Operator inequalities associated with A log A via Specht ratio

Transcription

1 Linear Algebra and its Applications 375 (2003) Operator inequalities associated with A log A via Specht ratio Takayuki Furuta Department of Mathematical Information Science, Faculty of Science, Tokyo University of Science, -3 Kagurazaka, Shinjukuku, Tokyo , Japan Received 3 March 2003; accepted 4 May 2003 Submitted by R.A. Brualdi Dedicated to the memory of Professor Shozo Koshi in deep sorrow Abstract An operator means a bounded linear operator on a Hilbert space H. We obtained the basic property between Specht ratio S() and generalized Kantorovich constant K(p) in Math. Inequal. Appl., in press], that is, Specht ratio S() can be expressed by generalized Kantorovich constant K(p): S() = e K (). We shall investigate several product type and difference type inequalities associated with A loga by applying this basic property to several Kantorovich type inequalities Elsevier Inc. All rights reserved. AMS classification: 47A63 Keywords: Kantorovich inequality; Specht ratio. Introduction An operator A is said to be positive operator (denoted by T 0) if (Ax, x) 0 for all x in H and also A is said to be strictly positive operator (denoted by A>0) if A is invertible positive operator. Definition. Let h>. S(h,p) is defined by S(h,p) = h p h p elogh p h p address: furuta@rs.kagu.tus.ac.jp (T. Furuta) /$ - see front matter 2003 Elsevier Inc. All rights reserved. doi:0.06/s (03) for any real number p (.)

2 252 T. Furuta / Linear Algebra and its Applications 375 (2003) and S(h,p) is denoted by S(p) briefly. Especially S() = S(h,) = h h elogh h is said to be Specht ratio and S() > is well known. Let h>. The generalized Kantorovich constant K(h, p) is defined by (h p ( h) (p ) (h p ) ) p K(h, p) = (p )(h ) p (h p for any real number p h) (.2) and K(h, p) is denoted by K(p) briefly. Basic property 3]. The following basic property among S(), K () and K (0) holds: S() = e K () = e K (0) ( ] ]) i.e., S() = exp lim p K (p) = exp lim K (p) p 0, (.3) K(0) = K() = ( ) i.e., lim K(p) = lim K(p) =, (.4) p 0 p S() = lim K(p) p = lim K(p) p. (.5) p p 0 Refer to Fig. for the relation between K(p) and S(p). The relation (.3) is quite important in this paper, so we state its proof for the sake of convenience. In fact K (p) can be written as follows: ( (p ) p (h p ) (h p h) ) p K (p) = (h )(h p ) h p (h p +p hp) log h+(h p )(h p h) log (p )(hp ) p(h p h) p By using L Hospital theorem to ( ), we have lim p K (p) = h h log h (h ) { 2 ]} h h log h(h log h + h) + (h )h log h log h log h = h ] h log h + log h h log h. ( )

3 T. Furuta / Linear Algebra and its Applications 375 (2003) Fig.. Relation between K(p) and S(p). ] h h = log eloghh = log S() so that we have S() = e K () and also S() = e K (0) by the same way. We remark that (.5) is an immediate consequence of (.3) by L Hospital theorem. Another nice relation between K(p) and S() is in 26]. Let A be strictly positive operator satisfying MI A mi > 0, where M> m>0. Put h = M m >. The celebrated Kantorovich inequality asserts that ( + h) 2 (Ax, x) (A x,x) (Ax, x) (.6) 4h holds for every unit vector x and this inequality is just equivalent to the following one ( + h) 2 (Ax, x) 2 (A 2 x,x) (Ax, x) 2 (.7) 4h holds for every unit vector x. We remark that K(h, p) in (.2) is an extension of (+h) 2 4h in (.6) and (.7), in fact, K(h, ) = K(h, 2) = (+h)2 4h holds.

4 254 T. Furuta / Linear Algebra and its Applications 375 (2003) Many papers on Kantorovich inequality have been published. Among others, there is a long research series by Mond Pečarić, we cite 2 23] for examples. We state the Jensen inequality as follows. (cf., Theorem 4;3,4;7, Theorem 2.].) Jensen inequality. Let f be an operator concave function on an interval I. If Φ is normalized positive linear map, then f(φ(a)) Φ(f (A)) for every self adjoint operator A on a Hilbert space H whose spectrum is contained in I. On the other hand, the relative operator entropy S(X Y) for X>0andY>0is defined in 7] as an extension of the operator entropy S(X I) = Xlog X S(X Y) = X 2 log ( X ) ] 2 YX 2 X 2. (.8) By using this S(X Y),wedefineT(X Y)for X>0andY>0; T(X Y) = (X Y )X S(X Y)X (X Y ), (.9) where X Y = X 2 (X 2 YX 2 ) 2 X 2. The power mean X p Y = X 2 (X 2 YX 2 ) p X 2 for p 0, ] is in 6] as an extension of X Y. We shall verify that T(X Y) = lim p (X p Y) in Proposition 3.2 and we remark that S(X Y) = lim p 0 (X p Y) shownin7]. In this paper lim p 0 F(p) means lim p +0 F(p) and also lim p F(p) means lim p 0 F(p), incidentally F (0) means F (+0) and F () means F ( 0) and so on. Next we state the following several Kantorovich type inequalities. Theorem A. Let A be strictly positive operator on a Hilbert space H satisfying MI A mi > 0, where M>m>0 and h = M m > and Φ be a normalized positive linear map on B(H). Let p (0, ). Then the following inequalities hold: (i) Φ(A) p Φ(A p ) K(p)Φ(A) p, (ii) Φ(A) p Φ(A p ) Φ(A) p g(p)i, where g(p) = m p h p h h + ( p) h p p ] p p(h ) and K(p) is defined in (.2). The right hand side inequalities of (i) and (ii) in Theorem A follow by 8, Corollary 2.6;23] and the left hand side one of (i) follows by Jensen inequality since f (A) = A p is operator concave for p 0, ]. More general forms than Theorem A are in 7] and related results to Theorem A are in 9,20].

