394 T. FURUTA AND Y. SEO An alternative roof of Theorem A in [5] and the best ossibility oftheoremaisshown in [3]. Recently a Kantorovich tye characte

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1 Scientiae Mathematicae Vol., No. 3(999), AN APPLICATION OF GENERALIZED FURUTA INEQUALITY TO KANTOROVICH TYPE INEQUALITIES TAKAYUKI FURUTA * AND YUKI SEO ** Dedicated in dee sorrow to the memory of Professor Szökefalvi-Nagy Bela Received Setember, 999 Abstract. As an alication of generalized Furuta ineuality, we shall show a characterization of chaotic order associated with oerator euation and a Kantorovich tye ineuality related to chaotic order which is an extension of Yamazaki and Yanagida [5] by using essentially the same idea of [0].. Introduction. In what follows, a caital letter means a bounded linear oerator on a comlex Hilbert sace H. An oerator T is said to be ositive ( in symbol: T 0) if (Tx;x) 0 for all x H. Also an oerator T is strictly ositive (insymbol: T > 0) if T is ositive and invertible. We obtained the following result. Theorem F (Furuta ineuality)([6]). If A B 0, then for each r 0, r B A B r (i) and (ii) B r B B r r A A A r A r B A r hold for 0 and with ( + r) + r. (0; r) ( + r) = + r 6 = (; ) A AA A A AAA AA AAA AA AA A (; 0) = Figure Alternative roofs of Theorem F have been given in [3], [], and one-age roof in [7]. The domain drawn for ; and r in Figure is the best ossible one [] for Theorem F. As a corollary of [9, Theorem.], we cite the following Theorem A which interolates Theorem F itself and an ineuality euivalent to main theorem of log majorization by Ando-Hiai []. - Theorem A ([9]). If A B 0 with A>0, then for each t [0; ], fa r (A t A A t ) s A r r g fa(a t B A t ) s A r g holds for any s 0, 0, and r t with (s )( ) 0 and ( t + r) ( t)s + r. 99 Mathematics Subject Classification. 47A30 and 47A63. Key words and hrases. Kantorovich ineuality, Furuta ineuality, generalized Furuta ineuality, chaotic order.

2 394 T. FURUTA AND Y. SEO An alternative roof of Theorem A in [5] and the best ossibility oftheoremaisshown in [3]. Recently a Kantorovich tye characterization of chaotic order is shown in [5]. Theorem B ([5]). Let A and B be invertible ositive oerators and M A m > 0. Then the following roerties are mutually euivalent: (I) (II) A fl B (i:e:; log A log B): (M + m ) 4M m A B holds for all 0.. Statement of results. Firstly we shall show the following characterization of chaotic order associated with oerator euation. Theorem. Let A and B be invertible ositive oerators. are mutually euivalent: Then the following roerties (I) A fl B (i.e., log A log B ). (II) For each ff [0; ], 0 and u 0, there exists the uniue invertible ositive contraction T such that (A ffu B A ffu ) s = TA (+ffu)s T holds for any s and ( + ffu)s ( ff)u. (III) For each ff [0; ] and u 0, there exists the uniue invertible ositive contraction T such that (A ffu B A ffu ) s = TA (+ffu)s T holds for any s. (IV) For each 0, there exists the uniue invertible ositive contraction T such that B = TA T: As an alication of Theorem, we obtain the following extension of Theorem B on a Kantorovich tye characterization of chaotic order. Theorem. Let A and B be invertible ositive oerators and M A m>0. Then the following roerties are mutually euivalent: (I) A fl B (i.e., log A log B ). (II) For each ff [0; ], 0 and u 0, (M (+ffu)s + m (+ffu)s ) 4M (+ffu)s m (+ffu)s A (+ffu)s A ffu B A ffu holds for any s and ( + ffu)s ( ff)u. (III) For each ff [0; ] and u 0, holds for any s. (M (+ffu)s + m (+ffu)s ) 4M (+ffu)s m (+ffu)s A (+ffu)s A ffu B A ffu

