ⟨ ⟩

)

(|⟨ ⟩| |⟨ ⟩|

)

( )

⟨| | ⟩ ⟨| | ⟩

( )

⟨| | ⟩ ⟨| | ⟩

( )

Thus,

|⟨ ⟩| ⟨ | | | | ⟩


‖ ‖ , we obtain

‖| | | | ‖

as required.

53



(

)‖| | | | ‖

‖| | | | ‖

Proof:

As in the proof of the first inequality in (2.1.4), we have

‖ ‖

Thus,

‖ ‖

‖| | ‖

and so

‖| | ‖ ‖| | ‖

‖| | | | ‖

Hence,

(

)‖| | | | ‖


To prove the second inequality in (2.2.4), let be any unit vector in Then

|⟨ ⟩| ⟨ ⟩ ⟨ ⟩

implies

54

|⟨ ⟩| ⟨ ⟩ ⟨ ⟩

⟨ ⟩ ⟨ ⟩

(

) |⟨ ⟩| |⟨ ⟩|

( [ )

⟨| | ⟩ ⟨| | ⟩ ( )

⟨| | ⟩ ⟨| | ⟩ ( )

⟨ | | | | ⟩

Thus,

|⟨ ⟩|

⟨ | | | | ⟩


‖ ‖ , we obtain

‖| | | | ‖

which proves the second inequality in (2.2.4), and completes the proof of the theorem.

Kittaneh, Moslehian and Yamazaki (2015) proved useful theorem concerning the

Cartesian decomposition and gave a new identity of the numerical radius of matrices in

as in the following theorem.

55

Theorem 2.2.4.(Kittaneh, Moslehian and Yamazaki, 2015):

Let be the Cartesian decomposition of Then for

‖ ‖

In particular,

‖ ‖

‖ ‖

Proof:

Since ( ⟨ ⟩) |⟨ ⟩| then

‖ ( )‖

( ( ))

On the other hand, let be the Cartesian decomposition of Then

( )

(( ) ( ) )

( )

( )

( ) ( )

Therefore, by putting in , we obtain (2.2.5).

Especially, by setting and we reach to inequalities

(2.2.6).

56

2.3. Spectral Radius Inequalities:

It is well-known that if then the spectral radius of defined as

{| | }

By using the inequality (2.1.23), Abu-Omar and Kittaneh (2013) proved a general

spectral radius inequality which improves the inequality (2.1.17).


Let . Then

( )

√( )

‖ ‖‖ ‖

Proof:

By using basic properties of the spectral radius, we have

(*

+)

(*

+ [

])

([

] *

+)

([

])

([

])

57

by the inequality (2.1.23), we have

( )

√( )

‖ ‖ ‖ ‖

The desired inequality follows by replacing and by and

respectively, in

the last inequality and then taking the infimum over

Corollary 2.3.1:

Let Then

( )

√( )

‖ ‖ ‖ ‖

Proof:

Letting , and in Theorem (2.3.1), we have

( )

√( )

‖ ‖

Similarly, letting , and in Theorem (2.3.1), we have

( )

√( )

‖ ‖

The desired inequality now follows from the inequalities (2.3.1) and ( ).

Corollary 2.3.2:

Let Then

( )

√( )

‖ ‖‖ ‖

58

Proof:

The desired inequality follows from Theorem (2.3.1) by letting ,

and .

Corollary 2.3.3:

Let Then

√ ‖ ‖‖ ‖ ‖ ‖‖ ‖

and

√ ‖ ‖‖ ‖ ‖ ‖‖ ‖

Proof:

Letting , , and in Theorem (2.3.1), we have

√‖ ‖‖ ‖

Similarly, letting , , and in Theorem (2.3.1), we have

√‖ ‖‖ ‖

Now, the inequality (2.3.3) follows from the inequalities (2.3.5) and (2.3.6). The

inequality (2.3.4) follows from inequality (2.3.3) by symmetry.