5 T. Furuta / Linear Algebra and its Applications 375 (2003) Theorem B. Let A and B be strictly positive operators on a Hilbert space H such that M I A m I>0and M 2 I B m 2 I>0.Putm = m m 2, M = M M 2 and h = M m = M M 2 m m 2 >. Letp (0, ). Then the following inequalities hold: (i) (A B) p A p B p K(p)(A B) p, (ii) (A B) p A p B p (A B) p g(p)i, where g(p) = m p h p h h + ( p) h p p ] p p(h ) and K(p) is defined in (.2). The right hand side inequalities of (i) and (ii) follow by 25, Theorem 6] and the left hand side one of (i) follows by 0;25, Theorem ]. Theorem C. Let A, B, C andd be strictly positive operators on a Hilbert space H such that M I A B m I>0and M 2 I C D m 2 I>0. Putm = m 2 M, M = M 2 m and h = M m = M M 2 m m 2 >. Letp (0, ). Then the following inequalities hold: (i) (A B) p (C D) (A p C) (B p D) K(p)(A B) p (C D), (ii) (A B) p (C D) (A p C) (B p D) (A B) p (C D) g(p)i(a B), where g(p) = m p h p h h + ( p) h p p ] p p(h ) and K(p) is defined in (.2). The right hand side inequalities of (i) and (ii) follow by 8, Corollary 4.4] and the left hand side inequality of (i) follows by 2, Theorem 4.] and also it follows by a corollary of 5, Theorem 5]. Theorem D. Let A and B be strictly positive operators on a Hilbert space H such that M I A m I>0and M 2 I B m 2 I>0. Putm = m 2 M,M = M 2 m and h = M m = M M 2 m m 2 >. Letp (0, ) and also let Φ be normalized positive linear map on B(H). Then the following inequalities hold: (i) Φ(A) p Φ(B) Φ(A p B) K(p)Φ(A) p Φ(B), (ii) Φ(A) p Φ(B) Φ(A p B) Φ(A) p Φ(B) g(p)φ(a), where g(p) = m p h p h h + ( p) h p p ] p p(h ) and K(p) is defined in (.2). The right hand side inequalities of (i) and (ii) follow by 8, Corollary 3.5] and the left hand side one of (i) follows by,6]. The following result is contained in 8, Corollary 4.] together with 5, Corollary 8].

6 256 T. Furuta / Linear Algebra and its Applications 375 (2003) Theorem E. Let A and B be strictly positive operators on a Hilbert space H such that M I A m I>0and M 2 I B m 2 I>0. Letp (0, ) and also p m = m M p p 2,M = M m p 2 and h = M m = M p M2 p m m 2 >. Then the following inequalities hold: (i) A p p ) I (B p p I A B K(p) A p p I (B p p I), (ii) A p p ) I (B p p I A B A p p I (B p I) p g(p)(b I), where g(p) = m p h p h h + ( p) h p p ] p p(h ) and K(p) is defined in (.2). In fact put A 3 = A p and B 3 = B p,thenm p I A 3 m p I>0andM p 2 I B 3 m p 2 I>0 under the hypotheses of Theorem E. By applying Theorem E ( p) p ( p) p to A 3 and B 3, put m 3 = m p p M 2 = m M 2, M 3 = M p p m 2 = M m 2 and h 3 = M 3 m 3 = M M 2 m m 2 >, so we have the following result as a variation of Theorem E. Theorem E. Let A and B be strictly positive operators on a Hilbert space H such that M I A m I>0and M 2 I B m 2 I>0. Putm = m M 2,M = M m 2 and h = M m = M M 2 m m 2 >.Letp (0, ). Then the following inequalities hold: (i) (A I) p (B I) p A p B p K(p)(A I) p (B I) p, (ii) (A I) p (B I) p A p B p (A I) p (B I) p g(p)(b p I), where g(p) = m p h p h h + ( p) h p p ] p p(h ) and K(p) is defined in (.2). We shall investigate several product type and difference type inequalities associated with A log A by applying the basic property to Theorems A E which are Kantorovich type inequalities. 2. Several product type and difference type inequalities associated with A loga In this section we shall state the following several product type and difference type inequalities associated with A log A. Theorem 2.. Let A and B be strictly positive operators on a Hilbert space H such that M I A m I>0and M 2 I B m 2 I>0. Putm = m 2 M,M= M 2 m and

