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1 Scientiae Mathematicae Vol. 2, No. 3(1999), 263{ OPERATOR INEQUALITIES FOR SCHWARZ AND HUA JUN ICHI FUJII Received July 1, 1999 Abstract. The geometric operator mean gives us an operator version of the Schwarz inequality for 2-positive maps. From this point of view, we show an operator inequality which is an extension for not only the Hua determinant inequality but also the classical Hua one. 0. Introduction. The Schwarz inequality is a fundamental tool to nd various inequality. Nowadays it may be represented as j'(b A)j 2 '(A A)'(B B) for operators A and B in a C*-algebra A and a state ' on it. One might observe that it comes from 2-positivity of'. A (not necessarily linear) map from A to another C*-algebra B is called 2-positive if A B (A) (B) 0 implies 0 C D (C) (D) for all operators A; B; C and D in A. For a state ', wehave '(A A) '(B A) '(A '(B A) '(B A) '(A B) B) '(B A) '(B 0 B) by A A B A A B B B Thus the positivity of its determinant is nothing but the Schwarz inequality. In this note, we estimate j(b A)j like the Schwarz inequality. To see this, we use the geometric mean ] introduced by Ando [1]: A]B max X 0 A X X B 0 : In the properties of the geometric operator mean in [1], we use the monotonicity and the subadditivity A C and B D imply A]B C]D A]B+C]D(A+B)](C+D): Then, we show an operator version of the Schwarz inequality for a 2-positive map : j(b A)j (A A) ] U (B B)U where (B A) Uj(B A)j. As an application, we show an operator version for the following Hua inequality for a state ': (1) j'(1, B A)j 2 '(1, A A)'(1, B B); 0: 1991 Mathematics Subject Classication. 47A63, 47C15. Key words and phrases. operator inequality, operator mean, Schwarz inequality, Hua inequality.

2 264 JUN ICHI FUJII which was used by Marcus [6] to show the determinant Hua inequality [4]. 1. Schwarz inequality. For the sake of completeness, we give a proof for the Schwarz operator inequality: Theorem 1. Let be a 2-positive map and (B A)Uj(B A)j the polar decomposition. Then, j(b A)j (A A) ] U (B B)U: Proof. Since is 2-positive, we have (A A) (B A) (B A) (B B) (A A) (A B) (B A) (B 0: B) It follows from U (B A)j(B A)j (B A) U that (A A) j(b A)j (A j(b A)j U (B A) (B A) U B)U U (B A) U (B B)U 1 0 (A A) (B A) U (B A) (B B) 0 U Therefore we havej(b A)j (A A) ] U (B B)U by the denition of the geometric mean. The above theorem leads us to an operator inequality for the modulus of operators: Corollary 1. If (X) Uj(X)j is the polar decomposition of (X) for a 2-potitive map, then j(x)j (jxj) ]U (jx j)u: In particular, if 'is a state, then j'(x)j p '(jxj)'(jx j). Proof. Let X V jxj be the polar decomposition of X. Since V jxjv jx j,wehave 0: j(x)j j(v jxj 1 2 jxj 1 2 )j(jxj) ]U (V jxjv )U (jxj)]u (jx j)u: 2. Two Hua inequalities. In this section we discuss the relation among inequalities by Hua. Though Marcus [6] used (1) as a passing result to show the Hua determinant inequality in [4]: (2) j det(1, B A)j 2 det j1, B Aj 2 det(1, A A) det(1, B B) for contractive matrices A and B. M.Nakamura pointed out the importance of (1). In fact (1) leads us to (2) and is a variation of the classical Hua inequality [5]: (3) nx 2, a k k1 nx 2 + n, a 2 k k1 for every positive numbers ; and real a k. By putting b k a k (n), n, '(X) (Tr X)n and B b1 0 C A ; 0 b n

3 OPERATOR INEQUALITIES FOR SCHWARZ AND HUA 265 it is equivalent to the following brief form: (4) j'(1, B)j 2 +1,'(B2 ) for every >0. According to Marcus' idea, we show (1) implies (2). Since we may assume that j1,b Aj is diagonal, we have det j1, B Aj 2 Y k jhj1, B Aje k ;e k ij 2 ; for a standard basis fe k g. Since operators 1, A A and 1, B B are positive, (1) implies Y k jhj1, B Aje k ;e k ij 2 Y h(1, A A)e k ;e k ih(1, B B)e k ;e k i: k Noting that each hhe k ;e k i is a diagonal entry of H, wehave Y k h(1, A A)e k ;e k ih(1, B B)e k ;e k idet(1, A A) det(1, B B) by the Hadamard theorem. (Marcus used Ky Fan's theorem instead of the Hadamard one to show his extended inequality of (2) in [6].) Next we see (1) implies (4). Let 0 < jj 1 and A B for B in (4). Then (1) implies j'(1, B)j 2 j'(1, A)j 2 (1,jj 2 )(1, '(A A)) (1,jj 2 ), 1,jj2 jj 2 '(B B): Now, putting (1,jj 2 )jj 2,wehave1,jj 2 ( + 1) and hence (4). Moreover, this argument shows the following Hua inequality for complex numbers: Theorem 2. If and a k are complex numbers, then (5) for all positive numbers. nx 2, a k k1 nx jj2 + n, ja k j 2 k1 3. Hua operator inequality. Noting that j1, Xj 1, jxjfor every normal operator X, we have a Hua type inequality: Theorem 3. Let A and B be operators on a Hilbert space and acontractive 2-positive map for a suitable algebra including these operators. If (B A)Uj(B A)j the polar decomposition for a normal operator (B A), then j1, (B A)j 1,j(B A)j 1,(A A) ]U (B B)U: In addition, if (1, A A) and (1, B B) are contractions and is linear, then (6) 1, (A A) ]U (B B)U (1, A A) ]U (1, B B)U:

