Some inequalities for operators

MINISTRY OF EDUCATION AND TRAINING HANOI PEDAGOGICAL UNIVERSITY 2——————–o0o——————— NGUYEN THI TUYET SOME INEQUALITIES FOR OPERATORS GRADUATION THESIS HA NOI, 2019... HANOI PEDAGOGICAL UN

MINISTRY OF EDUCATION AND TRAINING HANOI PEDAGOGICAL UNIVERSITY 2

——————–o0o———————

NGUYEN THI TUYET

SOME INEQUALITIES FOR OPERATORS

GRADUATION THESIS

HA NOI, 2019

HANOI PEDAGOGICAL UNIVERSITY 2

——————–o0o———————

NGUYEN THI TUYET

SOME INEQUALITIES FOR OPERATORS

GRADUATION THESIS

Major: Analysis Supervisor: HO MINH TOAN

HA NOI, 2019

Thesis Assurance

I assure that the data and the results of this thesis are true and not indentical to other topics I also assure that all the help for this thesis has been acknowledged and that the results presented in the thesis has been indentified clearly.

Ha Noi, May 5, 2019

Student

Nguyen Thi Tuyet

Thesis Acknowledgement

This thesis is conducted at the Department of Mathematics, HANOI GOGICAL UNIVERSITY 2 The lecturers have imparted valuable knowledge and facilitated for me to complete the course and the thesis.

PEDA-I would like to express my deep respect and gratitude to PhD Ho Minh Toan, who has direct guidance, help me to complete this thesis.

Due to time, capacity and conditions are limited, so the thesis can not avoid errors So I am looking forward to receiving valuable comments from teachers and friends.

Ha Noi, May 5, 2019

Student

Nguyen Thi Tuyet

3 A REVERSE CAUCHY INEQUALITY FOR

3.1 Operator monotone and operator convex funtions 273.2 Main results 313.3 Characterizations of the trace property 36

References 40

Bachelor thesis NGUYEN THI TUYET

NOTATION

AGM : Arithmetic-Geometric mean inequality

|||T ||| : Unitarily invariant norm of T

Φ: symmetric gauge function

Mn : square matrices n × n on the complex field

Mn+ : positive semidefinite matrices n × n matrices

A∗ : Conjugate operator of A

Diag(α1, , αn): Diagonal matrix with th element α1, , αn if ly on the

diagonal

I: identity matrix

Re(A) : the real part of complex A

Im(A) : The image part of complex A

λj(T ): the set of eigenvalues T

Inequalities is an important topic in mathematics and have variousapplications In particular, operator inequalities of Cauchy - type hasattracted much attention Applications of that inequalities in diversefields of mathematics have contributed to ones importance After itsdiscovery, the numerous authors studied, who either reproved it usingvarious techniques, or applying and generalizing it by many differentways Given the need for practicality, especially for students, I choosethe topic ”Some inequalities for operator ” to provide a relatively com-plete basis for the basic theory of some inequalities moreover, the basicknowledge of some operator on the matrixs

• T (u + v) = T u + T v

• T (cu) = cT u

Let u and v be vectors in a vector space V and let c be any scalar Aninner product on V is a map that associates a real number hu, vi witheach pair of vectors u and v and satisfies the following axioms

1 hu, vi = hu, vi

2 hu, v + wi = hu, vi + hu, wi

3 hcu, vi = chu, vi

4 hu, ui ≥ 0 and hu, ui = 0 if and only if u = 0

A vector space V with an inner product is called an inner productspace A linear map from V to itself is also called a linear operator or

simply, an operator In this thesis, we study some (Cauchy, Cauchy-type)inequalities of operator on V, where dimension of V is finite The fact

is that an n-dimensional vector space is isomorphic to kn Therefore,

we can assume that V = kn In this case, any linear operator on V

is continuous (hence, it is bouned) Let us denote by L(V) the set ofall linear operator on V Then L(V) is a ring the usual addition andcomposition If T is an operator on V, the adjoint of T , denoted by T∗,

where (tij) is an n×n-matrix which is the (standard) matrix of T relative

to the basis E and Mn is the ring of squares matrices of order n withcoefficients in k The map Θ is a ring isomorphism and Θ(T∗) = (tji).Hence, instead of study the inequalities of operators in L(V), we studythat in Mn

