NORM INEQUALITIES FOR THE GEOMETRIC MEAN AND ITS REVERSE. Ritsuo Nakamoto* and Yuki Seo** Received September 9, 2006; revised September 26, 2006

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1 Scientiae Mathematicae Japonicae Online, e-2006, NORM INEQUALITIES FOR THE GEOMETRIC MEAN AND ITS REVERSE Ritsuo Nakamoto* and Yuki Seo** Received September 9, 2006; revised September 26, 2006 Abstract. If two positive operators A and B commute, then A B = A B for all 0. In this note, we prove a norm inequality for the geometric mean A B and its reverse inequality: Let A and B be positive operators on a Hilbert space such that 0 <m A, B M for some scalars 0 <m<m and h = M. Then for each m 0 K(h 2,) A B A B A B, where K(h, ) is a generalized Kantorovich constant. Introduction. Let A and B be two positive operators on a Hilbert space. The arithmetric-geometric mean inequality says that () ( )A + B A B for all 0, where the -geometric mean A B is defined as follows: ( ) A B = A 2 A 2 BA (2) 2 A 2 for all 0. On the other hand, Ando [] proved the Matrix Young inequality: For positive semidefinite matrices A, B and p + q = (3) p Ap + q Bq U AB U for some unitary matrix U. By (3), for positive semi-definite matrices A, B (4) ( )A + B A B for all 0 and by () we have ( )A + B A B for 0 and A, B 0. Here we remark that McIntosh [6] proved that (4) for =/2 holds for positive operators. In this note, we prove a norm inequality and its reverse on the geometric mean. In other words, we estimate A B by A B as mentioned in the abstract. Moreover we discuss it for the case >. Our main tools are Araki s inequality [2] and its reverse one [4] Mathematics Subject Classification. 47A63 and 47A64. Key words and phrases. Positive operator, Araki s inequality, geometric mean, generalized Kantorovich constant, reverse inequality.

2 20 RITSUO NAKAMOTO AND YUKI SEO 2 Norm inequalities. First of all, we cite Araki s inequality [2]: Theorem A. If A and B are positive operators, then (5) BAB p B p A p B p for all p> or equivalently (6) B p A p B p BAB p for all 0 <p<. As seen in [3], it is equivalent to the Cordes inequality A p B p AB p for all 0 <p<. We show the following norm inequality, in which we use Theorem A twice: Theorem. Let A and B be positive operators. Then for each 0 (7) A B A B. Proof. It follows from (6) of Theorem A that ( ) A 2 BA 2 A 2 A 2 A 2 BA 2 A A 2 for 0. Furtheremore, if /2, then by (6) of Theorem A again 2 = A 2 A 2 BA 2 A B 2 A 2 = A B. BA 2 Hence, if /2, then we have the desired inequality (7). If </2, then by using A B = B A, it reduces the proof to the case /2 and so the proof is complete. (8) We use also the notation to distingush from the operator mean ; ( ) A B = A 2 A 2 BA 2 A 2 for all [0, ]. Theorem 2. Let A and B be positive operators. If 3/2 2, then (9) A B A B. Proof. Put =+ and /2. Then we have ( ) A B = B A = B 2 B 2 AB ( ) = B 2 B 2 A B B 2 B 2 A B 2 B 2 = B + 2 A B + 2 by (6) and /2 B + A 2 B + 2 by (6) and 0 < 2 = A B + = A B.

3 NORM INEQUALITIES FOR THE GEOMETRIC MEAN AND ITS REVERSE 2 Remark 3. In Theorem 2, the inequality ( ) A B A ( ) B does not always hold for 2 2 <<3/2. In fact, Let A = and B =. Then we have A 4 2 B = > A 3 B 4 3 = Also, A 7 B = < A 2 5 B 7 5 = Reverse type inequalities. In this section, we show reverse inequalities of the results obtained in the previous section. In order to prove our results, we need some preliminaries. For h>0, a generalized Kantorovich constant K(h, p) is defined by h p ( h p h p ) p (0) K(h, p) = (p )(h ) p h p h for any real numbers p R. We state some properties of K(h, p) (see [5, Theorem 2.54]): Lemma 4. Let h>0 be given. Then a generalized Kantorovich constant K(h, p) has the following properties. (i) K(h, p) =K(h,p) for all p R. (ii) K(h, p) =K(h, p) for all p R. (iii) K(h, 0) = K(h, ) = and K(,p)=for all p R. (iv) K(h r, p r ) p = K(h p, r p ) r for pr 0. The following theorem is reverse inequalities of Araki s inequality [4]. Theorem B. If A and B are positive operators such that 0 <m A M for some scalars 0 <m<m, then () B p A p B p K(h, p) BAB p for all p> or equivalently (2) K(h, p) BAB p B p A p B p for all 0 <p<, is a gener- where a generalized Kantorovich constant K(h, p) is defined by (0) and h = M m alized condition number of A in the sense of Turing [7]. We show the following reverse inequality for Theorem : Theorem 5. If A and B are positive operators such that 0 <m A, B M for some scalars 0 <m<m and h = M m, then for each 0 (3) K(h 2,) A B A B. Proof. Suppose that 0 2. Since m M A 2 BA 2 condition number of A 2 BA 2 is M m / m M = h2 and we have ( ) A B = (A 2 ) A 2 BA 2 (A 2 ) K(h 2,) A 2 A 2 BA 2 A 2 = K(h 2,) A 2 BA 2 M m, it follows that a generalized by (2) and 0 2 K(h 2,) A B 2 A 2 by (5) and 2 = K(h 2,) A B.

