; TeX output 2002.06.24:1627Ǝ9r卑l e1lcmssb8SOME FUNCTIONSPRESERVING53THE ORDEROFPOSITIVE\OPERATORSFۍJosip P=Zecaric1[yKtEo lcmss8FlacultyofTEextileTechnology,\UniversitlyofZagreb*9Jadrank=Za Misrc:36Slides.texcicmQTEechnicalCollegeZagreb,\UniversitlyofZagrebj>Yuki Seo@SeniolrHighschoxol,u#GOsaklaKyoikuUniversity Msrc:42Slides.texsrc:43Slides.tex src:45Slides.tex2KtEo lcmss81*Ǝ9r2cpKtEo1lcmss81. 9INTRODUCTION src:48Slides.texsrc:49Slides.tex:This repoO rtisbasedon[J.Misrc:51Slides.texO ?cic, 2J. PO ecaO ric,Y. GSeo, 1lcmssi8FO unctionorderofpositiveoperatoO rsbasedontheMond-PO esrc:53Slides.texO ?cari?cmethod,preprint].The Lsrc:55Slides.texowner-HeinztheoO remassertsthatthefunction 21cmmi8f1ѹ(t)[=ty2 cmmi8pkisoperatoO rmonotoneonlyfoO r src:57Slides.tex1FfK 1cmsy8p>0thoughitismonotoneincreas-ing PfoO rp '>0.PFuruta P[T.Furuta, Operatorinequalities associatedwithHsrc:58Slides.texolder-McCaO rthyand}KantoO rovichinequalities,J.Inequal. ʠAppl.2P(1998),(137{148.] +shoO wedseveralextensionsofsXtheKantoO rovichinequalityandappliedthemto shoO wthefollowingpreservingoperatoror-der. src:64Slides.texTheo=Zrem A (Furuta). ŽLetDAandsrc:65Slides.texB ͈bepos-itive8)operatoO rsonaHilbertspaceH satisfyingMA1zɿHBmv1zɿH>0, where `src:66Slides.texM W>m>0.IfAB^0,Kkthen43 >'muG cmex10#^)M) ;o-ܽm*m79pK  cmsy81\~ZAp?Ch(m;vM˹;p)ApBDp:foO rKkallp1,)qwhereKktheKyFO an-FurutaKkconstant&Jsrc:71Slides.texCh(m;vM˹;p)=M(p1)%'2 cmmi8pK cmsy81M֭M p "pFp(MZ%'p m%'p=)%'pV֭F4(MZm)(mMFp MmFp=)Fp1%׾: src:72Slides.tex22͠Ǝ9rLh src:74Slides.texIzuminoandNakO amoto[S.Izumino, R.Naka- moto, OFO unctional ordersofpositiveoperatoO rs induced %fromMond-PO esrc:76Slides.texO ?cari?c %convexinequali- ties,Scienticae Mathematicae,2(1999),195{ 200.]~discussed pYthisextensionfoO ranystrictly positive(convexdi erentiablerealvaluedfunc- tionKksrc:79Slides.texf1ѹ.: src:81Slides.texMoO reover, IY#amazaki[T.Yamazaki, IAnexten- sion iofSpecht'stheoO remviaKantorovichin- equalitO yandrelatedresults,-Math. #JInequl.Appl. 3 B@(2000), 89{96.]shoO wedthefollowingpre- servingKkoperatoO rorder. src:86Slides.tex7<2Theo=Zrem .B -O(Yamazaki).dӽLet AandB bepositive dLoperatoO rsonaHilbertspacesrc:87Slides.texH 쮽sati-sfyingMMA1zɿHƾB mv1zɿH>0,whereMsrc:88Slides.texM>m>0. 