Matrix Operations

Let A = [aij] and B = [bij] be matrices of size m × n, and let C = [cij]

Bachelor thesis NGUYEN THI TUYET

An a n × n matrix A is invertible ( or nonsingular) if there exist an

n × n matrix B such that AB = BA = In where In is the identitymatrix of order n The matrix B is called the ( multiplicative) inverse

of A A matrix that does not have an inverse is called non-invertible (orsingular)

If A is an invertible matrix, then its inverse is unique The inverse of

A is denoted by A−1

Proposition 1.0.2 Let A, B be square matrices of order n

a If A is an invertible matrix, m is a positive integer, and c is ascalar not equal to zero , then A−1, Am, cA, AT are invertible andthe following are true

Definition 1.0.3 An n × n matrix A is diagonalizable if A is similar to

a diagonal matrix that is, A is diagonalizable if there exists an invertiblematrix P such that P−1AP is a diagonal matrix

A diagonal matrix is a square matrix whose off-diagonal entries areequal to zero Hence, a diagonal matrix is at the same time :uppertriangular or lower triangular The matrix D is a diagonal matrix if andonly if Dij = 0 when i 6= j

Example 1.0.4 The 3 × 3 matrix D =

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0200

0040

000

Definition 1.0.6 (The scalar product)

The scalar product is also defined for column matrices

Let a = (a1, a2)T b = (b1, b2)T Then,ha, bi Multiply correspondingelements of each column matrix, then add up the products The result

is a scalar value

Definition 1.0.7 (The matrix Norms)

We consider matrix norms on (Cm,n, C) All results holds for (Rm,n, R)

A function k.k : Cm,n −→ C is called a matrix norm on Cm,n if for all

Suppose k, k1, , km ∈ R and A, A1, , Am are each n × n matrices.Then

(1) T r(A) = T r(A0)

Bachelor thesis NGUYEN THI TUYET

0 < T r(AB)m ≤ T r(AB))m f or all m ∈ N∗

Proof The equality takes place for n = 1 If n > 1, for B = I theinequality is true because 0 < T r(An) ≤ (T rA)n, become

where λ1, λ2, , λn are the eigenvalues of A

If A 7−→ AB, the result has been proved

Chapter 2

SOME CAUCHY INEQUALITIES

OF LINEAR OPERATORS

For positive semi-definite n × n matrices, the inequality 4|||AB||| 6

|||(A + B)2||| is shown to hold for every unitarily invariant norm Theconnection of this with some other matrix arithmetic-geometry meaninequalities and trace inequalities is discussed

2.1 Introduction to Cauchy inequalities

Some matrix versions of the classical arithmetic-geometric mean ity (AGM) were proved in [3-5], and seem to have aroused considerableintarsect See [2, Chapter IX;6] for a discussion and further references

inequal-In this note we prove one more inequality of this type, discuss itsconnection with the known results, and with some others that seemplausible but are yet unproved

For position real numbers a, b the AGM says that

√

ab 6 a + b

Replacing a, b by their squares, we could write this in the form

ma-in general, the matrix AB is not positive One way to get around this

is to compare not the matrices themselves but their singular values andnorms The second difficulty (that makes the problem more interesting)

is that the matrix square root and square functions have different tonicity properties, Thus each of the inequalities (2.1) - (2.3) leads todifferent matrix versions

mono-We label the singular values of an n × n matrix T as s1(T ) > >

sn(T ) If T has real eigenvalues, we label them as λ1(T ) > λn(T ) If T

is positive, we have sj(T ) = λj(T ) We use the notation |||T ||| to denoteany unitarily invariant norm of T A statement like sj(S) = sj(T ) willused to indicate that this inequality is true for all 1 ≤ j ≤ n Thisimplies the weakly majorisation sj(S) ≺w sj(T ), by which we mean thatthe sequence {sj(S)} is weakly majorised by {sj(T )} This is equivalent

to saying that |||S||| ≤ |||T |||, by which we mean that any unitarilyinvariant norm of S is dominated by the corresponding norm of T See[2] for details We use the symbol |T | for the operator absolute value(T ∗ T )1/2

