SOME INEQUALITIES FOR OPERTATORS

MINISTRY OF EDUCATION AND TRAINING HANOI PEDAGOGICAL UNIVERSITY ——————–o0o——————— NGUYEN THI TUYET SOME INEQUALITIES FOR OPERTATORS GRADUATION THESIS HA NOI, 2019 MINISTRY OF EDUCATION AND TRAINING HANOI PEDAGOGICAL UNIVERSITY ——————–o0o——————— NGUYEN THI TUYET SOME INEQUALITIES FOR OPERTATORS 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 Bachelor thesis NGUYEN THI TUYET Thesis Acknowledgement This thesis is conducted at the Department of Mathematics, HANOI PEDAGOGICAL UNIVERSITY The lecturers have imparted valuable knowledge a`n facilitated for me to complete the course and the thesis 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 i Contents Thesis Assurance Thesis Acknowledgement i Notation Preface 1 PRELIMINARIES SOME CAUCHY INEQUALITIES OF LINEAR OPERATORS 10 2.1 Introduction to Cauchy inequalities 10 2.2 Unitarily Invariant Norms on Operators 13 Proof of Cauchy inequality for operators 20 2.2.1 A REVERSE CAUCHY INEQUALITY FOR OPERATORS 26 3.1 Operator monotone and operator convex funtions 27 3.2 Main results 31 3.3 Characterizations of the trace property 36 ii Bachelor thesis NGUYEN THI TUYET References 40 iii Bachelor thesis NGUYEN THI TUYET NOTATION Bachelor thesis NGUYEN THI TUYET Preface Chapter PRELIMINARIES Linear Operators Let V, W be vector spaces over a filed k (k, in this thesis, is just the real or complex numbers) A linear map is a map T : V −→ W that satisfies the folowing properties for all u, v in V and for any number c • 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 An inner product on V is a map that associates a real number u, v with each pair of vectors u and v and satisfies the following axioms u, v = u, v u, v + w = u, v + u, w cu, v = c u, v u, u ≥ and u, u = if and only if u = A vector space V with an inner product is called an inner product space A linear map from V to itself is also called a linear operator or Bachelor thesis NGUYEN THI TUYET 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 k n Therefore, we can assume that V = k n In this case, any linear operator on V is continuous (hence, it is bouned) Let us denote by L(V) the set of all linear operator on V Then L(V) is a ring the usual addition and composition If T is an operator on V, the adjoint of T , denoted by T ∗ , is defined by x, T y = T ∗ x, y , ∀x, y ∈ V Fix a standard orthonormal basis E for V There is a natural one-toone and onto correspondence: Θ : L(V) →M(n) T →(tij )ni,j=1 , where (tij ) is an n×n-matrix which is the (standard) matrix of T relative to the basis E and M(n) is the ring of squares matrices of order n with coefficients 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 study that in M⋉ (k) Matrix Operations Let A = [aij ] and B = [bij ] be matrices of size m × n, and let C = [cij ] Bachelor thesis NGUYEN THI TUYET s ∈ [0, 1] function f (t) = ts (t ∈ [0, +∞)) is an operator monotone [7, Proposition 1.1] Recently, Ogata in [8] extended this inequality to standard von Neumann algebras Later, Hoa-Osaka-Ho generalize this inequality when the function f (t) = ts is replaced by any operator monotone function, then T r(A + B − |A − B|) may get smaller upper bound than what is used in quantum hypothesis testing Based on Ozawa’s proof I formulate Powers-Stormer’s inequality for an arbitrary operator monotone function on [0, +∞) Recall that ϕ(sup Ai ) = sup ϕ(Ai ) for every bounded increasing net Ai of positive elements