9IfKkAB^>0,then+\Apʹ+HKh(m;vM˹;p) 1zɿH"BDp<JfoO rKkallp>1,&where-Kh(m;vM˹;p)=#^MmMyp1HMmypM ;~)-ȾMHm(fCh(m;M;p)ᨍ 0133 s/ Nԍp1 U1g: src:93Slides.tex23 DƎ9r1W]m2. 9GENERALKkRESUL#TS src:100Slides.texsrc:101Slides.tex:FO orYconvenience,wede nesrc:103Slides.texf=Mf4(MZ)f(m)M֭do p mMZmsHfora?realvaluedfunctionsrc:104Slides.texf qontheinterval[m;vM˹].Also, !wO e C0introducethefollowingFurutacon-stantKkwithtO woKkparameters:3<捍src:106Slides.texsrc:106Slides.texCf (m;vM˹;qع)=ݍMf1ѹ(m)HfmM v--#U1HqM#O #^.n1Hq.n ;0s-%qݍfX v--f1ѹ(m)Hfm;O!IiП}qUS;src:107Slides.texe whereKksrc:108Slides.texq Cisarealnumbersuchthatº331q33 6 p qjGͲf$$ϟ׉[F ]5f4(m)f ܿm?>w0.nPO articularly#, Ksrc:109Slides.texC$tFpĹ(m;vM˹;p)= Ch(m;vM˹;p)KkfoO rthepowerfunctionsrc:110Slides.texf1ѹ(t)=typ _.%src:112Slides.tex5MTheo=Zrem 1:Let src:113Slides.texAandB Obepositiveopera-toO rs 1onaHilbertspaceH satisfying%qGcmss17Spg (BD) [m;vM˹], mwhere usrc:114Slides.texM >m>0. Letf 2CW([m;vM˹])be $aconvexfunctionandsrc:115Slides.texg t2 CW(U˹),Z:whereU[m;vM˹][Spm(A). SupposeVLthateitherofthefolloO wing kconditionsholds: Ozsrc:117Slides.tex(a)g "cisincreasingconvex0onU˽,=oO rsrc:118Slides.tex(b)gisdecreasingconcaveonU˽.@If 9src:119Slides.texABC>0, thenfoO ragiven 2(Nq cmbx12R+RMintheKkcase(a)oO rsrc:120Slides.tex 2R*,inKkthecase(b)+ gع(A)H+ W1zɿH"f1ѹ(BD)(1)&holds wfoO rsrc:124Slides.tex #͹=max2GzɿmtMvږff1ѹ(m)+f (tm) gع(t)g. src:126Slides.texsrc:127Slides.texk?4Ǝ9r0n src:130Slides.texT#oepO roveTheorem1,weneedthefollowingre- sults*[B.Mond,J.E.PO eO ?cari?c,ѽConvex*inequal- ities H&inHilbertspaces, UHoustonJ.Math.j19 (1993), 405{420.] and [J.Misrc:132Slides.texO ?cic, Y. Seo,S.- E. _T#akO ahasi, M.Tominaga,Inequalities _ofFO u- ruta ,andMond-PO ecaric, dMath. Ineq.Appl.2 (1999),Kk83{111.]: Lemma.ÿLet 2Bsrc:137Slides.texAbeapositiveoperatoO ron a . HilbertspaceH satisfyingSp (A) [m;vM˹], where#src:138Slides.texM>m>0andf2CW([m;vM˹])beacon- vexKkfunction. 9Then+jI(f1ѹ(A)x;vx)f((Ax;vx))r(2)& holds OfoO reveryunitvectorsrc:143Slides.texx: 2Hb. Moreover, letKkg2CW([m;vM˹]). 