Corollary 2.3.4:

Let Then

( )

√( )

‖ ‖‖ ‖ ‖ ‖‖ ‖

59

Proof:

Letting

,

, and in Theorem (2.3.1), we have

( )

√( )

‖ ‖‖ ‖

Similarly, letting , ,

and

in Theorem (2.3.1), we

have

( )

√( )

‖ ‖‖ ‖

The desired inequality follows from the inequalities (2.3.7) and (2.3.8).

Corollary 2.3.5:

Let Then

( )

√ ‖ ‖‖ ‖ ‖ ‖‖ ‖

‖ ‖

for every positive integer

Proof:

The desired inequality follows by letting ,where is a positive integer, in

Corollary (2.3.4).

60

Chapter Three

Two by Two Block Matrix Inequalities

Block matrices arise naturally in many aspects of matrix theory. If *

+,

where is block matrix (or partitioned matrix), then it is very useful to

explore the relations between various functions of the matrix and those of its block

entries and The matrix *

+ is called the diagonal part of *

+ and

*

+ is the off-diagonal part.

In this chapter we present general inequalities for block matrices. In section (3.1), we

present several bounds of numerical radius for the general block matrix.

In section (3.2), we present several bounds of numerical radius for the off-diagonal part

of block matrix. In section (3.3), we present and investigate inequalities for

special case of block matrices.

3.1. Numerical Radius Inequalities for General Block Matrices:

It is well-known ( ) that

(*

+) (*

+)

(*

+) (*

+)

and

(*

+) ( )

where

61

For with it is known that

‖ ‖

Since *

+

*

+

*

+ the subadditivity of the numerical radius

and the inequalities , together with the identities and ,

imply that

(*

+) ( ) ‖ ‖ ‖ ‖

and

(*

+) ( ( ) (*

+))

In particular, if then

(*

+) ‖ ‖ ‖ ‖

Hirzallah, Kittaneh and Shebrawi gave other upper bounds for the numerical

radius of the general block matrix *

+ as we will see.

Theorem 3.1.1.(Hirzallah, Kittaneh and Shebrawi, 2012):

Let *

+ be a matrix with Then

62

where

√‖ ‖ ‖ ‖

√

‖ ‖ ‖ ‖

√‖ ‖ ‖ ‖

√

‖ ‖ ‖ ‖

To prove Theorem (3.1.1), we need the following lemma.

Lemma 3.1.1:

Let Then

‖ ‖ ‖ ‖ ‖| | | | ‖ |‖ ‖ ‖ ‖ |

Proof:

We have

‖ ‖ ‖ ‖

‖ ‖ ‖ ‖ | ‖ ‖ ‖ ‖ |

‖ | | ‖ ‖ | | ‖ | ‖ ‖ ‖ ‖ |

‖ | | | | ‖ | ‖ ‖ ‖ ‖ |

‖ | | | | ‖ | ‖ ‖ ‖ ‖ |

Thus,

‖ ‖ ‖ ‖ ‖ | | | | ‖ | ‖ ‖ ‖ ‖ |

63

Replacing and by and in (3.1.8), respectively, we obtain

‖ ‖ ‖ ‖ ‖ | | | | ‖ | ‖ ‖ ‖ ‖ |

and so

‖ ‖ ‖ ‖ ‖ | | | | ‖ | ‖ ‖ ‖ ‖ |

as required.


We have

(*

+) ‖*

+‖

‖*

+ *

+‖

‖*| | | |

+‖

‖ | | | | ‖

and so

(*

+) ‖| | | | ‖

√ ‖ ‖ ‖ ‖ | ‖ ‖ ‖ ‖ |

( )

√‖ ‖ ‖ ‖

64

By taking a unitary matrix *

+ we obtain

(*

+) (*

+) (*

+)

(*

+) ( *

+ )

√‖ ‖ ‖ ‖

√

‖ ‖ ‖ ‖

Similarly,

(*

+) √‖ ‖ ‖ ‖

√

‖ ‖ ‖ ‖

√‖ ‖ ‖ ‖

√

‖ ‖ ‖ ‖

By observing that (*

+) (*

+) we have

(*

+)

as required.