4 22 RITSUO NAKAMOTO AND YUKI SEO Suppose that 2. Since 0 2, we have and so the proof is complete. A B = B A K(h 2, ) B ( ) A = K(h 2,) A B by (ii) of Lemma 4 We show the following reverse inequality for Theorem 2: Theorem 6. If A and B are positive operators such that 0 <m A, B M for some scalars 0 <m<m and h = M m, then for each K(h 2, )K(h, 2( )) 2 A B A B. Proof. Put =+ and /2. Then we have A B = B A ( ) = B 2 B 2 A B K(h 2,) B 2 B 2 A B 2 B 2 = K(h 2,) B + 2 ( K(h 2,) K(h 2, = K(h 2,)K(h 2, A B 2 = K(h 2,)K(h, 2) 2 A B. The last equality follows from K(h 2, by (i) and (iv) of Lemma 4. by (2) of Theorem B and /2 ) 2 ) B+ A 2 B + 2 by (2) and 0 < 2 2 ) A B + 2 ) = K(h 2, 2 ) = K(h, 2) 2 = K(h, 2) 2 As mentioned in Remark 3, we have no relation between A B and A B for 3 2. We have the following result: Theorem 7. If A and B are positive operators such that 0 <m A, B M for some scalars 0 <m<m and h = M m, then for each 3 2 K(h 2, ) A B A B K(h, 2( )) 2 A B.

5 NORM INEQUALITIES FOR THE GEOMETRIC MEAN AND ITS REVERSE 23 Proof. Put =+ and 0 2. Since a generalized condition number of A 2 is h 2, it follows that Also, we have ( ) A B = B A = B 2 B 2 AB ( ) = B 2 B 2 A B B 2 B 2 A B 2 B 2 by (6) and 0 = B + 2 A B + 2 ( ) K(h 2, 2 ) B+ A 2 B + 2 by () and 2 = K(h, 2( )) 2 A B by (i) and (iv) of Lemma 4. ( ) A B = B A = B 2 B 2 AB ( ) = B 2 B 2 A B and so the proof is complete. K(h 2,) B 2 B 2 A B 2 B 2 = K(h 2,) B + 2 A B + 2 by (2) and 0 K(h 2,) B + A 2 B + 2 by (5) and 2 = K(h 2, ) A B Finally, we consider the case of 2: Theorem 8. If A and B are positive operators such that 0 <m A, B M for some scalars 0 <m<m and h = M m, then for each 2 K(h, 2( )) 2 A B A B K(h 2, ) A B. Proof. Put =+ and. Then we have ( ) A B = B A = B 2 B 2 AB ( ) = B 2 B 2 A B K(h 2,) B 2 B 2 A B 2 B 2 = K(h 2,) B + 2 A B + 2 by () and K(h 2, ) A B by (6) and 0 < 2.

6 24 RITSUO NAKAMOTO AND YUKI SEO Also, it follows that ( ) A B = B A = B 2 B 2 AB ( ) = B 2 B 2 A B B 2 B 2 A B 2 B 2 by (5) and = B + 2 A B + 2 K(h 2, 2 ) B + A 2 B + 2 by () and 0 < 2 = K(h, 2( )) 2 A B by (i) and (iv) of Lemma 4. References [] T.Ando, Matrix Young inequalities, Operator Theory: Adv. and Appl., 75 (995), [2] H.Araki, On an inequality of Lieb and Thirring, Letters Math. Phys., 9 (990), [3] M.Fujii, T.Furuta and R.Nakamoto, Norm inequalities in the Corach-Recht theory and operator means, Illinois J. Math., 40 (996), [4] M.Fujii and Y.Seo, Reverse inequalities of Araki, Cordes and Löwner-Heinz inequalities, Nihonkai Math. J., 6 (2005), [5] T.Furuta, J.Mićić, J.E.Pečarić and Y.Seo, Mond-Pečarić Method in Operator Inequalities, Monographs in Inequalities, Element, Zagreb, [6] A.McIntosh, Heinz inequalities and perturbation of spectral families, Macquarie Math. Reports, 979. [7] A.M.Turing, Rounding off-errors in matrix processes, Quart. J. Mech. Appl. Math., (948), * Faculty of Engineering, Ibaraki University, Hitachi, Ibaraki 36, Japan. address : nakamoto@mxibaraki.ac.jp ** Faculty of Engineering, Shibaura Institute of Technology, 307 Fukasaku, Minumaku, Saitama-City, Saitama , Japan. address : yukis@sic.shibaura-it.ac.jp