7 T. Furuta / Linear Algebra and its Applications 375 (2003) h = M m = M M 2 m m 2 >.LetΦbe a normalized positive linear map on B(H).Thenthe following inequalities hold: (i) (ii) log S()]Φ(B) + T(Φ(A) Φ(B)) Φ(T (A B)) T(Φ(A) Φ(B)), mh log h (S() )Φ(A) + T(Φ(A) Φ(B)) h Φ(T (A B)) T(Φ(A) Φ(B)), (iii) log S()Φ(A) + Φ(S(A B)) S(Φ(A) Φ(B)) Φ(S(A B)), where S(X Y) and T(X Y) are defined in (.8) and (.9) and S() is defined in (.). We remark that the first inequality of (i) in Theorem 2. is the reverse inequality of the second one and also the first inequality of (ii) is the reverse inequality of the second one, and the first inequality of (iii) is the reverse inequality of the second one in 7, Theorem 7]. Corollary 2.2. Let A be strictly positive operator on a Hilbert space H satisfying MI A mi > 0, where M>m>0 and h = M m > and Φ be a normalized positive linear map on B(H). Then the following inequalities hold: (i) (ii) log S()]Φ(A) + Φ(A) log Φ(A) Φ(A log A) Φ(A) log Φ(A), mh log h (S() ) + Φ(A) log Φ(A) h Φ(A log A) Φ(A) log Φ(A), (iii) log S() + Φ(log A) log Φ(A) Φ(log A), where S() is defined in (.).

8 258 T. Furuta / Linear Algebra and its Applications 375 (2003) We remark that the first inequality of (i) in Corollary 2.2 is the reverse inequality of the second one which is known by, Theorem 4] and also the first inequality of (ii) is the reverse inequality of the second one, and the first inequality of (iii) is the reverse inequality of the second one which is known by Jensen inequality. Theorem 2.3. Let A, B, C and D be strictly positives operators on a Hilbert space H such that M I A B m I>0and M 2 I C D m 2 I>0. Put m = m 2 M,M = M 2 m and h = M m = M M 2 m m 2 >. Then the following inequalities hold: (i) log S()](C D) + T(A B C D) T(A C) D + C T(B D) T(A B C D), (ii) mh log h (S() )(A B) + T(A B C D) h T(A C) D + C T(B D) T(A B C D), (iii) log S()](A B) + S(A C) B + A S(B D) S(A B C D) S(A C) B + A S(B D), where S(X Y) and T(X Y) are defined in (.8) and (.9) and S() is defined in (.). We remark that the first inequality of (i) in Theorem 2.3 is the reverse inequality of the second one and also the first inequality of (ii) is the reverse inequality of the second one, and the first inequality of (iii) is the reverse inequality of the second one. Corollary 2.4. Let A and B be strictly positive operators on a Hilbert space H such that M I A m I>0and M 2 I B m 2 I>0.Putm = m m 2,M = M M 2 and h = M m = M M 2 m m 2 >. Then the following inequalities hold: (i) log S()](A B) + (A B)log(A B) A (B log B) + (A log A) B (A B)log(A B),

9 T. Furuta / Linear Algebra and its Applications 375 (2003) (ii) mh log h (S() ) + (A B)log(A B) h A (B log B) + (A log A) B (A B)log(A B), (iii) log S() + (log A) I + I (log B) log(a B) (log A) I + I (log B), where S() is defined in (.). We remark that the first inequality of (i) in Corollary 2.4 is the reverse inequality of the second one and also the first inequality of (ii) is the reverse inequality of the second one, and the first inequality of (iii) is the reverse inequality of the second one. Theorem 2.5. Let A and B be strictly positive operators on a Hilbert space H such that M I A m I>0and M 2 I B m 2 I>0. Putm = m M 2,M = M m 2 and h = M m = M M 2 m m 2 >. Then the following inequalities hold: (i) log S()](A I)+ A log B + (A I)log(A I) (A log A) I + (A I)log(B I) A log B + (A I)log(A I), (ii) mh log h (S() ) + A log B + (A I)log(A I) h (A log A) I + (A I)log(B I) A log B + (A I)log(A I), (iii) log S()](B I)+ (log A) B + (B I)log(B I) I (B log B) + (log(a I))(B I) (log A) B + (B I)log(B I), where S() is defined in (.). We remark that the first inequality of (i) in Theorem 2.5 is the reverse inequality of the second one and also the first inequality of (ii) is the reverse inequality of the second one, and the first inequality of (iii) is the reverse inequality of the second one.