Let *

+ be a matrix with Then

65

where

√ ‖ ‖ ‖ ‖

√ ‖ ‖ ‖ ‖

√ ‖ ‖ ‖ ‖

√ ‖ ‖ ‖ ‖

Proof:

We have

‖ ‖

( )

( )

‖ ‖

‖ ‖

‖ ‖

‖ ‖ ( )

and so

‖ ‖ ‖ ‖ ‖ ‖

By using the inequality , we have

(*

+) √ ‖ ‖ ‖ ‖

66

The proof of the general case can be obtained by an argument similar to that used in the

proof of Theorem (3.1.1).

Abu-Omar and Kittaneh, 2015,(b) improved and refined the inequalities (2.1.20) and

(2.1.22), respectively, as follows:


Let [ ] be a matrix with Then

([ ])

where

,

( )

([

])

To prove Theorem Abu-Omar and Kittaneh used the following lemma.

Lemma 3.1.2:

Let Then

(*

+)

‖ ‖

Proof:

By Theorem we have

(*

+)

‖ *

+ *

+

‖

67

and so

(*

+)

‖[

]‖

‖*

( )

+‖

‖ ‖


For any we have

‖ ( )‖

‖

‖

[ ( )

(

)

(

) ( )

(

)

(

)

(

)

(

) ( ) ]

‖

‖

‖

‖

[ ‖ ( )‖

‖(

)‖

‖(

)‖ ‖ ( )‖

‖(

)‖

‖(

)‖

‖(

)‖

‖(

)‖ ‖ ( )‖ ]

‖

‖

( )

‖[ ]‖

(by Lemma (3.1.2) and by the norm monotonicity of matrices with nonnegative entries)

68

Now, since the matrix [ ] is real symmetric, then we have

‖[ ]‖ ([ ])

Thus,

‖ ( )‖ ([ ])

as required.

Remark:

The inequality is sharper than the inequality . To see this, note that

[ ] is real symmetric, and so

([ ]) ‖[ ]‖

By the inequality and by the norm monotonicity of matrices with nonnegative

entries, we have

‖[ ]‖

‖[ ]‖

([ ])

([ ])

Corollary 3.1.1:

Let Then

(*

+)

( )

√( )

(*

+)

69

Proof:

By Theorem

(*

+) (* (*

+)

(*

+) +)

(* (*

+)

(*

+) +)

Since the matrix * (*

+)

(*

+) + is real symmetric, it follows that

(* (*

+)

(*

+) +) (*

(*

+)

(*

+) +)

( )

√( )

(*

+)

as required.

70

3.2. Inequalities for the off-Diagonal Part of 2×2 Block Matrices:

Recall that defines a vector norm on and

for any matrix and any unitary matrix

By applying the identity to the matrix *

+ and the unitary matrix

* ⁄

+ we get

*

+ [ ⁄ ⁄

]

( ⁄ [ ⁄ ⁄

]) ( | ⁄ | )

and so

(*

+) (*

+)

Also, by applying the identity to the matrix *

+ and the unitary matrix

√ *

+, we get

(*

+) ( )


Let Then

(*

+) ( )

71

and

(*

+)

Proof:

To prove the inequality (3.2.4), we have

(*

+)

(*

+ *

+)

(*

+) (*

+)

(*

+)

and so

(*

+)

By replacing by – in the inequality and then by using the inequality (3.2.2),

we have

(*

+)

(*

+)

Now, the inequality follows from the inequalities and

To prove the inequality consider the unitary matrix

√ *

+,

72

then

(*

+) ( *

+ )

([

])

([

] [

])

( ([

]) ([

]))

which proves the second inequality in and completes the proof of the theorem.