in M For all other notations used in the paper, we refer the reader to the monograph [9] 3.1 Operator monotone and operator convex funtions Let f be a real function defined on an interval I If D = diag(λ1 , λ2 ) is a diagonal matrix whose diagonal entries λj , we define f (D) = diag(f (λ1 ), , f (λn )) If A is a Hermitian matrix whose eigenvalues λj are in I, we choose a unitary U such that A = U DU ∗ , where D is diagonal and then defined f (A) = U f (D)U ∗ In this way we can defined f (A) for all Hermitian matrices (of any order) whose eigenvalues are in I In the rest of this chapter, it will always be assumed that our functions are real functions defined on an interval ( finite of infinite, closed or open) and are extended to Hermitian matrices in this way 27 Bachelor thesis NGUYEN THI TUYET We will use the notation A B to mean A and B are Hermitian matrices B −A is positive The relation is a partial order on Hermitian matrices A function f is said to be matrix monotone of order n if it is monotone with respect to this order on nxn Hermitian matrices, i,e if A B implies f (A) f (B) If f is a matrix monotone of order n for all n we say f is operator monotone A function f is said to be matrix convex of order n if for all nxn Hermitian matrices A and B for all real numbers f ((1 − λ) A + λB) λ (1 − λ) f (A) + λf (B) 1, (3.2) If f is matrix convex of all orders, we say that f is matrix convex or operator convex (Note that if the eigenvalues of A and B are all in an interval I, then the eigenvalues of any convex combination of A, B are also in I This is an easy consequence of result in chappter III.) We will consider continuous function only In this case, the condition (3.2) can be replaced by the more special condition f A+B f (A) + f (B) (3.3) (Function satifying (3.3) are called midpoint operator convex, and if they are continuous, then they are convex.) A function f is call operator concave if the function f is operator convex It is clear that the set of operator monotone functions and the set of 28 Bachelor thesis NGUYEN THI TUYET operator convex functions are both closed under positive linear combinations and also under (pointwise) limits In other words, if f, g are operator monotone, and if α,β are positive real numbers, then αf + βg is also operator monotone If fn are operator monotone, and if fn (x) → f (x), then f is also operator monotone The same is true for operator convex functions Example 3.1.1 The function f (t) = α + βt is operator monotone (on every interval) for every α ∈ R and β It is operator convex for all α, β ∈ R The first surprise is in the following example Example 3.1.2 The function f (t) = t2 on [0, ∞) is not operator monotone In other words, there exist positive matrices A,B such that B − A is positive but B − A2 is not To see this, take  A= Lemma 3.1.3 If B 1 1   ,B =  1   A, then for every operator X we have X ∗ BX X ∗ AX Proof For every vector u we have, u, X ∗ BXu = Xu, BXu Bu, AXu = u, X ∗ AXu (3.4) This prove the lemma An equally brief proof goes as follows let C be the positive square 29 Bachelor thesis NGUYEN THI TUYET root of the positive operator B − A Then X ∗ (B − A) X = X ∗ CCX = (CX)∗ CX Proposition 3.1.4 The functione f (t) = − 1t is operator monotone on (0, ∞) Proof Let B A > Then by Lemma (3.1.3), I B −1/2 AB −1/2 Since the map T → T −1 is order-reversing on commuting positive opB 1/2 A−1 B 1/2 Again, using lemma (3.1.3) we get erators, we have