10 260 T. Furuta / Linear Algebra and its Applications 375 (2003) Propositions to prove the results in Section 2 We prepare the following propositions to prove the results in Section 2. Proposition 3.. Let Φ be a normalized positive linear map on B(H). Then dφ(f (p)) df(p) = Φ holds for any real number p, (3.) dp dp where f(p)is a differentiable function of real number p. In particular dφ(a p ) dp ( da p ) = Φ = Φ(A p log A) holds for any real number p/= 0. dp (3.2) Proof. As Φ is a normalized positive linear map on B(H), we have dφ(f (p)) dp Φ(f (p + p)) Φ(f (p)) = lim p 0 p f(p+ p) f( p) = lim Φ p 0 p df(p) = Φ. dp by linearity of Φ Proposition 3.2 (i) lim p 0 (X p Y) = X 2 log ( X 2 YX )] 2 X 2 = S(X Y) for X>0and Y>0 where S(X Y)is defined in (.8), (ii) lim p (X p Y) = (X Y )X S(X Y)X (X Y ) = T(X Y) for X>0 and Y>0where T(X Y)is defined in (.9), (iii) S(I Y) = log Y and T(I Y) = Y log Y for Y>0, (iv) S(X I) = Xlog X and T(X I) = log X for X>0, (v) (X(p) Y(p)) = X (p) Y(p)+ X(p) Y (p) where X(p) and Y(p) are operator functions of real number p. Proof (i) lim (X p Y) = lim X ( ) ] 2 X 2 YX 2 px 2 p 0 p 0 = lim X 2 ( ) p (X ) ] ] X 2 YX 2 log 2 YX 2 X 2 p 0 = S(X Y) and (i) is shown in 6], we cite its proof for the sake of convenience.

11 (ii) T. Furuta / Linear Algebra and its Applications 375 (2003) lim (X p Y) = lim X ( ) ] 2 X 2 YX 2 px 2 p p = lim X ( ) p (X ) ] ] 2 X 2 YX 2 log 2 YX 2 X 2 p = X 2 ( ) (X ) ] ] ( ) X 2 YX 2 2 log 2 YX 2 X 2 YX 2 2 X 2 = X 2 ( ) X 2 YX 2 2 X 2 X X 2 (X ) ] ] log 2 YX 2 X 2 X X ( ) 2 X 2 YX 2 2 X 2 = (X Y )X S(X Y)X (X Y ) by (.8) and (.9). (iii) and (iv) are immediate consequence of (i) and (ii). (v) If U is the isometry of H into H H such that Ue n = e n e n,wheree n is fixed normal basis of H, then the Hadamard product A B of operators A and B on H is expressed in 5, Theorem ] as follows: A B = U (A B)U. (3.3) Then we have (X(p) Y(p)) = U (X(p) Y(p)) U by (3.3) = U (X(p) Y (p))u + U (X(p) Y(p) )U = X (p) Y(p)+ X(p) Y (p) by (3.3). Proposition 3.3. Let h>and let f(p)be defined by: ( f(p)= hp h h p ) p p + ( p) for p 0, ]. h p(h ) Then the following (i) (v) hold. (i) f(0) = lim f(p)= 0, p 0 (ii) f() = lim f(p)= 0, p (iii) f(p)= hp h h ( K(p) p (iv) f (0) = lim p 0 f (p) = log S(), (v) f () = lim p f (p) = ) 0 for all p 0, ], h log h h (S() ). Proof (i) and (ii) are obvious by L Hospital theorem.

12 262 T. Furuta / Linear Algebra and its Applications 375 (2003) (iii) f(p)= hp h h = hp h h = hp h h = hp h h ( h p + ( p) p(h ) ( + ( p) ( (h ) ( h p h h ) ( K(p) p 0 ) p p ( h p h p ) ( h p ) p ) p p(h ) ) p ( h p p ) ) p p and the inequality holds since hp h h 0and K(p) p 0forp 0, ] by K(p) forp 0, ] 4, Theorem ]. (iv) lim f h p log h ) (p) = lim ( K(p) p p 0 p 0 h h p h + lim p 0 h = log h h ( K(0) ( K(p) p ) by (iii) ) h + lim p 0 h ( ) K(p) p = 0 K (0) by using (.5) of basic property = log S(), (v) lim f h p log h ) (p) = lim ( K(p) p p p h + lim p h p h h = h log h ( S()) + 0 h h log h = (S() ). h ( K(p) p ) by (iii) by (.5) of basic property Proposition 3.4. Let h and m>0. Letg(p) be defined by: ( g(p) = m p h p ( h h p ) p ) p + ( p) for p 0, ]. h p(h ) Then the following (i) (v) hold: (i) g(0) = lim p 0 g(p) = 0,

