ISSN: 7-6X (Print ISSN: 735-855 (Online Special Issue of the ulletin of the Iranian Mathematical Society in Honor of Professor Heydar Radjavi s 8th irthday Vol. 4 (5, No. 7, pp. 85 94. Title: Submajorization inequalities associated with τ-measurable operators Author(s: J. Zhao and J. Wu Published by Iranian Mathematical Society http://bims.ims.ir
ull. Iranian Math. Soc. Vol. 4 (5, No. 7, pp. 85 94 Online ISSN: 735-855 SUMAJORIZATION INEQUALITIES ASSOCIATED WITH τ-measurale OPERATORS J. ZHAO AND J. WU (Communicated by Laurent Marcoux Dedicated to Professor Heydar Radjavi on his 8th birthday Abstract. The aim of this note is to study the submajorization inequalities for τ-measurable operators in a semi-finite von Neumann algebra on a Hilbert space with a normal faithful semi-finite trace τ. The submajorization inequalities generalize some results due to Zhang, Furuichi and Lin, etc.. Keywor: Submajorization, von Neumann algebra, τ-measurable operators. MSC(: Primary: 46L; Secondary: 47A63, 47C5.. Introduction First we recall the definition of majorization. Given a real vector x (x, x,..., x n R n, we rearrange its components as x [] x [] x [n]. For x (x, x,..., x n, y (y, y,..., y n R n, if k x [i] i k y [i], k,,..., n i then we say that x is weakly majorized by y and write x w y. If x w y and n x i n y i, then we say that x is majorized by y and write x y. i i Denote by M n the usual von Neumann algebra consisting of n n complex matrices. Let A M n. We always denote the singular values of A by s (A s (A s n (A, and denote s(a : (s (A, s (A,..., s n (A. If A is a Hermitian matrix, we denote the eigenvalues of A in decreasing order by Article electronically published on December 3, 5. Received: November 4, Accepted: 6 October 5. Corresponding author. 85 c 5 Iranian Mathematical Society
Submajorization inequalities 86 λ (A λ (A λ n (A and denote λ(a : (λ (A, λ (A,..., λ n (A. Recently, Zhang [5] proved the following result: ( A X Theorem.. Let H X be a positive semidefinite block matrix with square matrices A and of the same order. Then for any complex number z with z, λ(h λ([a + + Re(zX] + λ([a + Re(zX]. If, in addition, X is Hermitian, then for any real number r [, ] λ(h λ((a + + rx + λ((a + rx, while if X is skew-hermitian, then for any real number r [, ] λ(h λ((a + + irx + λ((a + irx, where Re(X X+X, i and is the zero matrix of compatible size. It is worth pointing that in the majorization theory, the submajorization (see definition in Section, is one of the most important orderings. Many authors have studied it. We refer to [, 3, 4, 7 3] for more related results on this topic. ( The purpose of this note is to extend Theorem. to the case that H A X X M (M is ˆτ-measurable positive operator in a semifinite von Neumann algebra. In addition, we also study the submajorization among (A + (A +, A and ( A +, where A, M (see below are τ-measurable operators.. Preliminaries Denote by M a semi-finite von Neumann algebra on the Hilbert space H with a normal faithful semi-finite trace τ. The closed densely defined linear operator T in H with domain D(T is said to be affiliated with M if U T U T for all unitary U that belong to the commutant M of M. If T is affiliated with M, then T is said to be τ-measurable if for every ε > there exists a projection P M such that P (H D(T and τ(p < ε (where for any projection P we let P I P, I here denotes the identity operator. The set of all τ-measurable operators will be denoted by M which is the closure of M with respect to the measure topology, which is the linear (Hausdorff topology whose fundamental system of neighborhoo around is given by V (ε, δ {T M T P εfor some projectionp M withτ(i P δ}
87 Zhao and Wu Here, ε, δ run over all strictly positive numbers and is the operator norm. For a positive self-adjoint operator T λde λ affiliated with M, we set τ(t sup n τ ( n λde λ λdτ(e λ. For < p <, L p (M; τ is defined as the set of all densely defined closed operators T affiliated with M such that T p τ( T p p <, where T (T T. In addition, we put L (M; τ M and denote by ( the usual operator norm. It is well known that L p (M; τ is a anach space under p ( p satisfying all the expected properties such as duality. Let T be a τ-measurable operator and t >. The t-th singular number (generalized s-number of T, denoted by µ t (T, is µ t (T inf{ T P P a projection in M/mboxwithτ(I P t}. Then µ t (T has the following properties (see [5, Lemma.5]. Lemma.. Let T, S, R be τ-measurable operators. (i The map: t (, µ t (T is non-increasing and continuous from the right. Moreover lim µ t(t T [, ]. t + (ii µ t (T µ t ( T µ t (T and µ t (αt α µ t (T for t > and α C. (iii µ t (T µ t (S, t >, if T S. (iv µ t (f( T f(µ t ( T, t > for any continuous increasing function f on [, ] with f(. (v µ t+s (T + S µ t (T + (S, t, s >. (vi µ t (ST R S R µ t (T, t >. (vii µ t+s (T S µ t (T (S, t, s >. From (vi of Lemma., we immediately know that µ t ( has unitary invariance property, i.e., µ t (U T U µ t (T for τ measurable operators T, U with U unitary. Next, we will give the definition of submajorization. The submajorization of operators is a generalization of the notion of submajorization for functions introduced by Hardy, Littlewood and Polya.