Corollary 3.2.1:

Let with the Cartesian decomposition Then

(*

+)

for any

Proof:

By replacing by in Theorem , we get

( )

(*

+)

73

Thus,

( )

(*

+)

( )

Since the result follows from the inequalities

Note that, for any we have

*

+

*

+

So by the identity we have

(*

+)

‖*

+‖

Take a unitary matrix

√ *

+ Then

‖*

+‖ ‖ *

+ ‖ ‖ ‖

and so

(*

+) ‖ ‖

In the following theorem Hirzallah, Kittaneh and Shebrawi gave upper

bound for the numerical radius of *

+ that involves and .

74


Let Then

(*

+) ( ) ‖ ‖ ‖ ‖

Proof:

Consider the unitary matrix

√ *

+ By the inequality (3.2.2) and the

identity (3.2.10), we have

(*

+) ( *

+ )

([

])

([

] *

+)

( ([

]) (*

+))

‖ ‖

Thus,

(*

+) ‖ ‖

By replacing by – in the inequality and by using the inequality we

have

(*

+) ‖ ‖

From the inequalities and , we have

75

(*

+) ‖ ‖ ‖ ‖

By interchanging and in the inequality we get

(*

+) ‖ ‖ ‖ ‖

Thus, the desired inequality follows from the inequalities and

Theorem 3.2.3:

Let Then

‖ ‖ (*

+) ‖ ‖ ‖ ‖

Theorem (3.2.3) was proved by Kittaneh, Moslehian and Yamazaki (2015). Now we

present our proof.

Proof :

For any we have

‖ ‖ ‖ ‖

(*

+) ( )

([

]) ( | | )

‖ ( ) ( ) ‖ ( )

‖ ( ) ( )

‖

‖ ‖

Replacing by we get

76

‖ ‖

(*

+) ‖ ‖ ‖ ‖

as required.

Abu-Omar and Kittaneh, 2015,(b) extended Theorem as follows:

Theorem 3.2.4.(Abu-Omar and Kittaneh 2015):

Let *

+ Then

√‖| | | | ‖

√‖| | | | ‖

Proof:

Let be a unit vector and let be a real number such that

⟨ ⟩ |⟨ ⟩|

Then we have

‖ ‖ ( )

‖( )

( )‖

√‖| | | | ‖

√|⟨(| | | | ) ⟩|

√|⟨ | | | | ⟩ ⟨ ⟩|

77

and so

√|⟨ | | | | ⟩ ⟨ ⟩ |

√|⟨ | | | | ⟩ |⟨ ⟩||

√|⟨ | | | | ⟩ |

Thus,

‖ ‖

√⟨ | | | | ⟩

√‖| | | | ‖

which proves the first inequality in .

To prove the second inequality in , we have

‖ ‖ ( )

‖( ) ( )‖

‖| | | | ( )‖

√ ‖| | | | ‖

‖ ‖

√‖| | | | ‖

which proves the second inequality in and completes the proof of the theorem.

78

Remark:

Note that

√‖ ‖

√‖ ‖ ‖ ‖ ‖ ‖

√‖ ‖ ‖ ‖ ‖ ‖‖ ‖

√ ‖ ‖ ‖ ‖

‖ ‖ ‖ ‖

Thus, the second inequality in is sharper than the inequality

In the following theorem, Abu-Omar and Kittaneh, 2015,(b) gave another upper

estimate for the numerical radius of the matrix *

+


Let *

+ Then

‖| | | |‖

‖| | | |‖

Proof:

Let *

+

be a unit vector. Then

|⟨ ⟩| |⟨ ⟩ ⟨ ⟩| |⟨ ⟩| |⟨ ⟩|

79

By Lemma (2.1.1), we have

|⟨ ⟩| |⟨ ⟩|

⟨| | ⟩ ⟨| | ⟩

⟨| | ⟩

⟨| | ⟩

⟨| | ⟩ ⟨| | ⟩ ⟨| | ⟩ ⟨| | ⟩

(by Cauchy-Schwarz inequality)

⟨ | | | | ⟩ ⟨ | | | | ⟩

‖| | | |‖ ‖| | | |‖

‖ ‖ ‖ ‖

‖| | | |‖ ‖| | | |‖

‖ ‖

‖ ‖

(by the arithmetic-geometric mean inequality)

‖| | | |‖

‖| | | |‖

and so

‖ ‖

|⟨ ⟩|

‖| | | |‖

‖| | | |‖

as required.