13 T. Furuta / Linear Algebra and its Applications 375 (2003) (ii) g() = lim g(p) = 0, p (iii) g(p) 0 for all p 0, ], (iv) g (0) = lim g (p) = log S(), p 0 (v) g () = lim p g (p) = mh log h h (S() ). Proof. As g(p) = m p f(p),wheref(p)is the same as in Proposition 3.3. (i), (ii) and (iii) are obvious by (i), (ii) and (iii) of Proposition 3.3 respectively. (iv) lim g (p) = lim (m p (log m)f (p) + m p f (p)) p 0 p 0 = f(0) log m + f (0) = log S() by (i) and (iv) of Proposition 3.3. (v) lim g (p) = lim (m p (log m)f (p) + m p f (p)) p p = mf () log m + mf () h log h = 0 + m (S() ) by (ii) and (v) of Proposition 3.3 h mh log h = (S() ). h 4. Proofs of the results in Section 2 For simplicity, F () means F ( 0) and F (0) means F (+0) andsoon. Proof of Theorem 2.. Applying basic property to Theorem D, we shall show Theorem 2.. Recall the following (4.) for S>0andT>0 S T = T and S 0 T = S, (4.) since S p T is defined by S p T = S 2 (S 2 TS 2 ) p S 2 for any p 0, ]. Define F(p)and G(p) by as follows: F(p) = Φ(A) p Φ(B) Φ(A p B) and G(p) = Φ(A p B) K(p)Φ(A) p Φ(B). Recall the following (4.2) by (3.) of Proposition 3. Φ(A p B)] = Φ(A p B) ]. (4.2) (i) As F() = Φ(A) Φ(B) Φ(A B) = Φ(B) Φ(B) = 0 by (4.) and F(p) 0forallp (0, ) by (i) of Theorem D, so F () 0, that is,

14 264 T. Furuta / Linear Algebra and its Applications 375 (2003) F () =Φ(A) p Φ(B)] p= Φ(A pb)] p= =Φ(A) p Φ(B)] p= Φ(A pb) p= ] by (4.2) = T(Φ(A) Φ(B)) Φ(T (A B)) by (ii) of Proposition 3.2 and we have the second inequality. As G() = Φ(A B) K()Φ(A) Φ(B) = Φ(B) K()Φ(B) = 0 by (4.) and K() = by (.4), and G(p) 0forallp (0, ) by (i) of Theorem D, so G () 0, that is, 0 G () =Φ(A p B)] p= K ()Φ(A) p Φ(B)] p= K()Φ(A) p Φ(B)] p= = Φ((A p B) p= ) K ()Φ(A) p Φ(B)] p= K()Φ(A) p Φ(B)] p= by (4.2) = Φ(T (A B)) log S()Φ(A) Φ(B)] T(Φ(A) Φ(B)) = Φ(T (A B)) log S()Φ(B) T(Φ(A) Φ(B)) by (ii) of Proposition 3.2, K() = by (.4), (4.) and basic property (.3), and we have the first inequality. (ii) We have only to show the first inequality since the second one is shown in (i). Define H(p)as follows H(p) = Φ(A p B) Φ(A) p Φ(B) + g(p)φ(a). As H() = Φ(A B) Φ(A) Φ(B)+g()Φ(A) = Φ(B) Φ(B)+g()Φ(A) = 0sinceg() = 0 by (ii) of Proposition 3.4 and H(p) 0forallp (0, ) by (ii) of Theorem D, 0 H () =Φ(A p B)] p= Φ(A) pφ(b)] p= + g (p) p= Φ(A) = Φ((A p B) p= ) Φ(A) pφ(b)] p= + g ()Φ(A) by (4.2) = Φ(T (A B)) T(Φ(A) Φ(B)) + g ()Φ(A) by (ii) of Proposition 3.2 and we have the desired inequality by (v) of Proposition 3.4 mh log h (S() )Φ(A) + T(Φ(A) Φ(B)) Φ(T (A B)). h (iii) As F(0) = Φ(A) 0 Φ(B) Φ(A 0 B) = Φ(A) Φ(A) = 0 by (4.) and F(p) 0forallp (0, ) by (i) of Theorem D, so F (0) 0, that is, 0 F (0) =Φ(A) p Φ(B)] p=0 Φ(A pb)] p=0 =Φ(A) p Φ(B)] p=0 Φ( (A p B) ) p=0 = S(Φ(A) Φ(B)) Φ(S(A B)) by (4.2)