9ThenfoO ranynumbersrc:145Slides.tex 2ROu gع((Ax;vx))H+ (f1ѹ(A)x;x)r(3)& holds VfoO reveryunitvectorsrc:149Slides.texxJ2H where V += maxzɿmtM`ff1ѹ(m)H+f (tm) gع(t)g. src:152Slides.tex25Proof WofTheoO rem1.KLetxY?2H be WanyunitvectoO r. єBy{theconvexityofsrc:153Slides.tex gع,xitfollowsfromsrc:154Slides.tex(2))in CLemmathatsrc:155Slides.tex (gع(A)x;vx)c g((Ax;vx)):By theincreaseofsrc:158Slides.tex gع, wO ehavesrc:159Slides.tex g((Ax;vx))̏ gع((BDx;vx)):pTherefoO re,jcombiningtwoinequal-itiesKkaboveandsrc:162Slides.tex(3)(0inLemmawO ehave src:163Slides.tex25Ǝ9rƎ src:166Slides.tex (gع(A)x;vx)+  g((Ax;x))+ ~ g((BDx;x))+- (f1ѹ(BD)x;vx):;J9src:169Slides.tex %src:172Slides.tex3,Rema=Zrk 2v>Assume thatconditionsofTheo-rem Mt1holdandletsrc:173Slides.tex g Lbestrictlyconvexdif-ferentiable =on[m;vM˹]. Thensrc:174Slides.tex Ccanbewrittenexplicitly !assrc:175Slides.texv A=;2f1ѹ(m)+f (t0m) gع(t0 h);where src:176Slides.text0iWisde nedastheuniquesolutionofg؟y0ѹ(t) 6/=j ibf ib׉ p . )`when Ksrc:177Slides.tex g؟y0(m)f* g؟y0(M˹), =oth-erwise 0t0isde nedassrc:179Slides.texM OboO rmaccordingasf> g؟y0ѹ(M˹)KkoO r g؟y0(m)>f :src:180Slides.texsrc:181Slides.tex/nLetusconsideracomplementaO ryresulttoThe-oO remsrc:182Slides.texKk1.src:184Slides.tex7R^Theo=Zrem 3:Let src:185Slides.texAandB Obepositiveopera-toO rs honaHilbertspaceH 1ʽsatisfyingSpC(A) :[m;vM˹], mwhere usrc:186Slides.texM >m>0. Letf 2CW([m;vM˹])be aconcavefunctionandsrc:187Slides.texg b2 CW(U˹), CwhereU[m;vM˹][Sp(BD). wSuppose2%thateitherofthefolloO wing kconditionsholds: Ozsrc:189Slides.tex(a)g "cisincreasingconcave0onU˽,=oO rsrc:190Slides.tex(b)gisdecreasingconvexonU˽.@If 9src:191Slides.texABC>0, thenfoO ragiven 2R+RMintheKkcase(a)oO rsrc:192Slides.tex 2R*,inKkthecase(b))Ѝf1ѹ(A) gع(BD)H+ W1zɿH$Ѝholds foO rsrc:196Slides.tex = min.V7zɿmtMrff1ѹ(m)]+f (tm) gع(t)g. src:198Slides.texsrc:199Slides.texk?6#Ǝ9r Ǝsrc:202Slides.texProof. By 1XtheconcavitO yoff1ѹ, jtheincreaseofvsrc:203Slides.tex g CandKktheconcavitO yof gitfolloO wsthatsrc:204Slides.tex(f1ѹ(A)x;vx) gع((Ax;x))1+  gع((BDx;x))+  (gع(BD)x;vx)H+ QLfoO rKkeveryunitvectorsrc:206Slides.texx2Hb.