Banach Journal of Mathematical Analysis ISSN: (electronic)

Transcription

1 Banach J. Math. Anal ), no., 87 9 Banach Jounal of Mathematical Analysis ISSN: electonic) 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 Mathematics Subject Classification. Pimay 47A6; Seconday 47A0, 47A64. Key wods and phases. Ando Hiai inequaqlity, positive opeato, geometic mean. 87

2 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 = ) Then we have since σa A 2 A B = 25 ) I 5 B) = {, 0.4}. On the othe hand, since B 2 = ) and σa 2 B = {.07, }, 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.

3 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

4 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, T. Fuuta, Extension of the Fuuta inequality and Ando Hiai log-majoization, Linea Algeba Appl ), 9 55.

5 ON A REVERSE OF ANDO HIAI INEQUALITY 9 4. T. Fuuta, Opeato inequalities associated with Hölde-McCathy and Kantoovich inequalities, J. Inequal. Appl ), T. Fuuta, J. Mićić, J.E. Pečaić and Y. Seo, Mond-Pečaić Method in Opeato Inequalities, Monogaphs in Inequalities, Element, Zageb, F. Kubo and T. Ando, Means of positive linea opeatos, Math. Ann ), R. Nakamoto and Y. Seo, A complement of the Ando Hiai inequality and nom inequalities fo the geometic mean, Nihonkai Math. J ), Faculty of Engineeing, Shibaua institute of Technology, 07 Fukasaku, Minuma-ku, Saitama-city, Saitama , Japan. addess: yukis@sic.shibaua-it.ac.jp