15 T. Furuta / Linear Algebra and its Applications 375 (2003) by (i) of Proposition 3.2 and we have second inequality. Next we have G(0) = Φ(A 0 B) K(0)Φ(A) 0 Φ(B) = Φ(A) K(0)Φ(A) = 0 by (4.) and K(0) = by (.4) and G(p) 0forallp (0, ) by (i) of Theorem D, so G (0) 0, that is, 0 G (0) =Φ(A p B)] p=0 K (0)Φ(A) p Φ(B)] p=0 K(0)Φ(A) p Φ(B)] p=0 = Φ((A p B) p=0 ) K (0)Φ(A) p Φ(B)] p=0 K(0)Φ(A) p Φ(B)] p=0 by (4.2) = Φ(S(A B)) + log S()Φ(A) 0 Φ(B)] S(Φ(A) Φ(B)) = Φ(S(A B)) + log S()Φ(A) S(Φ(A) Φ(B)) by (i) of Proposition 3.2, K(0) = by (.4), (4.) and basic property (.3) and we have the first inequality. Proof of Corollary 2.2. Put A = I in Theorem 2.. Then Φ(I) = I and (i) of Theorem 2. implies the following under the hypotheses of Theorem 2. log S()Φ(B) + T(I Φ(B)) Φ(T (I B)) T(I Φ(B)) and this can be rewritten as follows by (iii) of Proposition 3.2 log S()Φ(B) + Φ(B) log Φ(B) Φ(B log B) Φ(B) log Φ(B) so we have (i) of Corollary 2.2 replacing B by A, and (ii) of Corollary 2.2 is easily shown by the same way as (i). Also (iii) of Theorem 2. implies the following log S() + Φ(S(I B)) S(I Φ(B)) Φ(S(I B)), also this can be rewritten as follows by (iii) of Proposition 3.2 log S() + Φ(log B) log Φ(B) Φ(log B) so we have (iii) of Corollary 2.2 replacing B by A. Remark 4.. We remark that we can show an easy direct proof of Corollary 2.2 applying basic property to Theorem A. Proof of Theorem 2.3. Applying basic property to Theorem C, we shall show Theorem 2.3. Define f(p)and g(p) defined by F(p) = (A B) p (C D) (A p C) (B p D) and G(p) = (A p C) (B p D) K(p)(A B) p (C D).

16 266 T. Furuta / Linear Algebra and its Applications 375 (2003) (i) We have the following (4.3) F() = (A B) (C D) (A C) (B D) = C D C D = 0 by (4.). (4.3) As F() = 0 by (4.3) and F(p) 0forallp (0, ) by (i) of Theorem C, so F () 0, that is, 0 F () =(A B) p (C D)] p= (A pc) (B p D)] p= = T(A B) (C D) (A p C) (B p D)] p= (A p C) (B p D) ] p= = T(A B) (C D) T(A C) D C T(B D) by (ii) and (v) of Proposition 3.2 and (4.) and the second inequality holds. Also we have G() = (A C) (B D) K()(A B) (C D) = C D C D = 0, (4.4) since K() = by (.4) and (4.) holds. As G() = 0 by (4.4) and G(p) 0forall p (0, ) by (i) of Theorem C, so G () 0, that is, 0 G () =(A p C) (B p D)] p= K ()(A B) (C D) K()(A B) p (C D)] p= =(A p C) (B p D)] p= log S()(C D) T(A B C D) =(A p C)] p= (B D) + (A C) (B p D)] p= log S()(C D) T(A B C D) = T(A C) D + C T(B D) log S()(C D) T(A B C D) by (ii) and (v) of Proposition 3.2, (4.) and basic property (.3), and the first inequality holds. (ii) We have only to show the first inequality since the second one is shown in (i). Define H(p)as follows: H(p) = (A p C) (B p D) (A B) p (C D) + g(p)(a B), H() = (A C) (B D) (A B) (C D) + g()(a B) = C D C D + 0 = 0, (4.5) since g() = 0 by (ii) of Proposition 3.4 and (4.). As H() = 0 by (4.5) and H(p) 0forallp (0, ) by (ii) of Theorem C, so we have

17 T. Furuta / Linear Algebra and its Applications 375 (2003) H () =(A p C) (B p D)] p= (A B) p(c D)] p= + g ()(A B) =(A p C)] p= (B D)]+(A C)] (B p D)] p= (A B) p (C D)] p= + g ()(A B) = T(A C) D + C T ((B D) T(A B C D)) + g ()(A B) by (ii) and (v) of Proposition 3.2 and (4.), that is, we have the desired inequality by (v) of Proposition 3.4 mh log h (S() )(A B)+T(A B C D) T(A C) D+C T(B D). h (iii) We have F(0) = (A B) 0 (C D) (A 0 C) (B 0 D) = A B A B = 0 by (4.). (4.6) As F(0) = 0 by (4.6) and F(p) 0forallp (0, ) by (i) of Theorem C, so F (0) 0, that is, 0 F (0) =(A B) p (C D)] p=0 (A pc) (B p D)] p=0 = S(A B C D) (A p C) (B p D)] p=0 (A p C) (B p D) ] p=0 = S(A B C D) S(A C) B A S(B D) by (i) and (v) of Proposition 3.2 and (4.) and the second inequality holds. Also we have G(0) = (A 0 C) (B 0 D) K(0)(A B) 0 (C D) = A B A B = 0, (4.7) since K(0) = by (.4) and (4.). As G(0) = 0 by (4.7) and G(p) 0forallp (0, ) by (i) of Theorem C, so G (0) 0, that is, 0 G (0) =(A p C) (B p D)] p=0 K (0)(A B) 0 (C D) K(0)(A B) p (C D)] p=0 =(A p C) (B p D)] p=0 + log S()(A B) S(A B C D) =(A p C)] p=0 (B 0D) + (A 0 C) (B p D)] p=0 + log S()(A B) S(A B C D) = S(A C) B +A S(B D)+log S()(A B) S(A B C D).

