Spectral Radius, Numerical Radius and Unitarily Invariant Norm Inequalities in Hilbert Space

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 اإلهداء الر حم بس م هللا ن الر حيم إلى من أضاء ل الطر ق وغرس ف نفس حب المعرفة واإل مان أب إلى خ مة الحنان.. وغ مة المكان.. زهرة الح اة ونورها أم إلى شق ق روح.. سندي بعد هللا ورف ق درب.. خطوة بخطوة أخ وسف إلى القلب الطاهر.. مالك ف الح اة وقدوت.. أخت فاطمة وإلى بق ة أفراد عائلت الحب بة.. أهدي هذا الجهد المتواضع.

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 Acknowledgment. Abstract iii iv 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): Let be positive semidefinite matrices. Then * + * + for every unitarily invariant norm. We end this section with a matrix versions of the arithmetic-geometric mean.

20 Theorem 1.3.6.(Bhatia and Kittaneh, 1990): Let be positive semidefinite matrices. Then for every unitarily invariant norm. 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 for every unitarily invariant norm.

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: fact, The numerical radius norm and the matrix norm on are equivalent. In 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): every By Lemma (2.1.1) and by the arithmetic-geometric mean inequality, we have for

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, By taking the maximum on both sides in the above inequality over with 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 which proves the first inequality in (2.1.9). 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 Theorem 2.1.6.(El-Haddad and Kittaneh, 2007): 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 Proof of Theorem (2.1.5): For every unit vector we have ( ) ( ) ( ) ( ) ( )

40 Thus, ( ) taking the maximum on both sides in the above inequality over produces as required. Proof of Theorem (2.1.6): 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: Theorem 2.1.11.(Abu-Omar and Kittaneh, 2015): 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: Theorem 2.1.12.(Abu-Omar and Kittaneh, 2015): 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. Theorem 2.2.1.(El-Haddad and Kittaneh, 2007): 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 By taking the maximum on both sides in the above inequality over with, we obtain For the case we have ( ) ( ) as required.

52 Theorem 2.2.2.(El-Haddad and Kittaneh, 2007): Let with the Cartesian decomposition and let Then Proof: For any unit vector we have ( ) ( ) ( ) ( ) ( ) Thus, By taking the maximum on both sides in the above inequality over with, we obtain as required.

53 Theorem 2.2.3.(El-Haddad and Kittaneh, 2007): Let with the Cartesian decomposition and let Then ( ) Proof: As in the proof of the first inequality in (2.1.4), we have Thus, and so Hence, ( ) which proves the first inequality in (2.2.4). To prove the second inequality in (2.2.4), let be any unit vector in Then implies

54 ( ) ( [ ) ( ) ( ) Thus, By taking the maximum on both sides in the above inequality over with, 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). Theorem 2.3.1.(Abu-Omar and Kittaneh, 2013): 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. Proof of Theorem (3.1.1): We have (* +) * + * + * + * + and so (* +) ( )

64 By taking a unitary matrix * + we obtain (* +) (* +) (* +) (* +) ( * + ) Similarly, (* +) By observing that (* +) (* +) we have (* +) as required. Theorem 3.1.2.(Hirzallah, Kittaneh and Shebrawi, 2012): 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: Theorem 3.1.3.(Abu-Omar and Kittaneh, 2015): 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 (* +) [ ] * ( ) + Proof of Theorem (3.1.3): 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 (* +) ( ) Theorem 3.2.1.(Hirzallah, Kittaneh and Shebrawi, 2011): 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 Theorem 3.2.2.(Hirzallah, Kittaneh and Shebrawi, 2011): Let Then (* +) ( ) Proof: Consider the unitary matrix identity (3.2.10), we have * + By the inequality (3.2.2) and the (* +) ( * + ) ([ ]) ([ ] * +) ( ([ ]) (* +)) 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 * + Theorem 3.2.5.(Abu-Omar and Kittaneh, 2015): 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 Proof of Theorem (3.3.1): 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: Let be positive semidefinite matrices. Then * + for some contraction Proof of Theorem (3.3.5): 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 for every unitarily invariant norm. Theorem 3.3.7: Let such that * +. Then for every unitarily invariant norm. 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 for every unitarily invariant norm. 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: say, Since * + then we can write it as a square of positive semidefinite matrix,

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 * + If is Hermitian, then * + for every unitarily invariant norm.

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 * + If is Hermitian, then * + 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 offdiagonal 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.

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100 القطر نصف الطيفي, القطر نصف متباينات العددي والمعايير الال متغيرة همبرت فضاء في إعداد دعاء محمود السعافين المشرف د. عمياء عبد الجواد برقان المم خص في هذه األطروحة نعرض العديد من متباينات نصف القطر الطيفي, نصف القطر العددي والمعايير الال متغيرة لممصفوفات المربعة. كما نقدم لمتباينات مرتبطة بالمصفوفات المج أزة.