4 266 JUN ICHI FUJII Proof. The rst inequality follows from the normality of(b A) and the second from Theorem 1. We have the last inequality (6) by the monotonicity and the subadditivity of the geometric mean: (A A) ]U (B B)U + (1, A A) ]U (1, B B)U (A A) ]U (B B)U +, 1, (A A) ], 1, U (B B)U 1 ] 11; since (1) k(1)k 1 and U U 1. Remark. The normality of(b A) is necessary in the above theorem. In fact, consider the following example: Let be the identity map on the 2 2 matrices and put 1 0 A and B : Then, (B A)B AAis not normal and so that (1, B A) (1, B A) 1,1 1,1 j1, B Aj p 1 2,1 : 5,1 3 On the other hand, A A 1,BjAjimplies 1, A A B j1, A Aj and U 1,1 ;,1 2 : 1 0 It follows from 1, B B 1,Bthat U (1, B B)U B. Thus (1, A A) ]U (1, B B)U B]BB; which is not smaller than j1, B Aj. Here we consider the assumption that (B A) is normal particularly in case B A is nonnormal. If is a `pintching' map B(H H)! B(H) B(H) with V X W Y V 0 0 Y then we easily give such an example. For instance, taking W 6 0,we put 1 B W A : In particular, if the image of is abelian, we need not such an assumption. Moreover we can delete the above partial isometry U (precisely we can take it as a unitary operator in this case) like Corollary 1 and the condition that the operators are contractions. ; Corollary 2. If A and B are operators and ' is a state, then j'(1,b A)j 2 (1,j'(B A)j) 2 1, p '(A,A)'(B B) 2'(1,A A)'(1,B B): Proof. We have only to show the last inequality and it is shown by 2'(A A)'(B B) '(A A)+'(B B); that is, the arithmetic-geometric mean inequality.

5 OPERATOR INEQUALITIES FOR SCHWARZ AND HUA 267 In fact, this result (i.e., (1)) was our starting point and the author had shown it in 1996 [3]. As we see in the previous section, (1) is a dimension-free extension of the Hua inequality, but we did not notice it at that time. 4. Vector inequalities. Various extensions have been discussed for the Hua inequality. The operator norm and the determinant are nonlinear 2-positive maps. From the viewpoint of Theorem 3, we can discuss norm inequalities of Hua type. Finally we refer to vector inequalities of Hua type introdued by Dragomir and Yang [2]. Though such inequalities are usually in a real vector space, we need not `reality' any longer as we see in the previous section. As we see later, we have the complex version of the Dragomir-Yang inequality: Corollary 3 (Dragomir-Yang). For vectors y and x k in a Hilbert space, for every positive number. nx 2 nx y, x k + k1 kx k k 2 k1 + n kyk2 Instead of proving this result, we show a complex version of an extension by Radas and Sikic [7] (Precisely they formulate their statement for an operator A from a space to another, but it does not seem to be essential): Corollary 4 (Radas-Sikic). Let A be anoperator on a Hilbert space and x and y vectors in it. Then, for every >0, ky,axk 2 with equality if and only if (i) A 0and x 0,or + kak 2 kyk2, kxk 2 (ii) A 6 0; Ax kak2 + kak2 y and kaxk kakkxk: Proof. Since we have only to show (7) ky, Axk jkyk,kaxkj; jkyk,kaxkj 2 and, as we may assume y 6 0, it is equivalent to (8) j1,kaxkj 2 + kak 2 kyk2, kxk 2 ; + kak 2, kxk2 : A proof of (8) is parallel to the implication from (1) to (4). Along with Theorem 3, we have j1,kaxkj 2 j1,ka(x)kj 2 j1,kakkxkj 2 (1, 2 kak 2 )(1,kxk 2 2 ) 1, 2 kak 2, 1,2 kak 2 2 kxk 2 : Choosing with (1, 2 kak 2 ) 2,wehave1, 2 kak 2 ( + kak 2 ), and hence (8). The equality condition (ii) for A 6 0 follows from kaxk kakkxk and ky, Axk jkyk,kaxkj if and only if ty Ax for some scalar t

6 268 JUN ICHI FUJII under the assumption y 6 0. References [1] T.Ando: Topics on operator inequality, Hokkaido Univ. Lecture Note, [2] S.S.Dragomir and G.-S.Yang: On Hua's inequality in real inner product spaces, Tamkang J. Math., 27(1996), [3] J.I.Fujii: Private note dated December 14, [4] L.K.Hua: Inequalities involving determinants, Acta Math. Sinica, 5(1955), pp [5] L.K.Hua: Additive theory of prime numbers(tranlated by N.B.Ng), Translation of Mathematical Monographs, 13, Amer. Math. Soc., Providence, RI, [6] M.Marcus: On a determinant inequality, Amer. Math. Monthly, 65(1958), pp [7] S.Radas and T.Sikic: A note on the generalization of Hua's inequality, Tamkang J. Math., 28(1997), Department of Arts and Sciences (Information Science), Osaka Kyoiku University, Asahigaoka, Kashiwara, Osaka 582, Japan address: fujii@cc.osaka-kyoiku.ac.jp

A CHARACTERIZATION OF THE HARMONIC OPERATOR MEAN AS AN EXTENSION OF ANDO S THEOREM. Jun Ichi Fujii* and Masahiro Nakamura** Received December 12, 2005

Scientiae Mathematicae Japonicae Online, e-006, 7 3 7 A CHARACTERIZATION OF THE HARMONIC OPERATOR MEAN AS AN EXTENSION OF ANDO S THEOREM Jun Ichi Fujii* and Masahiro Nakamura** Received December, 005 Abstract.