%src:209Slides.tex2̍Rema=Zrk 4v>Assume thatconditionsofTheo-rem73holdandletsrc:210Slides.tex g bestrictlyconcavedif-ferentiable =on[m;vM˹]. Thensrc:211Slides.tex Ccanbewrittenexplicitly !assrc:212Slides.texv A=;2f1ѹ(m)+f (t0m) gع(t0 h);where -src:213Slides.text0̕isde nedasinRemaO rk2withtheoppositeKkconditions.src:214Slides.texsrc:215Slides.tex4>IfwO eputsrc:217Slides.tex =1inTheorem1andTheorem3,thenKkwO ehavethefollowingcorollary:src:220Slides.tex4>Co=Zrollary 5RLet\src:221Slides.texAandBWabepositiveoperatoO rsonaHilbertspaceH satisfyingsrc:222Slides.texSp۹(BD)[m;vM˹](resp.?src:223Slides.texSp۹(A) 7[m;vM˹]), where [M U>m>0.LetTUf2CW([m;vM˹])beaconvexsrc:224Slides.tex(resp. concave)function *andsrc:225Slides.texg _2 \CW(U˹)anincreasingconvexsrc:226Slides.tex(resp. wkconcave) ѽfunction, L+whereU ![m;vM˹]'[Sp۹(A)H[Sp#(BD). 9IfKksrc:227Slides.texAB^>0,then)c΍tgع(A)H+ W1zɿH"f1ѹ(BD)vZ%S(resp.I!}f1ѹ(A)gع(BD)H+ W1zɿHA)$c΍holdsMfoO r =max0 +zɿmtMtf(f1ѹ(m)+f (tm))gع(t)g src:232Slides.texwL(resp. }-=min-ʵzɿmtMr]Nf(f1ѹ(m)+f (tm))Hgع(t)g): src:234Slides.texsrc:235Slides.texk?7-?Ǝ9rSn src:238Slides.texIfzwO echoose suchthat k;=eZ0inTheoremsrc:239Slides.tex1 and TheoO rem3, ! wehavethefollowingcorol- laO ry: src:241Slides.tex7<2Co=Zrollary 6RLet\src:242Slides.texAandBWabepositiveoperatoO rsonaHilbertspaceH satisfyingsrc:243Slides.texSp۹(BD)[m;vM˹](resp.src:244Slides.texSp۹(BD) 6[m;vM˹]), 6where G@M 9>m>0.LetTUf2CW([m;vM˹])beaconvexsrc:245Slides.tex(resp. concave)function :andsrc:246Slides.texg 2 -CW(U˹), -whereU [m;vM˹][Sp۹(A)ڤ[Sp(BD). ߚSupposethateitherofthefol-loO wingKkconditionsholds::src:248Slides.tex(i)kgdCisincreasingconvexonsrc:249Slides.texU˽,8g>0on[m;vM]andKkf1ѹ(m)>0;vf(M˹)>0,src:250Slides.tex(ii)"g pis increasingconvexonU˽, $src:251Slides.texg W< ݹ0on[m;vM˹]Kkandf1ѹ(m)<0;f1ѹ(M˹)<0,src:252Slides.tex(iii) ?g isdecreasingconcaveonsrc:253Slides.texU˽, 3g v`> 0on[m;vM˹]Kkandf1ѹ(m)<0;f1ѹ(M˹)<0,src:254Slides.tex(iv) ?g isdecreasingconcaveonsrc:255Slides.texU˽, 3g va< 0on[m;vM˹]Kkandf1ѹ(m)>0;f1ѹ(M˹)>0.%src:256Slides.texsrc:257Slides.texk?8 6̠Ǝ9rƎ src:260Slides.texIfKkAB^>0,then(TVN +gع(A)f1ѹ(BD)B(resp.6Uf(A) *5g(BD))#TV holdsKkfoO r@? +.