I from this B −1 A−1 Lemma 3.1.5 If B Proof If B A A and B is invertible , then A1/2 B −1/2 then I hence A1/2 B −1/2 ∗ B −1/2 AB −1/2 = A1/2 B −1/2 A1/2 B −1/2 , Proposition 3.1.6 The function f (t) = t1/2 is operator monotone on (0, ∞] Proof Let B A A1/2 B −1/2 Suppose B is invertible Then by Lemma (3.1.5), spr A1/2 B −1/2 = spr B −1/4 A1/2 B −2/4 Since B −1/4 A1/2 B −1/4 is positive, this imples that I Hence, by Lemma (3.1.3), B 1/2 B −1/4 A1/2 B −1/4 A1/2 This proves the proposition un- der the assumption that B is invertible.If B is not stricly positive, then for every ε > 0, B + εI is strictly positive So,(B + εI)1/2 ε → This shows that B 1/2 A1/2 30 A1/2 Let Bachelor thesis NGUYEN THI TUYET Theorem 3.1.7 The function f (t) = tr is operator monotone on [0, ∞) for 3.2 r Main results Let n ∈ N and Mn be the algebra of n × n matrices let I be an interval in R and f : I −→ R be a continuous function We call a function f matrix monotone of order n or n-monotone in short whenever the inequality A B =⇒ f (A) f (B) for an arbitrary self-adjoint matrices A, B ∈ Mn such that A B and all eigenvalues of A and B are contained in I Let H be a separable infinite dimensional Hilbert space and B(H) be the set of all bounded linear operators on H We call a function f operator monotone whenever the inequality A B =⇒ f (A) f (B) for an arbitrary self-adjoint matrices A, B ∈ B(H) such that A B and all eigenvalues of A and B are contained in I We denote the space of operator monotone functions by P∞ (I) The space for n-monotone functions are written as Pn (I) We have then P1 (I) ⊇ ⊇ Pn−1 (I) ⊇ Pn (I) ⊇ Pn+1 (I) ⊇ ⊇ P∞ (I) Here we note that ∩∞ n=1 Pn (I) = P∞ (I) and each inclusion is proper [12,13] 31 Bachelor thesis NGUYEN THI TUYET The following result is proved in [11] Lemma 3.2.1 Let f be a strictly positive, continuous function on [0, ∞) If the function f is 2n-monotone, then for any positive semi-definite A and a contraction C in Mn we have C ∗ f (A)C f (C ∗ AC) The following result is essentially proved i.n [14,Theorem 2.4], but for the reader’s convenience we will include a proof Proposition 3.2.2 Let f be a strictly positive, continuous function on t [0, ∞) If the function f is 2n-monotone, the function g(t) = is f (t) n-monotone on [0, ∞) Proof Let A, B be positive matrices in Mn such that < A C = B − A− Then C −1 B Let Since f is 2n-monotone, −f satisfies the Jensen type inequality from Lemma 3.2.1, that is, −f (A) = −f (C ∗ BC) −f (A) −A−1/2 f (A)A−1/2 −A−1 f (A) −C ∗ f (B)C −A1/2 B −1/2 f (B)B −1/2 A1/2 −B −1/2 f (B)B −1/2 −B −1 f (B) Therefore, since −1/t is operator monotone, −1/(−f (t)t) = t/f (t) is n-monotone Remark 3.2.3 The condition of 2n-monotonicity of f is needed to guarantee the n-monotonicity of g Indeed, it is well-known that t3 is 32 Bachelor thesis NGUYEN THI TUYET monotone, but not 2-monotone In this case the function g(t) = 1 = t3 t2 is obviously not 1-monotone Corollary 3.2.4 Let f be a 2n-monotone, continuous function on [0, ∞) such that f ((0, ∞))  ⊂ (0, ∞), and let g be a Borel function on [0, ∞)  t (t ∈ (0, ∞)) f (t) defined by g(t) = Then for any pair of positive  (t = 0) matrices A, B ∈ Mn with A B, g(A) g(B) Proof Since f is 2n-monotone, continuous function on [0, ∞) such that f ((0, ∞)) ⊂ (0, ∞), from Proposition 3.2.4 g is n-monotone on (0, ∞) Since g(A + ε) function g that g(A) g(B + ε) for ε > it follows by continuity of the g(B) Theorem 3.2.5 Let T r be a