Bachelor thesis NGUYEN THI TUYET

In [4] we proved that, if A, B are positive, then

sj(AB) 6 sj

 A2 + B22

|||AXB||| 6 1

2|||A2X + XB2|||, (2.6)and it was noted that a corresponding generalisation of (2.4) fails tohold Another proof of (2.6) was given [5]

If instead of (2.2) we were to start with (2.1) of (2.3) as the scalarAGM, we are led to the following question If A, B are positive matrices,then which of the following inequalities are true:

The square function on Hermitian matrices is matrix convex [2], i.e.,

 A + B2

2

6 A

2 + B2

2 .Hence, the statement (2.7) is stronger than (2.4)

Our main result is the following

Theorem 2.1.1 The inequality (2.9) is true for all positive matricesA,B

This is proved in Section 2 We have remarked that this says thatthe inequality (2.8) is true for all Q-norms (and hence for all Schattenp-norms for p > 2) We will see that (2.8) is also true for the trace norm(which is not a Q-norms) This leads us to conjecture that this is truefor all unitarily invariant norms.We will observe also that when n = 2,the inequality (2.7) is true Again, this leads us to believe that it might

be true in all dimensions

2.2 Unitarily Invariant Norms on Operators

In this section, we will sudy the norms of operators on the Hilbert space

Cn with the usual inner product h., i and the associated norm k.k If A is

a linear operator on Cn, we will denote by kAk the operator ( bound)

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as s1(A) > s2(A) > > sn(A) We have

kAk = k|A|k = s1(A) (2.11)

Now, if U, V are unitary operators on Cn, then |U AV | = V∗|A|V andhence

kAk = kU AV k (2.12)

for all unitary operators U, V This last property is call unitary ance Several other norms have this property These are frequentlyuseful in analysis, and we will study them in some detail We will usethe symbol k|.|k to mean a norm on n × n matrices that satisfies

invari-k|U AV |k = k|A|k (2.13)

for all A and for unitary U, V We will call such a norm a unitarilyinvariant norm on the space Mn of n × n matrices We will normalisesuch norms so that they all take the value 1 on the matrix diag(1, 0, , 0).There is an intimate connection between these norms and symmetricgauge functions on Rn; the link is provided by singular values

Theorem 2.2.1 Given a symmetic gauge function Φ on Rn, define a

in-s(A + B) ≺w s(A) + s(B) f or all A, B ∈ Mn

and then use the fact that Φ is strongly isotone and monotone (SeeExample II.3.13 and Problem II.5.11-[2] ) To prove th converse, notethat (2.15) clearly gives a norm on Rn Since diagonal matrices of theform diag(eiθ1, , eiθn) and permutation matrices are all unitary, thisnorm is absolute and permutation invariant, and hence it is a symmetricgauge function

Symmetric gauge functions on Rn constructed in the proceding sectionthus lead to several examples of unitarily invariant norms on Mn Twoclasses of such norms are specially important The first is the class of

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Schatten p-norms defined as

is also called the Hilbert-Schmidt norm or the Frobenius norm (and

is sometimes written as kAkF) for that reason) It will play a basic role

in our analysis For A, B ∈ Mn let

hA, Bi = T rA∗B (2.20)

This defines an linear product on Mn and the norm associated with thisinner product is kAk2,i.e.,

kAk2 = (T rA∗A)1/2 (2.21)

If the matrix A has entries aij, then

Thus the norm kAk2 is the Euclidean norm of the matrix A when it

is thought of as an element of Cn2 This fact makes this norm easilycomputable and geometrically tractable