=^max7kmtMMy))nXJ(f1ѹ(m)H+f (tm))=gع(t))o6 O t%resp.5z *۹=?min7kmtMMy))nXJ(f1ѹ(m)H+f (tm))=gع(t))oVv]O! !퍑 inKkcasesrc:268Slides.tex(i)and(iii),oO r@? +.=?min7kmtMMy))nXJ(f1ѹ(m)H+f (tm))=gع(t))o O t%resp.5z *۹=^max7kmtMMy))nXJ(f1ѹ(m)H+f (tm))=gع(t))oVv]O! !퍑 inKkcasesrc:274Slides.tex(ii)and(iv). %src:276Slides.tex0䤍Rema=Zrk 7v>Assume ~thatconditionsofCoO rol-yVlaO ry G6hold.cMoreover, supposethateitherofthe JfolloO wingadditionalconditionsholds:src:278Slides.tex(a)g Iis)strictlyconvexdi erentiableincasesrc:279Slides.tex(i)oO r(ii), oO r |(bsrc:280Slides.tex)g 3isstrictlyconcavedi erentiablein case(iiisrc:281Slides.tex)oO r(iv).EThenavalueof *赽maybeLdeterminedmoO repreciselyasfollows: src:282Slides.tex *۹=(f1ѹ(m)U+f (t}oZm))=gع(t}o ); .!wheresrc:283Slides.text}o2 [m;vM˹]is f de nedastheuniquesolutionofsrc:284Slides.texf gع(t)*=g؟y0ѹ(t)(f1ѹ(m)WB+f (tm) Nifsrc:285Slides.texf(m)g؟y0ѹ(m)=gع(m) z f Tf1ѹ(M˹)g؟y0ѹ(M)=gع(M),Hotherwiset}oŽisde nedas5src:286Slides.texM "oO rmaccordingasf>f1ѹ(M˹)g؟y0ѹ(M)=gع(M)oO rKksrc:287Slides.texf1ѹ(m)g؟y0ѹ(m)=gع(m)<f .src:288Slides.texsrc:289Slides.texk?9 =Ǝ9r-sэ$¹3. 9FUNCTIONKkPRESERVINGTHEkOPERA#TORKkORDER src:297Slides.texsrc:298Slides.tex:Asapplicationsofourgeneralresults,0,thenfoO ragiven 2R++ξ f1ѹ(A)H+ W1zɿH"f(BD)&holdsfoO rsrc:315Slides.tex =f1ѹ(m)+f (t0 0m) f(t0 h)andt0visde nedĆastheuniquesolutionofsrc:316Slides.texf1џy0yʹ(t)=f = when !]src:317Slides.texf1џy0yʹ(m) f = f1џy0(M˹), otherwise !]t0Žisde ned assrc:318Slides.texM oO rmaccordingasf1џy0yʹ(M˹)N0<f = oO rKkf = 0,thenfoO ragiven 2R+%=dϾf1ѹ(A) f(BD)H+ W1zɿHholdsfoO rsrc:335Slides.tex =f1ѹ(m)+f (t0 0m) f(t0 h)andsrc:336Slides.text0vis=dde nedĆastheuniquesolutionofsrc:337Slides.texf1џy0yʹ(t)=f = when !]f1џy0yʹ(M˹) f = f1џy0(m), otherwise !]src:338Slides.text0Žisde ned asM oO rmaccordingassrc:339Slides.texf1џy0yʹ(M˹)N0>f = oO rKkf = >f1џy0yʹ(m).src:340Slides.texsrc:341Slides.texsrc:343Slides.tex/Rema=Zrk 10\IfwO eput =1inTheorem8andTheoO rem;9,>thenwehavethefollowingclaim:Let w^src:344Slides.texf %2CW([m;vM˹])beastrictlyconvexsrc:345Slides.tex(resp.concave) increasingdi erentiablefunction.FIfsrc:346Slides.texMA1zɿHC[EAB