80

3.3 On Unitarily Invariant Norm Inequalities and Hermitian Block Matrices:

The well-known arithmetic geometric mean inequality for singular values due to

Bahatia and Kittaneh (1990) says that

for any

On the other hand, Zhan (2000) has proved

Zhan (2002) has proved that the two inequalities (3.3.1) and (3.3.2) are equivalent. Tao

(2006) gave an equivalent form of the two inequalities which is in the following

theorem.

Theorem 3.3.1.(Tao, 2006):

Let such that *

+ Then

*

+

for .

To prove Theorem Tao used the following lemma which can be found in

Bhatia (1997).

Lemma 3.3.1:

The Hermitian matrix *

+ where with rank has eigenvalues

81


Consider the unitary matrix *

+ Then

*

+ *

+ *

+

*

+

*

+ *

+

Thus,

*

+ *

+

By Weyl's monotonicity principle, we have

*

+ *

+

By using Lemma we get

*

+

as required.

Theorem 3.3.2.(Tao, 2006):

The following statements are equivalent:

(i) Let be positive semidefinite matrices. Then

82

(ii) For any

(iii) Let such that *

+ Then

*

+

Proof:

(i)(ii) Zhan (2002) proved this part as follows:

Let *

+ *

+ Then is unitary, and so

* + *

+

[ ]

*

+

[

]

Thus, we have

83

(ii)(iii) Since *

+ then there exist such that

*

+ [ ] [ ]

Now, from (ii) and since [ ] [ ] *

+, we have

[ ][ ]

[ ] [ ]

*

+

*

+

(iii)(i) Let be positive semidefinite matrices and let

√ *

+ be a

unitary matrix. Then *

+ and so

*

+ ( *

+ )

*

+

From (iii), we have

*

+

84

Corollary 3.3.1:

Let such that is Hermitian, and Then

( )

Proof:

Let *

+ and

√ *

+ be a unitary matrix.

Then

*

+

Since so is positive semidefinite and by Theorem (3.3.1) we have

*

+

( )

as required.

Bhatia and Kittaneh obtained that if such that is Hermitian,

and then

Recently, Audeh and Kittaneh employed the previous inequality as in the

following theorem.

85

Theorem 3.3.3.(Audeh and Kittaneh, 2012):

Let such that *

+ Then

Proof:

Consider the unitary matrix *

+

then

*

+ *

+ *

+ *

+ *

+ *

+

Thus,

*

+ *

+

By applying the inequality we get

( )

and so

as required.

Audeh and Kittaneh proved that the inequalities and are

equivalent as follows:

86

Theorem 3.3.4.(Audeh and Kittaneh, 2012):

The following statements are equivalent:

(i) Let where is Hermitian, and Then

for

(ii) Let such that *

+ Then

for

Proof:

(i)(ii) This follows from the proof of Theorem

(ii)(i) Let where is Hermitian, and Since *

+ is

unitarily equivalent to *

+ (by the identity (3.3.4), then *

+

Thus, by (ii) we have

for

By Theorem (3.3.1), we have

‖ ‖

‖*

+‖

for all such that *

+

87

The following theorem gives another upper bound for ‖ ‖

Theorem 3.3.5:

Let such that *

+ Then

‖ ‖ ‖ ‖ ‖ ‖

To prove Theorem we need the following lemma which can be found in

Zhang (1999).

Lemma 3.3.2:


*

+

for some contraction ‖ ‖


By Lemma (3.3.2), we have

And so

‖ ‖ ‖ ‖ ‖ ‖‖

‖

‖ ‖ ‖ ‖

as required.

88

The following theorem gives another upper bound for ‖ ‖ in case and are

commute.