The main importance of the Ky Fan norms lies in the following:Theorem 2.2.2 (Fan Dominance Theorem) Let A,B be two n × n ma-trices If

kAk(k) 6 kBk(k) f or k = 1, 2, , nthen

|||A||| 6 |||B||| for all unitarily invariant norms

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k

X

j=1

sj − ksk = kAk(k) − ksk,kCk = sk,

Proof If v is a symmetric norm, then for unitary U, V we have

v(U AV ) 6 v(A) and v(A) = v(U−1U AV V−1) 6 v(UAV ) So, v is tarily invariant.Conversely, by Problem III.6.2-[2], sj(BAC) 6 kBkkCksj(A)for all j = 1, 2, , n So, if Φ is any symmetric gauge function, thenΦ(s(BAC)) 6 kBkkCkΦ(s(A)) and hence the norm associated with Φ

uni-is symmetric

In particular, this implies that every unitarily invariant norm is

|||AB||| 6 |||A||| |||B||| for all A, B

Theorem 2.2.5 If A, B are n × n matrices, then

sr(AB) ≺w sr(A)sr(B) f or all r > 0 (2.25)

Corollary 2.2.6 (Holder’s Inequality for Unitarily Invariant Norms)For every unitarily invariant norm and for all A, B ∈ Mn

|||AB||| ≤ ||| |A|p|||1/p||| |A|q|||1/q (2.26)

for all p > 1 and 1

||| |AB|r|||1/r ≤ ||| |A|p|||1/p ||| |A|q|||1/q (2.27)Choosing p = q = 1, one gets from this

||| |AB|1/2||| ≤ (|||A||| |||B|||)1/2 (2.28)This is the Cauchy-Schwarz inequality for unitarily invariant norms

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Example 2.2.8 Given a unitarily invariant norm |||.||| on Mn, define

|||A|||(p) = ||| |A|p|||1/p 1 ≤ p < ∞ (2.29)Show that this is a unitarily invariant norm Note that

kAk(p2 )

p1 = kAkp1p2 f or all p1, p2 ≥ 1 (2.30)and

2.3 Proof of Cauchy inequality for operators

Proof We give a proof of (2.9) for the case of the Hilbert-Schmidt nius) norm k.k2 first As is often the case, this is simpler For any matrix

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Now note that

Re A2 + B2 ± 2AB
= (A ± B)2,

Im A2 + B2 ± 2AB
= ±1

i (AB − BA) Here we have used the notations ReT and Im T for the matrices (T +

T∗)/2 and (T − T∗/2i), respectively Since kT k22 = kReT k22 + kImT k22,

we obtain from (2.35)

(A + B)2

2 2

≥ (A − B)2

2 2

+ 16 kABk22 (2.37)This shows that

4kABk2 ≤ (A + B)2

2

and there is equality here if and only if A = B

Now for the proof of Theorem (2.1.1) in full generality Using (2.6)

Bachelor thesis NGUYEN THI TUYET

a well-known result [2, Theorem IV.2.5] we have the weak majorisation

s1/2j (AB) ≺w s1/2j (A)s1/2j (B) (2.42)

By the AGM (2.1), the quantity on the right-hand side is bounded

by 1/2(sj(A) + sj(B)) Hence, in particular

T r |AB|1/2 6 1

2T r(A + B). (2.43)

This is just the inequality (2.8) for the trace norm We observed thisalready in [4].

Note that in the case of the operator norm, inequalities (2.8) and (2.9)both reduce to

A and B are invertible Then

sn(AB) = λ1/2n (BA2B) = λ−1/21 (B−1A−2B−1)

Since

λ−1/21 (B−1A−2B−1) = s1(A−1B−1) ≥ λ1(A−1B−1),this gives

This proves the inequality (2.45)

Thus, when n = 2, the inequality (2.7) is valid for all values of j

Chapter 3

A REVERSE CAUCHY

INEQUALITY FOR OPERATORS

A reverse of the classical Cauchy inequality can be stated as follows Forany positive real numbers a, b, we have

ab + |a − b|

2 ≥ a + b

2 .

In this chapter, we will formulate this inequality for operators (on Cn.)

A tracial version of a reverse Cauchy inequality for operator is namedPowers-Stormer’s inequality (see, for example.[10 , Lemma 2.4.3, Theo-rem 11.19]) For s ∈ [0, 1], the following inequality

2T r(AsB1−s) > T r(A + B − |A − B|) (3.1)

holds for any pair of positive semi-definite matrices A, B

This is a key inequality to prove the upper bound of Chernoff bound,

in quantum testing theory [6] This inequality was first proven in [6],using an integral representation of the function ts After that, Ozawagave a much simpler proof for the same inequality, using fact that for