Banach J. Math. Anal. 4 200), no., 87 9 Banach Jounal of Mathematical Analysis ISSN: 75-8787 electonic) www.emis.de/jounals/bjma/ ON A REVERSE OF ANDO HIAI INEQUALITY YUKI SEO This pape is dedicated to Pofesso Las-Eik Pesson Communicated by M. Fujii Abstact. In this pape, we show a complement of Ando Hiai inequality: Let A and B be positive invetible opeatos on a Hilbet space H and α [0, ]. If A α B I, then A α B A α B) I fo all 0 <, whee I is the identity opeato and the symbol stands fo the opeato nom.. Intoduction A bounded linea) opeato A on a Hilbet space H is said to be positive in symbol: A 0) if Ax, x) 0 fo all x H. In paticula, A > 0 means that A is positive and invetible. Fo some scalas m and M, we wite mi A MI if mx, x) Ax, x) Mx, x) fo all x H. The symbol stands fo the opeato nom. Let A and B be two positive opeatos on a Hilbet space H. Fo each α [0, ], the weighted geometic mean A α B of A and B in the sense of Kubo-Ando [6] is defined by ) α A α B = A 2 A 2 BA 2 A 2 if A is invetible. In fact, the geometic mean A 2 of XA X = B. B is a unique positive solution Date: Received: 6 Septembe 2009; Accepted: 7 Febuay 200. 2000 Mathematics Subject Classification. Pimay 47A6; Seconday 47A0, 47A64. Key wods and phases. Ando Hiai inequaqlity, positive opeato, geometic mean. 87
88 Y. SEO To study the Golden-Thompson inequality, Ando Hiai in [] developed the following inequality, which is called Ando Hiai inequality: Let A and B be positive invetible opeatos on a Hilbet space H and α [0, ]. Then o equivalently A α B I = A α B I fo all, AH) A α B A α B fo all. Löwne Heinz inequality assets that A B 0 implies A B fo all 0. As compaed with Löwne Heinz inequality, Ando Hiai inequality is ephased as follows: Fo each α [0, ] A /2 B A /2) α A = A /2 BA /2) α A fo all 0 <..) Now, Ando Hiai inequality does not hold fo 0 < in geneal. In fact, put = /2, α = / and A = ) 2 and B = ) 45 + 4 5 5 7 5 5 25 5 7 5 50 4. 5 Then we have since σa A 2 A B = 25 ) 5 + 2 5 5 5 5 5 20 2 I 5 B) = {, 0.4}. On the othe hand, since B 2 = 0.86602 0.8700 0.8700 0.77068 ) and σa 2 B = {.07, 0.62547}, we have A 2 B 2 I. Thus, in [7], Nakamoto and Seo showed the following complement of Ando Hiai inequality AH): Theoem A. Let A and B be positive opeatos such that mi A, B MI fo some scalas 0 < m < M, h = M and α [0, ]. Then m A α B I = A α B Kh 2, α) I fo all 0 <, whee the genealized Kantoovich constant Kh, α) is defined by ) h p h p h p p Kh, p) = fo all p R, p )h ) p h p h see [5, 2.79)]. We emak that Kh 2, α) in the case of =, though Kh 2, α) = in the case of α = 0, in Theoem A. Theeby, in this pape, we conside anothe complement of Ando Hiai inequality AH) which diffe fom Theoem A.