18 268 T. Furuta / Linear Algebra and its Applications 375 (2003) By (i) and (v) of Proposition 3.2, (4.) and basic property (.3), and the first inequality holds. Proof of Corollary 2.4. Put A = B = I in Theorem 2.3. Then (i) of Theorem 2.3 implies the following under the hypotheses of Theorem 2.3 log S()(C D) + T(I I C D) T(I C) D + C T(I D) T(I I C D) and this can be rewritten as follows by (iii) of Proposition 3.2 log S()(C D) + (C D) log(c D) (C log C) D + C (D log D) (C D) log(c D) so we have (i) of Corollary 2.4 replacing C and D by A and B, and (ii) of Corollary 2.4 is easily shown by the same way as (i). Also (iii) of Theorem 2.3 implies the following log S()(I I)+ S(I C) I + I S(I D) S(I I C D) S(I C) I + I S(I D), also this can be rewritten as follows by (iii) of Proposition 3.2 log S() + (log C) I + I (log D) log(c D) (log C) I + I (log D) so we have (iii) of Corollary 2.4 replacing C and D by A and B. Remark 4.2. We remark that we can show an easy direct proof of Corollary 2.4 applying basic property to Theorem B. Proof of Theorem 2.5. Applying basic property to Theorem E, we shall show Theorem 2.5. Define F(p)and G(p) by as follows: and F(p) = (A I) p (B I) p A p B p G(p) = A p B p K(p)(A I) p (B I) p.

19 and T. Furuta / Linear Algebra and its Applications 375 (2003) Recall the following by (v) of Proposition 3.2 F (p) = (A I) p log(a I)](B I) p (A I) p (B I) p log(b I) (A p log A) B p + A p (B p log B) G (p) = (A p log A) B p A p (B p log B) K (p)(a I) p (B I) p K(p)(A I) p log(a I)](B I) p + K(p)(A I) p (B I) p log(b I). (i) As F() = 0andF(p) 0forallp (0, ) by (i) of Theorem E, so F () 0, that is, 0 F () = (A I)log(A I) (A I)log(B I) (A log A) I + A (log B) and the second inequality holds. On the other hand, As G() = 0sinceK() = by (.4) and G(p) 0forallp (0, ) by (i) of Theorem E, so G () 0, that is, 0 G () = (A log A) I A log B K ()(A I) K()(A I)log(A I)]+K()(A I)log(B I) and the first inequality holds since K () = log S() by (.3) and K() = by (.4), so we have (i). (ii) We have only to show the first inequality of (ii) since the second one is shown in (i). Define H(p)as follows H(p) = A p B p (A I) p (B I) p + g(p)(b p I). Recall the following by (v) of Proposition 3.2 H (p) = (A p log A) B p A p (B p log B) (A I) p log(a I)](B I) p + (A I) p (B I) p log(b I) + g (p)(b p I) g(p)((b p log B) I). As H() = 0sinceg() = 0 by (ii) of Proposition 3.4 and H(p) 0forallp (0, ) in (ii) of Theorem E, so 0 H () = (A log A) I A log B (A I)log(A I)] + (A I)log(B I)+ g ()(I I) g()((log B) I), that is, we have the desired result by (v) of Proposition 3.4 since g() = 0 by (ii) of Proposition 3.4

20 270 T. Furuta / Linear Algebra and its Applications 375 (2003) mh log h (S() ) + A log B + (A I)log(A I) h (A log A) I + (A I)log(B I). (iii) As F(0) = 0andF(p) 0forallp (0, ) by (i) of Theorem E, so F (0) 0, that is, 0 F (0) =log(a I)](B I) (B I)log(B I) (log A) B + I (B log B) and the second inequality holds. On the other hand, As G(0) = 0andG(p) 0for all p (0, ) by (i) of Theorem E, so G (0) 0, that is, 0 G (0) = (log A) B I (B log B) K (0)(B I) K(0)log(A I)](B I)+ K(0)(B I)log(B I) and the first inequality holds since K (0) = log S() by basic property (.3) and K(0) = by (.4), so we have (iii). Whence the proof is complete. 5. Parallel results to Section 2 and related remarks We state an extension of Kantorovich inequality. Theorem F. Let A be strictly positive operator satisfying MI A mi > 0, where M>m>0.Puth = M m >. Then the following inequalities (i) (iii) hold for every unit vector x and follow from each other: (i) K(h, p)(ax, x) p (A p x,x) (Ax, x) p (ii) (Ax, x) p (A p x,x) K(h, p)(ax, x) p (iii) K(h, p)(ax, x) p (A p x,x) (Ax, x) p for any p>. for any >p>0. for any p<0. We remark that the latter half inequality in (i) or (iii) of Theorem F and the former half one of (ii) are called Hölder McCarthy inequality and the former one of (i) or (iii) and the latter half one of (ii) can be considered as generalized Kantorovich inequality and the reverse inequalities to Hölder McCarthy inequality. (i) and (iii) are in ] and the equivalence relation among (i) (iii) is shown in 4, Theorem 3] and several extensions of Theorem F are shown, for example, 7, Theorem 3.2]. Related results to Theorem F and operator inequalities associated with Kantorovich type inequalities are in Chapter III of 2]. In this section we sum up the following results which are obtained as applications of basic property and they are parallel results to Sections and 2.