Vmv1zɿH>0, {where src:347Slides.texM z>m>0, b9then Csrc:348Slides.texf1ѹ(A)+ W1zɿH6>8f(BD)B(resp.kf(A)f1ѹ(BD)+ W1zɿHA)holdsfoO rsrc:349Slides.tex =f(m)+f (t0 }m)f1ѹ(t0 h) Xandt07isexactlyonesolutiontheequa-tionKksrc:350Slides.texf1џy0yʹ(t)=ffintheinterval[m;vM˹].src:351Slides.texsrc:352Slides.texk?11 OdƎ9r7s src:355Slides.texIfywO echoose suchthat =0inTheorem8, wO e havethefollowingcorollary(cf.Izumino- NakO amoto,KkTheorem2.2]). %src:359Slides.tex5MCo=Zrollary 11Let AandB /bepositiveopera-toO rs 1onaHilbertspacesrc:360Slides.texH satisfyingSpg (BD) [m;vM˹]3src:361Slides.tex(resp. Sp m(A)Q![m;M˹],%where3src:362Slides.texMo>m>0. PLetf2,CW(U˹)beaincreasingconvex(resp.concave)Jfunction,wheresrc:363Slides.texU[m;vM˹][Sp(A)[Sp۹(BD).Suppose JthateitherofthefolloO wingconditionsnholds: ?src:366Slides.tex(i)fҚ>ɹ0 Yon[m;vM˹]oO r-Rsrc:367Slides.tex(ii)f<0Kkon [m;vM˹]. 9IfAB^>0,then+ +f1ѹ(A)f(BD)(resp.Ef(A) *5f(BD))&holdsKkfoO rO2 +.=^max7kmtMMy))nXJ(f1ѹ(m)H+f (tm))=f(t))o6O resp.Cm *۹=?min7kmtMMy))nXJ(f1ѹ(m)H+f (tm))=f(t))oX ͟O! "4퍽inKkcase(i),oO rO2 +.=?min7kmtMMy))nXJ(f1ѹ(m)H+f (tm))=f(t))oO resp.Cm *۹=^max7kmtMMy))nXJ(f1ѹ(m)H+f (tm))=f(t))oX ͟O! "4퍽inKkcase(ii).%src:384Slides.texsrc:385Slides.texk?12 YWƎ9rƎ src:388Slides.texWO e fshallshowafunctionalorderversionof܍ TheoO remKkA(Furuta). %src:390Slides.tex.Co=Zrollary 12Let mAandB bepositiveinvert-ible N:operatoO rsonaHilbertspacesrc:391Slides.texH ֜satisfyingSp۹(BD)ג[m;vM˹], where csrc:392Slides.texM ]>m>0.WLetf c2CW(U˹)beastrictlyconvexincreasingtO wicedif-ferentiablefunction,wwheresrc:394Slides.texU[m;vM˹][Spʹ(A)[Sp۹(BD). Letxsrc:395Slides.tex(i)f>0on[m;vM˹]oO r(ii)f<0on[m;vM˹]. 9IfKkAB^>0,then1#^_7Xf1џy0yʹ(M˹)_7X ;;l-ܽf1џU0yʹ(m)/f1ѹ(A) f(A)f(BD);6where /Psrc:397Slides.tex L=j +f4(m)+f ܻ(tg-0 #5m) +׉* ]50jf4(tg-0 #5)Q,andt0ִ2hm;vMiisnmthe \uniquesolutionofsrc:398Slides.texf f1ѹ(t)=fy0yʹ(t)(f(m)/+f (tHm)).src:400Slides.texsrc:401Slides.texsrc:402Slides.tex+Proof. \By!theassumptionoff1ѹ,wO ehaveff1џy0yʹ(M˹) -andsrc:403Slides.tex0 M +m. DIfsrc:420Slides.texABD, 3thenfoO ragivenKk 2R++ﲾ eAgD+H W1zɿH"eB&whereD =]捑8Z>/>><>>>:x 1inTheorem1, Zthenwehavethe folloO