canonical trace on M(n) and f be a 2nmonotone function on [0, ∞) such that f ((0, ∞)) ⊂ (0, ∞) Then for any pair of positive matrices A, B ∈ Mn where g(t) =   t f (t)  2T r f (A) g(B)f (A) , T r(A) + T r(B) − T r(|A − B|) (3.5) (t ∈ (0, ∞)) (t = 0) Proof let A, B be any positive matrices in Mn For operator (A-B) let us denote by P = (A − B)− its positive and negative part, respectively The we have A − B = P − Q and |A − B| = P + Q, 33 (3.6) Bachelor thesis NGUYEN THI TUYET from that it follows that A + Q = B + P (3.7) On account of (3.7) the inequality (3.5) is equivalent to the following 1 T r(A) − T r f (A) g(B)f (A) Since B + P g(A) B and B + P T r(P ) A+Q A 0, we have g(B + P ) by Corollary 3.2.4 and 1 T r(A) − T r f (A) g(B)f (A) 1 1 = T r f (A) g(A)f (A) − T r f (A) g(B)f (A) 1 1 T r f (A) g(B + P )f (A) − T r f (A) g(B)f (A) 1 = T r f (A) (g(B + P ) − g(B))f (A) 1 T r f (B + P ) (g(B + P ) − g(B))f (B + P ) 1 = T r f (B + P ) g(B + P )f (B + P ) 1 − T r f (B + P ) g(B)f (B + P ) 1 T r(B + P ) − T r f (B) g(B)f (B) = T r(B + P ) − T r(B) = T r(P ) Hence, we have the conclusion Remark 3.2.6 (i) When given positive matrices A, B in Mn satisfies 34 Bachelor thesis NGUYEN THI TUYET the condition A B, the inequality (3.5) becomes 1 T r f (A) g(B)f (A) T r(A) (ii) As pointed in Proposition 3.2.2, 2-monotonicity of f is needed to guarantee the inequality (3.5) Indeed, let f (t) = t3 and n = Then, for any a, b ∈ (0, ∞), the inequality 3.5 would imply a 1 f (a) g(b)f (a) , that is, a f (a) b f (b) t is, however, not 1-monotone, the later inequality is imf (t) possible As an application we get Powers-Stormer’s inequality Since Corollary 3.2.7 (6,Theorem1) Let A and B be positive matrices, then for all s ∈ [0, 1] T r(A + B − |A − B|) T r As B 1−s Proof Let f (t) = ts (s ∈ [0, 1]) Then f is operator monotone with f (0, ∞) ⊂ (0, ∞) and g(t) = t1−s Hence, we have the conclusion from Theorem 3.2.5 Remark 3.2.8 For matrices A, B ∈ Mn+ let us denote Q(A, B) = T r A(1−s)/2 B s A(1−s)/2 s∈[0,1] 35 (3.8) Bachelor thesis NGUYEN THI TUYET and 1 QF2n (A, B) = inf T r f (A) g(B)f (A) , (3.9) f ∈F2n where F2n is the set of all 2n-monotone functions on [0, +∞) satisfy condition of the Theorem 3.2.5 and g(t) = tf (t)−1 (t ∈ [0, +∞)) Note that the function f (t) = ts (s ∈ [0, +∞)) satisfies the condition of the Theorem 3.2.5 Since the class of 2n-monotone functions is large enough [13], we know that QF2n (A, B) < Q(A, B) Hence, we hope on finding another 2n-monotone function f on [0, +∞) such that 1 T r f (A) g(B)f (A) < Q(A, B) (3.10) If we can find such a function, the we may get smaller upper bound than what is used in quantum hypothesis testing [1] For example, conT r(|A − B|) sidering the trace distance T (A, B) = , we might have the following better estimate T r(A+B)−QF2n (A, B) T (A, B) T r(A + B) 2 −QF2n (A, B)2 (See the estimate (3.2.8) in [1].) 3.3 Characterizations of the trace property In this section the generalized Powers-Stormer inequality in the previous section implies the trace property for a positive linear functional on operator algebras 36 Bachelor thesis NGUYEN THI TUYET Lemma 3.3.1 Let ϕ be a positive linear funtional on Mn and f be a contiuous function on [0, ∞) such that f (0) = and f ((0, ∞)) ⊂ (0.