3 AN APPLICATION OF GENERALIZED FURUTA INEQUALITY 395 (IV) (M + m ) 4M m A B holds for all 0. We cite the following two lemmas in order to give roofs of our results. Lemma 3. Let T be a nonsingular ositive oerator. X 0 and Y 0, thenx = Y. If XTX = YTY holds for some Proof. If XTX = YTY holds for some X; Y 0, then we have (T XT ) =(T YT ), so that T XT = T YT holds and the nonsingularity oft ensures X = Y. Lemma 4. If A is a ositive oerator such that M A m > 0 and B is a ositive contraction, then (M + m) 4Mm A BAB: Proof. By the Kantorovich ineuality, wehave (ABx; Bx)(A Bx;Bx)» KkBxk 4 for any unit vector x H, where K = (M+m) 4Mm. Hence it follows that so the roof is comlete. (ABx; Bx)(A Bx;Bx)» K(B x; x)» K(Bx;x) by I B 0 = K(A Bx;A x)» K(A Bx;Bx)(Ax; x); Remark 5. In Lemma 4, one might conjecture the following (Λ) (Λ) A BAB holds for any ositive oerator A and any ositive contraction B (M+m) 4Mm instead of A BAB. But we can give a counterexamle to this conjecture as follows. Take A and B as follows: 0 A = : 0 Then A 0 and I B 0, but we have and B = A BAB = This Remark 5 is closely related to (II),(III) and (IV) of Theorem (see Remark 6). The following characterization of chaotic order is shown in [4] and [8] :

4 396 T. FURUTA AND Y. SEO Theorem C. Let A and B be invertible ositive oerators. Then the following roerties are mutually euivalent: (I) A fl B (i.e., log A log B ). (II) A A B A holds for all 0. (III) A u A u B A u u +u holds for all 0 and u 0. (I) ()(II) is shown in []. Recently a simle and excellent roof of (I) =)(III) is shown in [4] by only alying Theorem F. Here we cite the following simlifed imlication since (III) =)(II) is trivial. Simlified roof of (II) =)(I) of Theorem C. A (A B A ) I ψ A (B I)A = (II) yields! + A I f(a B A ) + Ig and tending # 0, so we have log A (log B +loga), that is, log A log B. 3. Proofs of the results. Proof of Theorem. (I) =)(II). For each 0 and u 0, ut A = A u and B = (A u B A u ) u +u in (III) of Theorem C. Then we have A B 0. By Theorem A, it follows that for each t [0; ], () ( t)s+r A fa r (A t B A t ) s A r g holds for any s,,, and the following conditions () and (3) () (3) r t; ( t + r) ( t)s + r: Put = +u u incaseu>0, =,r =( t)s and also ut ff = t in () and (3). Then (3) is satisfied, so the only reuired condition () is euivalent to the following (4) ( + ffu)s ( ff)u: Therefore, () imlies that for each ff [0; ], 0 and u 0, (5) I A (+ffu)s fa (+ffu)s (A ffu B A ffu ) s A (+ffu)s g A (+ffu)s holds for s and the condition (4). Let T be defined by the right hand side of (5). Then it turns out that T is an invertible ositive contraction by (5), so that we have (6) Taking suare both sides of (6), we obtain A (+ffu)s TA (+ffu)s = fa (+ffu)s (A ffu B A ffu ) s A (+ffu)s g : A (+ffu)s TA (+ffu)s TA (+ffu)s = A (+ffu)s (A ffu B A ffu ) s A (+ffu)s :

5 AN APPLICATION OF GENERALIZED FURUTA INEQUALITY 397 That is, we have the following euation (7) TA (+ffu)s T = A ffu B A ffu holds for s and ( + ffu)s ( ff)u in case u > 0. Next we check (7) in case u =0. In fact (II) of Theorem C ensures I T = A A B A A for all 0, so TA s T = B s holds for 0, s andthiseuation is just (7) in case u =0. The uniueness of T in (7) follows by Lemma 3. (II)=)(III). Put u 0 in (II). Then the reuired condition ( + ffu)s ( ff)u is satisfied, so we have (III). (III)=)(IV). Put u =0orff =0ands = in (III). (IV)=)(I). Assume (IV). Then we have A TA = A TA TA = A B A by (IV): By raising each sides to ower, it follows from Löwner-Heinz ineuality that (8) A A TA (A B A ) ; and the first ineuality holds since I T 0andwehave (I) by Thereom C. Whence the roof of Theorem is comlete. Proof of Theorem. (I)=)(II). The hyothesis M A m>0ensuresm +ffu)s A (+ffu)s m (+ffu)s > 0 for the hyothesis on ff; ; u and s, so the roof is comlete by (II) of Theorem and Lemma 4. (II)=)(III). Put u 0 in (II). Then the reuired condition ( + ffu)s ( ff)u is satisfied, so we have (III). (III)=)(IV). We have onlytoutu =0orff = 0 and s = in (III). (IV)=)(I) is shown by Theorem B. Whence the roof of Theorem is comlete. Remark 6. If (Λ) in Remark 5 holds, by scrutinizing the roof of Theorem it turns out that (II),(III) and (IV) in Theorem might be extended as follows: Let A and B be invertible ositive oerators such that A fl B (i.e., log A log B). Then we might exect (II-c) For each ff [0; ], 0 and u 0, A (+ffu)s A ffu B A ffu holds for any s and ( + ffu)s ( ff)u. (III-c) For each ff [0; ], u 0, holds for any s. A (+ffu)s A ffu B A ffu (IV-c) A B holds for all 0.