wingKkcorollary#. src:447Slides.tex7<2Co=Zrollary 15Let AandB /bepositiveopera-toO rs 1onaHilbertspacesrc:448Slides.texH satisfyingSpg (BD) [m;vM˹],wheresrc:449Slides.texM>#m>0. IfABg>0,thenfoO rKkagiven 2R++ Aq + W1zɿH"BDp Zƾ; ffoO rKkallžp2RHn[0;v1iandq>1,&whereKksrc:451Slides.tex =>"ˍּ"8Z>/<>:\+src:454Slides.tex (qj H1)v)܃ 1 M p q$$tFpğ)*UqKbЉ+ Nԍq~1l+$tFpH:if"Ⱦqmyqa1&PcMi'tApM LT p } &rqM˟yqa1#*;src:457Slides.texn\+max9m>0. 9IfKksrc:483Slides.texAB^>0,then4#^EM˟yp1E ;7⭟-mUqa1~Aq'C+(m;vM˹;p;qع)AqBDp(4)*holdsfoO rallsrc:487Slides.texp>1andq>1,-whereC+(m;vM˹;p;qع)=K]捑TC8ZTC>/TC>TC>TC<TC>TC>TC>TC:⿲nCh(m;vM˹;p;qع)ifxdqmyp1'S܃MMZ%'p m%'pM?˟ p =MZmMeqM˟yp1znmypqif܃ʫMZ%'p m%'pʫ?˟ p =MZm<qmyp1nM˟ypqifxdqM˟yp1)r<܃MMZ%'p m%'pM?˟ p =MZm;(5)]and src:498Slides.texCh(m;vM˹;p;qع) -=C$tFpĹ(m;vM;qع) istheFO urutaconstant.src:499Slides.texsrc:500Slides.texk?16zƎ9r %src:503Slides.texProof. 9First,KkwO eprovethattheinequality3Ӎ&,fhp1'S#^M(qj H1)yqa1M ;d^-(q؟Uq#^'Ź(hypH1)yqnS ;-(hH1)(hUph)Uqa1(6),Rholds dfoO rallsrc:509Slides.texpQ>1, "q)>1 dandhQ>1 dwithsrc:510Slides.texq܃33h%'p=133* Ÿ p h12gBqhyp1!s.:src:512Slides.texThen,8wO ekprovethe rstinequalityin(4)'0. ݎPutsrc:513Slides.texh.=܃2aM2a p Mm#׾>1. o&If qmyp1(r܃2aMZ%'p m%'p2a?˟ p =MZmP,qM˟yp1#v, HthenthisKkinequalitO yfollowsfromsrc:514Slides.tex(6)4#^B*M˟yp1B* ;7⭟-mUqa1 $=mqapPhp1'SC$tFpĹ(m;vM˹;qع):2Otherwise,-wO eseethatsrc:516Slides.tex܃33MZ%'p133.D4ϿmFq~16܃Mm%'p1M+4+mFq~19Xǹandsrc:517Slides.tex܃33MZ%'p133.D4ϿmFq~1#܃33MZ%'p133.D4+MZFq~11 .!TherefoO re ׸wehavethe rstinequal-itO y zinsrc:519Slides.tex(4).&eWehavethesecondinequalityinsrc:520Slides.tex(4))dif wO echoose suchthatsrc:521Slides.tex .=M0inCorol-laO ry15. ?Thenitfollowsthatsrc:522Slides.tex coincideswithC+(m;vM˹;p;qع)Kkin(4)(0.src:523Slides.texsrc:526Slides.tex3%Rema=Zrk 18\If wO eputq |= pin L(6),, ɖthenitassumption pisautomaticallysatis edandsothehconstantsrc:527Slides.texC+(m;vM˹;p;qع)hcoincideswithKyFO an-Furuta tQconstantsrc:528Slides.texCh(m;vM˹;p).TherefoO rewO eKkhaveTheoremA(Furuta).