Theorem 3.3.6:

Let such that *

+ If then

‖ ‖ ‖ ‖

To prove Theorem (3.3.6) we need the following lemma which can be found in

Zhang

Lemma 3.3.3:

Let such that *

+ If then

Our proof of Theorem (3.3.6):

We have

‖ ‖ ‖ ‖ ‖

‖

‖ ‖ ( )

‖ ‖ ‖ ‖

‖ ‖ ( )

as required.

89

It follows from the inequality that if such that is Hermitian,

and then

‖ ‖ ‖ ‖


Theorem 3.3.7:

Let such that *

+ . Then

‖ ‖

‖ ‖


Proof:

Since *

+ then

*

+ *

+ *

+ *

+

By using the fact that a matrix is positive semidefinite if and only if the matrix

*

+ is positive semidefinite, we get

Similarly, since

*

+ *

+ *

+ *

+

then

90

Thus,

So the desired inequality follows from the inequality

Corollary 3.3.2:

Let such that *

+ If is Hermitian, then

‖ ‖

‖ ‖


Now, we present a remarkable decomposition lemma noticed in Bourin and Lee

(2012).

Lemma 3.3.4:

Let such that *

+ Then

*

+ *

+ *

+

for some unitary matrices

Proof:

Since *

+ then we can write it as a square of positive semidefinite matrix,

say,

91

*

+ *

+ *

+

where

Let *

+ and *

+ Then

*

+ *

+ *

+ *

+

Since *

+ *

+ and by using the fact that and are unitary

equivalent to and respectively, we get the desired inequality.

This decomposition turned out to be an efficient tool and it also plays a major role

below.

Theorem 3.3.8.(Bourin, Lee and Lin, 2012):

Let such that *

+ Then

*

+ [

] [

]

and

*

+ [

] [

]

for some unitary matrices

Proof:

It is easy to see that *

+ and *

+ are unitary equivalent.

92

In fact,

*

+ *

+ *

+ *

+

Now, if we take the unitary matrix

√ *

+ then we observe that

*

+ [

]

and

*

+ [

]

where stands for unspecified entries. Now the desired inequalities follow from Lemma

(3.3.4).

The following lemma can be found in Bourin, Lee and Lin (2012).

Lemma 3.3.5:

Let such that *


‖*

+‖ ‖ ‖


93

In the following theorem, we give an upper and lower bounds for the spectral norm

of positive semidefinite block matrices.

Theorem 3.3.9:

Let such that *


‖ ‖ ‖*

+‖ ‖ ‖

Proof:

Since *

+ , then by the inequality , we have

‖ ‖ ‖*

+‖

and by Lemma we have

‖*

+‖ ‖ ‖

The desired inequality follows from the inequalities and

Our next theorem gives an upper bound for the spectral norm of the off-diagonal part

of positive semidefinite block matrices

Theorem 3.3.10:

Let such that *

+ Then

‖ ‖ (‖ ‖

‖ ‖

)

94

Proof:

Since *

+ then

*

+ *

+ *

+ *

+

and

*

+ *

+ *

+ * +

By using Corollary and by the inequalities and , we deduce

‖ ‖ ‖ ‖ ‖ ‖

and

‖ ‖ ‖ ‖ ‖ ‖

respectively. So the desired inequality follows from the inequalities and

At the end of this section, we give an estimate for the numerical radius of the off-

diagonal part of positive semidefinite block matrices. In fact, our result improve

the inequality (3.3.8) for the spectral norm.

Theorem 3.3.11:

Let such that *

+ Then

‖ ‖

95

Proof:

Since *

+ it follows that [

] In fact, if

we take *

+ then is unitary and

[

] *

+

Thus, by the inequality we have

‖ ( )‖

‖ ‖

and so by Theorem (1.4.9), we have

‖ ( )‖

‖ ‖

as required.

96

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100

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المشرف

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الممـخــــص

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Spectral Radius, Numerical Radius and Unitarily Invariant