6 398 T. FURUTA AND Y. SEO But (Λ) does not hold as seen in Remark 5, so that the corresonding (II-c), (III-c) and (IV-c) are all false. In order to verify that (II-c) and (III-c) are both false, we cite the following best ossibility of Theorem C. Theorem D ([6]). Let > 0 and u > 0. If ff >, then there exist invertible ositive oerators A and B such that A fl B (i.e., log A log B) and A ffu 6 A u B A u ffu +u : Theorem D asserts that the outside exonent +u [0; ] on the right hand side of (III) in Theorem C is the best ossible. Theorem D yields that (II-c) and (III-c) are both false since s is reuired in (II-c) and (III-c). Acounterexamle to (IV-c) is as follows. Take A and B as follows: 3 0 A = and B = : 0 Then A B 0, so that log A log B since log t is oerator monotone, but A 6 B since 6 5 A B = 6 0: 5 4 We would like to remark that (II-c), (III-c) and (IV-c) are false even if A B 0, which is more stronger hyothesis than A fl B, because (II-c), (III-c) are false by Tanahashi's result [] stated in Introduction and a counterexamle to (IV-c) is A and B stated above. u References. T.Ando, On some oerator ineuality, Math. Ann., 79(987), T.Ando and F.Hiai, Log-majorization and comlementary Golden-Thomson tye ineualities, Linear Alg. and Its Al., 97,98(994), M.Fujii, Furuta's ineuality and its mean theoretic aroach, J.Oerator Theorey, 3(990), M.Fujii, T.Furuta and E.Kamei, Furuta's ineuality and its alication to Ando's theorem, Linear Alg. and Its Al., 79(993), M.Fujii and E.Kamei, Mean theoretic aroach to the grand Furuta ineuality, Proc. Amer. Math. Soc., 4(996), T.Furuta, A B 0 assures (B r A B r ) = B (+r)= for r 0, 0, with (+r) +r, Proc. Amer. Math. Soc., 0(987), T.Furuta, Elementary roof of an order reserving ineuality, Proc. Jaan Acad., 65(989), T.Furuta, Alications of order reserving oerator ineualities, Oerator Theorey: Advances and Alications, 59(99), T.Furuta, Extension of the Furuta ineuality and Ando-Hiai log-majorization, Linear Alg. and Its Al., 9(995), T.Furuta, Generalizations of Kosaki trace ineualities and related trace ineualities on chaotic order, Linear Alg. and Its Al., 35(996), E.Kamei, A satellite to Furuta's ineuality, Math. Jaon., 33(988), K.Tanahashi, Best ossibility of the Furuta ineuality, Proc. Amer. Math. Soc., 4(996), K.Tanahashi, The best ossibility of the grand Furuta ineuality, to aear in Proc. Amer. Math. Soc. 4. M.Uchiyama, Some exonential oerator ineualities, Mathematical Ineuality and Alications, (999), T.Yamazaki and M.Yanagida, Characterizations of chaotic order associated with Kantorovich ineuality, Sci. Math., (999), M.Yanagida, Some alications of Tanahashi's result on the best ossibility of Furuta ineuality, Mathematical Ineuality and Alications, (999),

7 AN APPLICATION OF GENERALIZED FURUTA INEQUALITY 399 * Deartment of Alied Mathematics, Faculty of Science, Science University of Tokyo, Kagurazaka, Shinjukuku, Tokyo 6-860, Jaan. furuta@rs.kagu.sut.ac.j ** Tennoji Branch, Senior Highschool, Osaka Kyoiku University, Tennoji, Osaka , Jaan. yukis@cc.osaka-kyoiku.ac.j