Submajorization inequalities 88 Definition.. If f and g are measurable positive decreasing functions on (,, then we say that g is submajorized by f and write g f if g(s f(s for all t >. Given A, M, we say that is submajorized by A and write A if µ( µ(a. For example, it has been established that µ(a + µ(a + µ( and µ(a µ(aµ( for A, M (see Theorems 4.4 and 4. in [5]. We remark that if M M n and τ is the standard trace, i.e., τ(a tr(a n a ii for A (a ij M n, then i µ t (A s j (A, t [j, j, j,,..., n, and if A, M, then A is equivalent to k s i (A i k s i (, k,,..., n. i 3. Submajorization inequalities Let M (M be the von Neumann algebra M (M { ( Aij A A } M, i, j, A A on the Hilbert ( space H H with trace ˆτ tr τ, i.e., ˆτ(A τ(a + A A A for A M A A (M. For A M (M, let ( ( A A (3. C(A C A A ( I where U I positive contraction. ( A A (U k AU k, k. Then C : M (M M (M is a trace-preserving In this section, we mainly study submajorization inequalities. To achieve our goal, we need the following lemma. Its matrix version was given by ourin and Lee in [].
89 Zhao and Wu ( A X Lemma 3.. Let H X M (M be ˆτ-measurable positive operator. Then there exist partial isometries U, V M (M such that (3. ( A X X U ( A ( U + V V. Proof. ( Since H is positive, then there exists a positive ˆτ-measurable operator C Y Y M D (M such that ( ( ( A X C Y C Y X Y D Y D ( ( ( C C Y Y Y + D where T ( C Y T T + S S (, S Y D. ( Y D Let T U T, S V S be the polar decompositions, respectively. It is easy to check that ( T A T U U and S S V ( V. ased on Lemma 3., we obtain the main result. ( A X Theorem 3.. Let H X M (M be a ˆτ-measurable positive operator. Then for any complex number z with z, (3.3 µ([a + Y ] [A + + Y ] µ(h µ([a + Y ] + µ([a + + Y ] where Y Re(zX, and is the zero operator.
Submajorization inequalities 9 Proof. y Lemma., 3. and [5, Theorem 4.4], we have ( ( A X X ( ( A t ( ( + µ s (3.4 + ( ( A ( ( ( I I ( ( A + for t >. ( Taking J I I and J I I z unitaries and (3.5 J J z HJ z J where K A +(zx zx. ( I I ( ( ( I zi ( A+ Y K K A++Y, then J and J z are Noting that (H (J J z HJ z J for s >, then the second inequality in (3.3 follows from (3.4. On the other hand, by Lemma., equality (3., [5, Theorem 4.4], we have for t >, which is equivalent to (C(H (3.6 µ(a µ(h. (H, According to (3.5 and (H (J J z HJz J for s >, the relation (3.6 yiel µ([a + Y ] [A + + Y ] µ(h. This completes the proof of the first inequality in (3.3. Corollary 3.3. Let X, Y M be τ-measurable operators. complex number z with z, Then for any (3.7 µ(xx + Y Y µ(x X + Y Y Z + µ(x X + Y Y + Z where Z zx Y + zy X.