ON A REVERSE OF ANDO HIAI INEQUALITY 89 Fist of all, we state the main esult: 2. Main esults Theoem 2.. Let A and B be positive invetible opeatos and α [0, ]. Then A α B I = A α B A α B I fo all 0 <, o equivalently A α B A α B A α B fo all 0 <. We emak that A α B = in the case of =. We need the following lemmas to give a poof of Theoem 2.. Lemma 2.2 is egaded as a evesal of Löwne Heinz inequality: Lemma 2.2. Let A and B be positive invetible opeatos. Then A B = A p 2 B p A p 2 B p A p fo all 0 < p. Poof. This lemma follows fom Löwne Heinz inequality. In fact, A B implies A p B p fo all 0 < p and then I A p 2 B p A p 2 A p 2 B p A p 2. Lemma 2. []). Let A be a positive invetible opeato and B an invetible opeato. Fo each eal numbes BAB ) = BA 2 A 2 B BA A 2 B. Poof of Theoem 2.. If we put C = A 2 BA 2, then the assumption implies A C α. By Lemma 2.2 and 0 < <, we have A = A 2 A A 2 A 2 C α ) A 2 A 2 C α ) A 2. On the othe hand, it follows that A C α implies C α C 2 AC. By Lemma 2.2, we have C 2 AC 2 C α ) ) C 2 AC 2 C α ) ) C 2 AC. Futhemoe, by Lemma 2., we have B = A 2 CA = A 2 C 2 C 2 AC C 2 A 2 C 2 AC 2 C α ) ) C 2 AC 2 A 2 C 2 C α ) ) C 2 A 2. Hence, by Aaki-Codes inequality [2, Theoem IX.2.0], we have C 2 AC 2 C α ) ) C 2 AC 2 C 2 AC 2 C α C 2 AC 2
90 Y. SEO since 0 < <. Let A) be the spectal adius of A. Then we have C 2 AC 2 C α C 2 AC 2 = C 2 AC 2 C α C 2 AC Theefoe, it follows that A α B = C 2 AC C α ) = A C α ) = A 2 C α A A 2 C α A 2. A 2 C α ) A 2 α C 2 AC 2 C α ) ) C 2 AC 2 α ) A 2 C )α A 2 α A 2 C α ) )+ A 2 A 2 C α A 2 ) α) C 2 AC 2 C α C 2 AC 2 )α A 2 C )α α C α ) )+ )A 2 = A 2 C α A 2 A α B A α B) I by C )α α C α ) )+ = C α and the assumption of A α B I. Hence the poof is complete. By Theoem 2., we immediately have the following coollay in the case of. Coollay 2.4. Let A and B be positive invetible opeatos on H. Then A α B A α B A α B fo all. Finally, Fuuta [4] showed the following Knatoovich type opeato inequality in tems of the condition numbe: Let A and B be positive invetible opeatos. Then B A = B B B ) A fo all. 2.) By Theoem 2., we have the following Kantoovich type inequality of.) which coesponds to 2.): Theoem 2.5. Let A and B be positive invetible opeatos and α [0, ]. Then ) A 2 B A α ) 2 A = A α 2 BA 2 A α B A fo all. Refeences. T. Ando and F. Hiai, Log-majoization and complementay Golden-Thompson type inequalities, Linea Algeba Appl. 97/98 994),. 2. R. Bhatia, Matix Analysis, Spinge, New Yok, 997.. T. Fuuta, Extension of the Fuuta inequality and Ando Hiai log-majoization, Linea Algeba Appl. 29 995), 9 55.
ON A REVERSE OF ANDO HIAI INEQUALITY 9 4. T. Fuuta, Opeato inequalities associated with Hölde-McCathy and Kantoovich inequalities, J. Inequal. Appl. 2 998), 7 48. 5. T. Fuuta, J. Mićić, J.E. Pečaić and Y. Seo, Mond-Pečaić Method in Opeato Inequalities, Monogaphs in Inequalities, Element, Zageb, 2005. 6. F. Kubo and T. Ando, Means of positive linea opeatos, Math. Ann. 246980), 205 224. 7. R. Nakamoto and Y. Seo, A complement of the Ando Hiai inequality and nom inequalities fo the geometic mean, Nihonkai Math. J. 8 2007), 4 50. Faculty of Engineeing, Shibaua institute of Technology, 07 Fukasaku, Minuma-ku, Saitama-city, Saitama 7-8570, Japan. E-mail addess: yukis@sic.shibaua-it.ac.jp