%src:530Slides.tex src:531Slides.tex217ԠƎ9rb src:534Slides.texAs oageneralizationofTheoO rem17, weshow theKkfolloO wingtheorem. src:535Slides.tex7<2Theo=Zrem 19LetAandB ^bepositiveopera-toO rs 1onaHilbertspaceH satisfyingsrc:536Slides.texSp۹(BD) [m;vM˹],KkwhereM>m>0. 9IfKksrc:537Slides.texAB^>0,then0]C+(mr;vM˟r n;#^pH1+r ;W-& r];#^qj H1+r ;WV-&]r]Ck)Aq'BDp(holds 'foO rallsrc:541Slides.texp >1, q >1 'andr C> ڹ1,whereC+(m;vM˹;p;qع)Kkisde nedassrc:542Slides.tex(4)Ž.src:543Slides.texsrc:544Slides.tex3%T#o ~pO roveTheoremsrc:545Slides.tex19, ˢweneedthefollowingFO uruta >inequality[T.Furuta, EA @B ;L0as-suresԹ(BDyr Ayp _BDyr)y1=q"BDy(p+2r}Y)=qUfoO rsrc:546Slides.texrv0,%Yp0,q } 1 Swith(1+2ri)q p+2ri, MProc.Amer.Math. 9Soc.101Kk(1987),85{88.]:%src:549Slides.tex5MTheo=Zrem eF c(theFurutainequality). IfGqA$ܠB  ƹ0, Ithen IfoO reachsrc:550Slides.texr /0, I)B𑖍r.w 5 Nԍ2 ߾Ayp _B𑖍r.w 5 Nԍ2)guI1uI 5 Nԍq)+J;) QB𑖍r.w 5 Nԍ2 ߾BDyp ZƾB𑖍r.w 5 Nԍ2)giм1iП 5 Nԍqziand|src:553Slides.tex) QA𑖍rr33 5 Nԍ2 Ayp _A𑖍rr33 5 Nԍ2)ge1e 5 Nԍqv)2A𑖍rr33 5 Nԍ2 BDyp ZƾA𑖍rr33 5 Nԍ2)glƼ1lƟ 5 Nԍq~holdfoO rKksrc:556Slides.texp0andq1with(1H+ri)qp+r. src:557Slides.tex218͠Ǝ9r msrc:560Slides.texProofofTheoO rem19. ŹByFurutainequality#,Tit EfolloO wsthatsrc:561Slides.texABensuresthatsrc:562Slides.tex(B𑖍r.w 5 Nԍ2 ߾Ayq B𑖍r.w 5 Nԍ2)n331+r33MC Nԍq~+r%BDy1+r1,foO r allsrc:563Slides.texq > 01andr %>0.nPutsrc:564Slides.texA1$="(B𑖍r.w 5 Nԍ2 ߾Ayq B𑖍r.w 5 Nԍ2)n331+r33MC Nԍq~+r(TandRsrc:565Slides.texB1<=ԾBDy1+r$lc,TthenA1ԾB1>0@ andM˟y1+r)`1zɿH"B1my1+r&1zɿH>0. ApplyingTheoO remsrc:566Slides.texKk17toA1)ӹandsrc:567Slides.texB1 h,wehave+E)"C+(m1+r#q;vM˟1+r%;p1 h;q1)Ayqg-1͍1aByDpg-1͍1&foO r Eallsrc:568Slides.texp1;>;e1andq1>;e1. Putsrc:569Slides.texp1=º . p+r n" p 1+r5O>;e1 andKkq1=ºqa+rM" p 1+r1>1,thenwO ehave6m@C+(m1+r#q;vM˟1+r%;#^dpH+r ;3T,-1H+r9A;#^qj +Hr ;3T,-1H+r)B𑖍r.w 5 Nԍ2 ߾Aq B𑖍r.w 5 Nԍ2dBDp+r#hNuUfoO r Sallsrc:574Slides.texpN>1 Sandq&>N1. fMultiplyB𑖍.wr.w tt Nԍ蠼2iqonbothsidesandreplacesrc:575Slides.texr>PbO yr1,thenitfollowsthat/sxC+(mr;vM˟r n;#^pH1+r ;W-& r];#^qj H1+r ;WV-&]r]Ck)Aq'BDp(foO rKkallsrc:579Slides.texp>1,Kkq>1andrv>1. 9 src:580Slides.tex219;(Nq cmbx12%qGcmss17uG cmex10K cmsy8K  cmsy8K 1cmsy82 cmmi82 cmmi821cmmi81lcmssi8KtEo lcmss8KtEo lcmss8 e1lcmssb8KtEo1lcmss8