21 T. Furuta / Linear Algebra and its Applications 375 (2003) Theorem G (3]). Let A be strictly positive operator satisfying MI A mi > 0, where M>m>0. Put h = M m >. Then the following inequalities hold for every unit vector x: (i) log S()](Ax, x) + (Ax, x) log(ax, x) ((A log A)x, x) (Ax, x) log(ax, x). (ii) (iii) mh log h (S() ) + (Ax, x) log(ax, x) h ((A log A)x, x) (Ax, x) log(ax, x). log S()]+((log A)x, x) log(ax, x) ((log A)x, x). Theorem H (5]). Let A j be strictly positive operator satisfying MI A j mi > 0 for j =, 2,...,n, where M>m>0and h = M m >.Alsoλ,λ 2,...,λ n be any positive numbers such that n j= λ j =. Then the following inequalities hold: n n n (i) log S()] λ j A j + λ j A j log λ j A j (ii) (iii) j= n λ j A j log A j j= j= j= n n λ j A j log λ j A j. j= j= j= mh log h n n (S() ) + λ j A j log λ j A j h n λ j A j log A j j= n n λ j A j log λ j A j. j= log S()]+ j= j= j= k k λ j log A j log λ j A j j= k λ j log A j. j=

22 272 T. Furuta / Linear Algebra and its Applications 375 (2003) We remark (iii) for n = 2 of Theorem H is shown in 9]. The following interesting result is shown in 6]. Theorem I. Let A be strictly positive operator satisfying MI A mi > 0.Also let h = M m >. Then the following inequality holds for every unit vector x: S() x (A) (Ax, x) x (A). where x (A) for strictly positive operator A at a unit vector x is defined by x (A) = exp ((log A)x, x). x (A) is defined in 8]. We remark that (ii) of Theorem F implies Theorem I via basic property. In fact (ii) of Theorem F ensures (Ax, x) (A p x,x) p K(h, p) p (Ax, x) for any >p>0. (5.) and is easily verified that lim p 0 (A p x,x) p = x (A) and lim p 0 K(h, p) p = S() by (.5), so that (5.) implies Theorem I. Interesting closely related results to Theorems G and H are in 24]. References ] T. Ando, Concavity of certain maps on positive definite matrices and applications to Hadamard products, Linear Algebra Appl. 26 (979) ] J.S. Aujla, H.L. Vasudeva, Inequalities involving Hadamard product and operator means, Math. Japon. 42 (995) ] M.D. Choi, A Schwarz inequality for positive linear maps on C -algebras, Illinois J. Math. 8 (974) ] C. Davis, A Schwarz inequality for convex operator functions, Proc. Amer. Math. Soc. 8 (957) ] J.I. Fujii, The Marcus Khan theorem for Hilbert space operators, Math. Japon. 4 (995) ] J.I. Fujii, S. Izumino, Y. Seo, Determinant for positive operators and Specht s theorem, Sci. Math. (998) ] J.I. Fujii, E. Kamei, Relative operator entropy in noncommutative information theory, Math. Japon. 34 (989) ] J.I. Fujii, Y. Seo, Determinant for positive operators, Sci. Math. (998) ] M. Fujii, Y. Seo, M. Tominaga, Golden Thompson type inequalities related to a geometric mean via Specht s ratio, Math. Inequal. Appl. 5 (2002) ] T. Furuta, Hadamard product of positive operators, circulated note in 995. ] T. Furuta, Operator inequalities associated with Hölder McCarthy and Kantorovich inequalities, J. Inequal. Appl. 2 (998) ] T. Furuta, Invitation to Linear Operators, Taylor & Francis, London, ] T. Furuta, Specht ratio S() can be expressed by Kantorovich constant K(p): S() = exp dk(p) dp ] p= and its application, Math. Inequal. Appl., in press. 4] T. Furuta, Basic property of generalizd Kantorovich constant K(h, p) = (h p ( h) (p ) h p ) p (p )(h ) p (h p h) and its applications, preprint. 5] T. Furuta, J. Pečarić, An operator inequality associated with the operator concavity of operator entropy A log A, Math. Inequal. Appl., in press.

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