FURTHER EXTENSION OF AN ORDER PRESERVING OPERATOR INEQUALITY. (communicated by M. Fujii)

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1 Journal of Mahemaical Inequaliies Volume, Number 4 (008), FURTHER EXTENSION OF AN ORDER PRESERVING OPERATOR INEQUALITY TAKAYUKI FURUTA To he memory of Professor Masahiro Nakamura in deep sorrow (communicaed by M. Fujii) Absrac. A capial leer means a bounded linear operaor on a Hilber space H. The celebraed Löwner-Heinz inequaliy assers ha A B 0 ensures A α B α for any α 0,], bua p B p does no always hold for p >. From his poin of view, we shall prove he following resul. Le A B 0 wih A > 0, 0,] and p, p,...,p n for naural number n. Then he following inequaliy holds for r : A +r A A r A ( A... n imes and A n imes by urns B p ) p A p3 ] p4 A... n imes and A n imes by urns ] pna r +r ϕn;r,] where ϕn;r,] =...(p )p +]p 3 p 4 +]p 5... p n + r appears n imes and appears n imes by urns n n n n = r + p i +( p i + p i p i p n p n ) i= i=3 i=5 i=7 n erms n n n ( p i + p i + p i p (n ) p n p n + p n ). i= i=4 i=6 nerms This resul is furher exension of he following previous one: if A B 0 wih A > 0, hen for 0,] and p, A +r A r ( B p ) s A r (p )s+r +r holds for r and s, in paricular, A +r (A r B p A r ) +r p+r for p and r 0.. Inroducion An operaor T is said o be posiive (denoed by T 0) if (Tx,x) 0forall x H,andT is said o be sricly posiive (denoed by T > 0) if T is posiive and inverible. Mahemaics subjec classificaion (000): 47A63. Keywords and phrases: Löwner-Heinz inequaliy, order preserving operaor inequaliy. This work was suppored by Gran-in-Aid for Scienific Research (C). c D l,zagreb Paper JMI

2 466 TAKAYUKI FURUTA THEOREM LH. (Löwner-Heinz inequaliy, denoed by (LH) briefly). If A B 0 holds, hen A α B α for any α 0,]. (LH) This was originally proved in L] and hen in H]. Many nice proofs of (LH) are known. We menion P] and B]. Alhough (LH) assers ha A B 0 ensures A α B α for any α 0,], unforunaely A α B α does no always hold for α >. The following resul has been obained from his poin of view. THEOREM A. If A B 0, hen for each r 0, p ( + r)q = p + r (i) (B r A p B r ) q (B r B p B r ) q q = p = q and (ii) (A r A p A r ) q (A r B p A r ) q (, ) hold for p 0 and q wih.( + r)q p + r (0, r) (, 0) Figure q The original proof of Theorem A is shown in F], an elemenary one-page proof is in F] and alernaive ones are in MF], K]. I is shown in T] ha he condiions p, q and r in Figure are bes possible. THEOREM B. If A B 0 wih A > 0, hen for 0,] and p, holds for r and s. A +r A r ( B p ) s A r +r (p )s+r (.) The original proof of Theorem B is in F3], and an alernaive one is in MF-K]. An elemenary one-page proof of (.) is in F4]. We menion ha furher exensions of Theorem B and relaed resuls o Theorem A are in MF-K-N], F5], F-W], F-Y- +r (p )s+r of Y], K] and Y-G]. I is originally shown in T] ha he exponen value he righ hand of (.) is bes possible and alernaive ones are in MF-M-N], Y]. I is known ha he operaor inequaliy (.) inerpolaes Theorem A and an inequaliy equivalen o he main resul of Ando-Hiai log majorizaion A-H] by he parameer 0,].. Definiions of C A,B n] and qn] Le A > 0, B 0, 0,] and p, p,..., p n,..., p (n ), p n, p n fora naural number n.

3 FURTHER EXTENSION OF AN ORDER PRESERVING OPERATOR INEQUALITY 467 Le C A,B n]=c A,B n; p, p,..., p (n ), p n, p n ] be defined by C A,B n]=c A,B n; p, p,..., p (n ), p n, p n ] = A A... A (A B p ) p A and A alernaely n imes ] p3 p4...a ] pn pn A and A alernaely n imes (.). For examples, C A,B ]=A ( B p ) p A and ] p4a C A,B 4]=A A A ( B p ) p A p 3. Nex we define For examples, qn]=qn; p, p,..., p n,..., p (n ), p n, p n ] =...(p )p + ]p 3 p 4 + ]p 5... p n +. (.) and alernaely n imes appear q]=(p )p + and q4]= (p )p + p 3 ] p 4 +. The following Lemma is easily shown by (.) and (.). LEMMA.. For A > 0 ( and B 0 and any naural ) number n p(n+)a (i) C A,B (n + )] = A (C A,B n]) p n+ (.3) (ii) q(n + )] = (qn]p n+ )p (n+) +. (.4) Also we remark ha (.) and (.) easily imply C A,A n]=a qn] holds for any naural number n. (.5) qn]= n i= n n n p i +( p i + p i + p i p n p n + ) i=3 i=5 i=7 nerms n ( i= p i + n i=4 p i + n i=6 p i p (n ) p n p n + p n ). (.6) nerms