9 Zhao and Wu ( X Y Proof. Let T ( T T X Y and ( T X T Y. Then ( X Y ( X Y ( XX + Y Y ( X X X Y Y X Y Y for t >. On the other hand, let T U T be a polar decomposition. Then Thus, (T T T T U (T T U. (U T T U y Theorem 3., we get the desired results.. (T T. Putting z in Z. Then inequality (3.7 gives µ(xx + Y Y µ(x X + Y Y (X Y + Y X + µ(x X + Y Y + (X Y + Y X. This is a generalization of a matricial result obtained by Turkman, Paksoy and Zhang in [4]. Corollary 3.4. Let S, T M be τ-measurable positive operators. Then for any real number r [, ], (3.8 µ(t + ST S µ(t + T S T + rt ST + µ(t + T S T rt ST. Proof. Let X T and Y ST. Then zx Y + zy X Re(zT ST. Since Re(z ranges over [-,] as we vary z over all complex number of modulus, the result follows from Corollary 3.6. Remark 3.5. The matrix form of 3.8 when r was obtained by Furuichi and Lin in [6]. Corollary 3.6. Let A, M be τ-measurable operators. Then for any real number r [, ], the following inequality hol, ( ( A µ A + A + A A + A A A + µ([a A + rc] (3.9 where C A + A. + µ([a A + + rc]
Submajorization inequalities 9 ( ( A Proof. Since both A A A and A A A are ˆτ-measurable A ( A positive operators, so is A + A + A A + A A A +. Then by Theorem 3., we obtain the result (3.9. 4. Further results In this section, we mainly study the submajorization among (A + (A +, A and ( A +, where A, M are τ-measurable operators. We have the following results. Theorem 4.. Let A, M be τ-measurable operators. Then the following hol: (4. µ([a + ] [A + ] µ(a µ([ A + ]. Proof. Since (4. ( A + A + ( A ( I + I ( A ( I I by [5, Theorem 4.4] and Lemma., the first submajorization of (4. hol at once. On the other hand, let A U A, U be polar decompositions. Then (4.3 ( A + ( A ( A, and (4.4 ( A ( A ( A A A. There exists a partial isometry U 3 such that (4.5 ( A A A U 3 ( A ( A U 3.
93 Zhao and Wu Then by Lemma., (3., (4., (4.4, (4.5 and [5, Theorem 4.4], we have ( ( A ( ( A ( ( A A A ( ( ( A A U 3 ( ( A + for t >. y Definition., the second submajorization of (4. also hol. Acknowledgements The authors would like to thank the editors and anonymous referees for their careful reading and detailed comments which have considerably improved this paper. This work is supported by the National Natural Science Foundation of China (Grant Nos. 7873 and 64. References [] T. ekjian and D. Dauitbek, Submajorization inequalities of τ-measurable operators for concave and convex functions, Positivity 9 (5, no., 34 345. [] J. ourin and E. Lee, Unitary orbits of Hermitian operators with convex and concave functions, ull. London Math. Soc. 44 (, no. 6, 85. [3] P. Dod, T. Dod and. Pagter, Noncommutative anach function spaces, Math. Z. (989, no. 4, 583 597. [4] P. Dod and F. Sukochev, Submajorisation inequalities for convex and concave functions of sums of measurable operators, Positivity 3 (9, no., 7 4. [5] T. Fack and H. Kosaki, Generalized s-numbers of τ-measurable operators, Pacific. J. Math. 3 (986, no., 69 3. [6] S. Furuichi and M. Lin, A matrix trace inequality and its applications, Linear Algebra Appl. 433 (, no. 7, 34 38. [7] T. Harada, Majorization inequalities related to increasing convex functions in a semifinite von Neumann algebra, Math. Inequal. Appl. (8, no. 3, 449 455. [8] F. Hiai, Majorization and stochastic maps in von Neumann algebras, J. Math. Anal. Appl. 7 (987, no., 8 48. [9] F. Hiai and Y. Nakamura, Closed convex hulls of unitary orbits in von Neumann algebras, Trans. Amer. Math. Soc. 33 (99, no., 38. [] F. Hiai and Y. Nakamura, Majorizations for generalized s-numbers in semifinite von Neumann algebras, Math. Z. 95 (987, no., 7 7. [] M. Lin and H. Wolkwicz, An eigenvalue majorization inequality for positive semidefinite block matrices, Linear Multilinear Algebra 6 ( no. -, 365 368. U 3
Submajorization inequalities 94 [] M. Marsalli and G. West, Noncommutative H p spaces, J. Operator Theory 4 (998, no., 339 355. [3] F. Sukochev, On the A. M. ikchentaev conjecture, Russian Math. 56 (, no. 6, 57 59. [4] R. Turkmen, V. Paksoy and F. Zhang, Some inequalities of majorization type, Linear Algebra Appl. 437 (, no. 6, 35 36. [5] Y. Zhang, Eigenvalue majorization inequalities for positive semidefinite block matrices and their blocks, Linear Algebra Appl. 446 (4 6 3. (Jianguo Zhao College of Mathematics and Statistics, Chongqing University, Chongqing, 433, P.R. China, and College of Science, Shihezi University, Shihezi, Xinjiang, 833, P. R. China. E-mail address: jgzhao dj@63.com (Junliang Wu College of Mathematics and Statistics, Chongqing University, Chongqing, 433, P.R. China. E-mail address: jlwu678@63.com