∞) If the following inequality ϕ(A + B) − ϕ(|A − B|) 1 2ϕ f (A) g(B)f (A) (3.11) holds true for all A, B ∈ Mn+ , then ϕ should be  a positive scalar multiple  t (t ∈ (0, ∞)) f (t) of the canonical trace Tr on Mn , where g(t) =  (t = 0) Proof As is well known, every positive linear functional ϕ on Mn can be represented in the form ϕ(.) = T r(Sϕ ) for some Sϕ ∈ M+ n It is easily seen that without loss of generality we can assume that Sϕ = diag(α1 , α2 , , αn ), and we have to prove that αi = αj for all i, j = 1, , n Clearly, it is sufficient to prove that α1 = α2 By assumption, the inequality (3.11) holds true , in particular, for any positive matrices X = [Xij ]ni,j=1 , Y = [Yij ]ni,j=1 from M+ n such that = xij = yij if or j i n n Thus, it suffices to consider the case n = Assume that Sϕ = diag(d, 1)(d ∈ [0, 1]) and ϕ(D) = T r(Sϕ D), ∀D ∈ M2 We show that d = For arbitrary positive numbers λ, µ such that λ < µ we consider the following matrices  λ A=√ λµ and  A= √ λµ µ √    λ − λµ  √ − λµ µ It is clear that these are positive scalar multiple of projections of rank 37 Bachelor thesis NGUYEN THI TUYET one In addition, µ−λ µ+λ µ−λ =2 µ+λ µ−λ =2 µ+λ 2 f (A) g(B)f (A) = A We have 2ϕ f (A) g(B)f (A) T r(Sϕ A) (dλ + µ) By direct calculation,   √ λu |A − B| =  √  λu Consequently, ϕ(A + B) − ϕ(|A − B|) = d 2λ − λµ + 2µ − Then the inequality (3.11) becomes µ−λ µ+λ (dλ + µ) Dividing two side by d+ √ √ λ √ µ− λµ + µ − d λ− √ λµ λ , we get √ √ √ µ− λ µ+ λ √ √ √ λ µ+ λ Tending λ to µ from the left we obtain d 38 (dλ + µ) µ λ λµ Bachelor thesis NGUYEN THI TUYET Since d ∈ [0, 1], d = This means that ϕ is a positive scalar multiple of the canonical trace T r on Mn Remark 3.3.2 Let ϕ be a positive linear functional on Mn and s ∈ [0, 1] From Lemma 3.3.1 it is clear that if the following inequality ϕ(A + B) − ϕ(|A − B|) ≤ 2ϕ A 1−s B sA 1−s (3.12) holds true for any A, B ∈ Mn+ , then ϕ is a tracial In particular, when s = the following inequality characterizes the trace property ϕ(B) − ϕ(A) ≤ ϕ(|A − B|) (A, B ∈ Mn+ ) (3.13) Corollary 3.3.3 Let ϕ be a positive linear functional on Mn and the following inequality ϕ(A + B) ≤ ϕ(|A|) + ϕ(|B|) (3.14) holds true for any self-adjoint matrices A, B ∈ Mn Then ϕ is a tracial Proof From the assumption we have ϕ(|B − A|) ϕ(|B|) − ϕ(|A|) for any pair of self-adjoint matrices A, B in Mn Moreover, for any pair of positive matrices A, B ∈ Mn we have ϕ(|B − A|) ϕ(B) − ϕ(A) On account of Remark 3.3.2 , it follows that ϕ should be a tracial Corollary 3.3.4 Let ϕ be a positive linear functional on Mn and the 39 Bachelor thesis NGUYEN THI TUYET following inequality |ϕ(A)| ϕ(|A|) (3.15) holds true for any self-adjoint matrix A ∈ Mn Then ϕ is a tracial Proof Let A, B ∈ Mn be a arbitrary positive matrices Then C = B −A is a self-adjoint matrix Since A, B 0, the value ϕ(A) and ϕ(B) are real From the assumption, we have ϕ(B) − ϕ(A) ≤ |ϕ(B) − ϕ(A)| = |ϕ(B − A)| ≤ ϕ(|B − A|) On account of Remark 3.3.2 , it follows that ϕ should be a tracial 40 Bibliography [1] [2] [3] [4] 41 ... norm The connection of this with some other matrix arithmetic-geometry mean inequalities and trace inequalities is discussed 2.1 Introduction to Cauchy inequalities Some matrix versions of the classical... equivalent to saying that the inequalities (2.8) is valid for all Q-norms (see [2] for the definition) Note also that we have written three inequalities instead of four because the inequalities (2.7) is... i=1 Bin Tr i=1 Chapter SOME CAUCHY INEQUALITIES OF LINEAR OPERATORS For positive semi-definite n × n matrices, the inequality 4|||AB||| |||(A + B)2 ||| is shown to hold for every unitarily invariant