4 468 TAKAYUKI FURUTA 3. Saemen of resuls THEOREM 3.. Le A B 0 wih A > 0, 0,] and p, p,..., p n. Then he following inequaliy holds, A A A... A (A and A alernaely n imes where qn] is in (.). B p ) p A p3 ] p4...a ] pn pna and A alernaely n imes ] qn] COROLLARY 3.. If A B 0 wih A > 0, 0,] and p, p, p 3, p 4,hen he following inequaliy holds, A (C A,B 4]) q4], (3.) ha is, A A A A (A B p ) p A p 3 ] p 4 A q4]. THEOREM 3.3. Le A B 0 wih A > 0, 0,] and p, p,..., p n for naural number n. Then he following inequaliy holds for r, A +r A r A... A ( n imes and A n imes by urns where ϕn; r,]=qn]+r. B p ) p A p3 ] p4 A... n imes and A n imes by urns ] pna r +r ϕn;r,] (3.) COROLLARY 3.4. F6] If A B 0 wih A > 0, 0,] and p, p, p 3, p 4, A +r A r holds for r. We need he following lemma. A ( B p ) p A ] +r p3 p4a r (p )p +p 3 ]p 4 +r LEMMA A. F3, Lemma ] Le X be a posiive inverible operaor and Y be an inverible operaor. For any real number λ, (YXY ) λ = YX (X Y YX ) λ X Y. Proof of Theorem 3.. The case n =. (3.) for n = is shown by puing r = in (.) of he Theorem B, ha is, if A B 0 wih A > 0, hen for 0,] and p, p A ( B p ) p A (p )p + A (3.3)

5 FURTHER EXTENSION OF AN ORDER PRESERVING OPERATOR INEQUALITY 469 holds. We shall sae he following proof of (3.3) because he mehod of he proof is very useful o give a proof of he general case for naural number n of (3.) (see Remark 3.). Firs sep. The case p. As p, (p )p + 0,] and A B by LH since 0,] and aking inverses of boh sides, B,wehave B = A (A B p ) p A (p )p + by Lemma A = B p (B p B p ) p B p B p (B p B B p ) p B p (p )p + Second sep. Repeaing (3.4) for A B > 0, we have (p )p + = B A = A. (3.4) (p )p + (3.5) A A ( Bp )p A holds for p, p. Taking p =(p )p + in (3.5) i yields A A A A (A B p ) p A A ] p A (p )p p + = A ( B p ) p p A (p )p p +, for 4 p p. (3.6) Repeaing his process from (3.4) o (3.6), we obain (3.) for any p, in he case n =. The general case for some naural number n. Le A B 0 wih A > 0, 0,] and p, p,..., p n. Assume (3.) holds for some naural number n. We shall show (3.) for n +, ha is, if A B 0 wih A > 0, hen for 0,] and p, p,..., p n, p n+, p (n+) by (.3) i follows C A,B (n + )] q(n+)] = A ( A (C A,B n]) p n+ For his, we noe ha q(n + )] and we have ) p(n+) A q(n+)] A. (3.7) C A,B (n)] qn] (3.8) by he assumpion of inducion, and LH since 0,] and aking inverses of boh sides. Firs sep. The case p (n+). Denoe v = p n+ and w = p (n+) briefly. We have C A,B (n + )] = A q(n+)] ( (CA,B n]) v ) wa q(n+)] by (.3) ( ) = (C A,B n]) v (C A,B n]) v (C A,B n]) v w (CA,B n]) v q(n+)]

6 470 TAKAYUKI FURUTA (by Lemma A) ) (C A,B n]) ((C v A,B n]) v (CA,B n]) qn] (C A,B n]) v w (CA,B n]) v q(n+)] = ((C A,B n]) v+(v )(w )) qn] q(n+)] = ( (C A,B n]) (qn]v )w+ ) qn] q(n+)] =(C A,B n]) qn] because q(n + )] = (qn]v )w + holds by (.4) A by (3.) (3.9) The firs inequaliy follows by LH since w 0,]. Namely (3.9) ensures ha (3.7) holds under he assumpion p, p,..., p n,v and w, ha is, C A,B (n + )] q(n+)] = A (CA,B n]) v ] wa q(n+)] A (3.0) A holds for any w. Second sep. Pu A = A and B = A ( A (C A,B n]) v Since A B 0 wih A > 0 by (3.0), and (3.3) implies ) wa q(n+)] in (3.0). A p (A B ) w A (p )w + A (3.) for any 0,], p andw. In (3.), pu p = q(n + )] = (qn]v )w + by (.4). Then we have A (A A ( (C A,B n]) v ) ) wa w A qn]v )ww + ( ) = A (CA,B n]) v ww A qn]v )ww + A. (3.) Since w in (3.0), and any w in (3.), (3.) ensures ha (3.7) holds for any p (n+). REMARK 3.. In fac (3.3) iself can be shown by MF-K, Theorem ], here we sae he proof of 3.., F5] for he sake of readers convenience. Proof of Corollary 3.. We have only o pu n = in Theorem 3.. Proof of Theorem 3.3. Pu A = A and = B =(C A,B n]) qn] A A... A (A and A alernaely n imes A p ) p A p3 ] p4...a ] pn A pna and A alernaely n imes ] qn]

7 FURTHER EXTENSION OF AN ORDER PRESERVING OPERATOR INEQUALITY 47 in (3.) in Theorem 3.. Then A B by (3.) holds for 0,] and p, p,...p n, by applying Theorem A, r (A A +r Bs r A ) +r s +r holds for s andr 0. (3.3) In (3.3) we have only o pu r = r 0ands = qn] o obain (3.) since s + r = qn]+r = ϕn,r,]. Proof of Corollary 3.4. Pu n = in Theorem 3.3. REMARK 3.. Corollary 3.4 yields Theorem B by puing p = p 3 =. Acknowledgemen. We would like o express our cordial hanks o he referee for nice and kind commens o he firs version. REFERENCES A-H] T. ANDO AND F. HIAI, Log majorizaion and complemenary Golden-Thompson ype inequaliies, Linear Alg. and Is Appl.,97, 98 (994), 3 3. B] R. BHATIA, Posiive Definie Marices, Princeon Univ. Press, 007. MF] M. FUJII, Furua s inequaliy and is mean heoreic approach, J. Operaor Theory, 3 (990), MF-K] M. FUJII AND E. KAMEI, Mean heoreic approach o he grand Furua inequaliy, Proc. Amer. Mah. Soc., 4 (996), MF-K-N] M. FUJII, E. KAMEI AND R. NAKAMOTO, Grand Furua inequaliy and is varian, J. Mah. Inequal., (007), MF-M-N] M. FUJII, A.MATSUMOTO AND R. NAKAMOTO, A shor proof of he bes possibiliy for he grand Furua inequaliy, J. of Inequal. and Appl., 4 (999), F] T. FURUTA, A B 0 assures (B r A p B r ) /q B (p+r)/q for r 0, p 0,q wih ( + r)q p + r, Proc. Amer. Mah. Soc.,0 (987), F] T. FURUTA, Elemenary proof of an order preserving inequaliy, Proc. Japan Acad., 65 (989), 6. F3] T. FURUTA, An exension of he Furua inequaliy and Ando-Hiai log majorizaion, Linear Alg. and Is Appl., 9 (995), F4] T. FURUTA, Simplified proof of an order preserving operaor inequaliy, Proc, Japan Acad., 74 (998), 4. F5] T. FURUTA, Inviaion o Linear Operaors, Taylor & and Francis, 00, London. F6] F-W] T. FURUTA, An order preserving operaor inequaliy, Mah. Inequal. Appl. (o appear) T. FURUTA AND D. WANG, A decreasing operaor funcion associaed wih he Furua inequaliy, Proc. Amer. Mah. Soc., 6 (998), F-Y-Y] T. FURUTA, M. YANAGIDA AND T. YAMAZAKI, Operaor funcions implying Furua inequaliy, Mah. Inequal. Appl., (998), H] E. HEINZ, Beiräge zur Sörungseorie der Spekralzerlegung, Mah. Ann.,3 (95), K] E. KAMEI, A saelie o Furua s inequaliy, Mah. Japon., 33 (988), K] E. KAMEI, Paramerized grand Furua inequaliy, Mah. Japon., 50 (999), L] K. LÖWNER, Über monoone MarixFunkionen, Mah.Z.,38 (934), P] G. K. PEDERSEN, Some operaor monoone funcions, Proc. Amer. Mah. Soc., 36 (97), T] K. TANAHASHI, Bes possibiliy of he Furua inequaliy, Proc. Amer. Mah. Soc., 4 (996), T] K. TANAHASHI, The bes possibiliy of he grand Furua inequaliy, Proc. Amer. Mah. Soc., 8(000),5-59.

8 47 TAKAYUKI FURUTA Y] T. YAMAZAKI, Simplified proof of Tanahashi s resul on he bes possibiliy of generalized Furua inequaliy, Mah. Inequal. Appl., (999), Y-G] J. YUAN AND Z. GAO, Complee form of Furua inequaliy, Proc. Amer. Mah. Soc., 36 (008), (Received July 0, 008) Takayuki Furua Deparmen of Mahemaical Informaion Science Tokyo Universiy of Science -3 Kagurazaka, Shinjukuku Tokyo Japan furua@rs.kagu.us.ac.jp Journal of Mahemaical Inequaliies jmi@ele-mah.com