ESTIMATESONSINGULARVALUESOFFUNCTIONSOFPERTURBEDOPERATORSByQinboLiuADISSERTATIONSubmittedtoMichiganStateUniversityinpartialentoftherequirementsforthedegreeofAppliedMathematics{DoctorofPhilosophy2016ABSTRACTESTIMATESONSINGULARVALUESOFFUNCTIONSOFPERTURBEDOPERATORSByQinboLiuInthisthesiswestudythebehavioroffunctionsofoperatorsunderperturbations.Weprovethatiffunctionfbelongstotheclass!def=ff:!f()const!()gforanarbitrarymodulusofcontinuity!,thensj(f(A)f(B))c!(1+j)1pkABkSlpkfk!forarbitraryself-adjointoperatorsA,Bandall1jl,where!(x)def=xR1x!(t)t2dt(x>0).Theresultisthengeneralizedtocontractions,maximaldissipativeoperators,normaloperatorsandn-tuplesofcommutingself-adjointoperators.ACKNOWLEDGMENTSIwouldliketoexpressmyspecialappreciationandthankstomyadvisorProfessorDr.VladimirV.Peller,youhavebeenatremendousmentorforme.Iwouldliketothankyouforencouragingmyresearchandforallowingmetogrowasaresearchscientist.Youradviceonbothresearchaswellasonmycareerhavebeeninvaluable.Iwouldalsoliketothankmycommitteemembers,professorAlexanderL.Volberg,professorNikolaiV.Ivanov,professorYanBaishengandprofessorZhanDapengforservingasmycommitteemembersevenathardship.Ialsowanttothankyouforlettingmydefensebeanenjoyablemoment,andforyourbrilliantcommentsandsuggestions,thankstoyou.iiiPREFACEItiswellknownthataLipschitzfunctiononthereallinedoesnothavetobeoperatorLipschitz.Thesituationchangesdramaticallyifweconsidertheolderclassoffunctions.In[1]and[3],itwasprovedthatiffbelongstotheolderclass(R)with0<<1,thenkf(A)f(B)kconstkfkkABkforallpairsofself-adjointorunitaryoperatorsAandB.Theauthorsalsogeneralizedtheirresultstotheclass!,andobtainedestimatekf(A)f(B)kconstkfk!!kABk.In[2],itwasshownthatforfunctionsfintheolderclass(R)with0<<1andif1<p<1,theoperatorf(A)f(B)belongstoS,wheneverAandBarearbitraryself-adjointoperatorssuchthatAB2Sp.Inparticular,itwasprovedthatif0<<1,thenthereexistsaconstantc>0suchthatforeveryl0,p2[1;1),f2(R),andforarbitraryself-adjointoperatorsAandBonHilbertspacewithboundedAB,thefollowinginequalityholdsforeveryjl:sj(f(A)f(B))ckfk(R)(1+j)pkABkSlp(see(3.1.1)for:Insectionx3.2,wegeneralizethisestimatetotheclass!.Weprovethatiffunctionfbelongstotheclass!foranarbitrarymodulusofcontinuity!,thensj(f(A)f(B))c!(1+j)1pkABkSlpkfk!forarbitraryself-adjointoperatorsA,Bandall1jl.Theresultisthengeneralizedtocontractions,maximaldissipativeoperators,normaloperatorsandn-tuplesofcommutingself-adjointoperators.Wealsoobtainsomelower-boundestimatesforrankoneperturbationswhichalsoextendtheresultsin[2].Insectionx3.3,similarestimatesaregivenwithoutproofsincaseofcontractions,maximaldissipativeivoperators,normaloperatorsandn-tuplesofcommutingself-adjointoperators.Inchapter1,wegiveabriefintroductiontothetheoryofdoubleoperatorintegralsandtheirapplicationstotheperturbationtheory.Wereferthereaderto[21]formoredetails.NecessaryinformationonfunctionspacesBsp;qand!aregiveninsectionx2.2.Wereferthereaderto[1]formoredetailedinformation.Theresultsobtainedinsectionx3.2andx3.3wereprovedin[14],submittedtotheIndianaUniversityMathematicsJournalinApril,2016.vTABLEOFCONTENTSChapter1Abriefnoteondoubleoperatorintegrals.............11.1Introduction....................................11.1.1Formalofdoubleoperatorintegrals.............11.1.2Functionsofnon-commutingoperators.................31.2DOIonS2.....................................41.3DOIonS1andB.................................71.3.1ClassM..................................71.3.2Criterionof2M............................91.4Transformersonotherclasses..........................101.5ApplicationsofDOItotheperturbationtheory................111.5.1TransformersZ˚.............................111.5.2Testsfor˚2MS.............................131.6DOIwithrespecttosemi-spectralmeasures..................15Chapter2OperatorolderFunctionsandarbitrarymoduliofcontinuity192.1Introduction....................................192.2Functionspaces..................................192.2.1Besovclasses...............................192.2.2Spaces!.................................252.3olderestimatesforself{adjointoperators...................272.4olderestimatesforotherclassesofoperators.................312.4.1Thecaseofunitaryoperators......................322.4.2Thecaseofcontractions.........................332.4.3Thecaseofmaximaldissipativeoperators...............342.4.3.1Dissipativeoperators......................342.4.3.2olderEstimates........................382.4.4Thecaseofnormaloperators......................392.4.5Thecaseofn{Tuplesofcommutingself{adjointoperators.......402.5Arbitrarymoduliofcontinuity..........................41Chapter3Estimatesonsingularvalues.....................473.1ResultsforperturbationofclassSp.......................473.2Estimatesonsingularvaluesoffunctionsofperturbedself-adjointandunitaryoperators.....................................503.3Estimatesforothertypesofoperators......................57BIBLIOGRAPHY....................................60viChapter1Abriefnoteondoubleoperatorintegrals1.1Introduction1.1.1FormalofdoubleoperatorintegralsFormally,DoubleOperatorIntegrals(DOI)areobjectsoftheformT=ZXZYx;y)dE1(x)QdE2(y):(1.1.1)In(1.1.1)(X;E1())and(Y;E2())aretwospaceswithspectralmeasure.ThevaluesofthemeasureE1()areorthogonalprojectionsinaseparableHilbertspaceH1,andsimilarforthemeasureE2()intheHilbertspaceH2.Thescalar-valuedfunctionx;y)(thesymboloftheDOI)isonXY.Finally,QisalinearboundedoperatoractingfromH2toH1,orQ2B(H2;H1).UnderreasonabletheresultTisalsoanoperatoractingfromH2toH1.Hence,theintegral(1.1.1)alinearmapping(transformer)TE1;E2:Q7!T:(1.1.2)1TE1;E2isoftenwrittenasTforshort,especiallywhenthespectralmeasuresE1;E2areSometimeswewriteT:=TE1;E2Q(1.1.3)IfE1;E2arethespectralmeasuresofself-adjointoperatorsA;B(E1=EA1;E2=EB2),theninsteadof(1.1.3)wewriteT:=TA;BQ(1.1.4)Rigorousoftheintegral(1.1.1)dependsonthespaceofoperatorswewishtodealwithandtheclassofadmissiblesymbolsisalsodeterminedbythechoiceofthisspace.InthecaseofthespaceS2ofHilbert{Schmidtoperators,theintegral(1.1.1)canbewellforanarbitraryboundedandmeasurablesymbolwithrespecttoanappropriatemeasureonXY.ThemeasureisdeterminedbythegivenspectralmeasuresE1andE2;theoperatorTisalsoHilbert{Schmidtandmoreover,kTkS2()-supjjkQkS2:(1.1.5)Allthis,includingtheconstructionofthemeasure˙,willbeexplainedinsectionx1.2.Forotherspacesofoperatorsthesituationismorecomplex.Oneofthemostimportantcasesiswhentheintegral(1.1.1)canbewellforanyboundedoperatorQandtheresultingoperatorTisalsobounded.ThenthetransformerTE1;E2actsinthespaceB(H2;H1)andisboundedbyClosedGraphTheorem.Theorem1.3.1givesafulldescriptionoftheclassMofalladmissiblesymbolsofthistype.If2M,thenthetransformerTE1;E2isalsoboundedinthespaceS1ofalltraceclassoperatorsandinthespaceS1ofallcompactoperators.Itispossibletoconsidertheactionoftheintegral(1.1.1)betweenotherspaces2ofoperators,andthespacesforQandTmayfromeachotherandtheexhaustivedescriptionoftheclassofadmissiblesymbolsforthemostofcasesisnotknow.However,therearemanytconditionswhichallowonetoapplythegeneralresultsofthetheoryofDOI.1.1.2Functionsofnon-commutingoperatorsSupposethatH2=H1andin(1.1.1)X=Y=R,E1=EA1,E2=EB2whereA;Bareself-adjointoperators.ThenitisnaturaltoregardTasthefunctionofthepair(A;B),separatedbytheoperatorQ.TheoperatorsAandBarenotassumedcommuting,sincethepresenceoftheoperatorQpreventsanypossiblegainswhichmightcomefromthecommutationofAandB.Forthesimplecasewhenx;y)=˚(x) (y)where˚and areboundedfunctions,wehavebySpectralTheorem˚(A)Q (B)=Z˚(x)dE1(x)QZ (y)dE2(y):Formally,thiscanbere-writtenasT=˚(A)Q (B)=Z˙(A)Z˙(B)˚(x) (y)dE1(x)QdE2(y):(1.1.6)Moreover,wehavekTkk˚kL1(A;E1)k kL1(B;E2)kQk:(1.1.7)3Theequality(1.1.6)canserveastheionoftheintegral(1.1.1)forthefunctionx;y)=˚(x) (y).Thisextendsnaturallytothesumsx;y)=X1kN˚k(x) k(y);inparticulartothecasewhenisapolynomialinx;yandtheoperatorsA;Barebounded.However,theestimatesimilarto(1.1.7),i.e.kTkkkL1kQkisnolongervalid.Theorem1.3.1willgiveanestimateoftheoperatornorminamoregeneralsituation.IfoneisonlyinterestedintheHilbert{Schmidtnorm,theestimate(1.1.5)givesthedesiredresult.1.2DOIonS2Let(X;E1)and(Y;E2)betwospectralmeasuresinthespaceH1andH2respectively.TheHilbert{SchmidtclassS2=S2(H2;H1)isaHilbertspace,withrespecttothescalarproductQ;R=tr(QR)=tr(RQ):(1.2.1)WewillconstructacertainspectralmeasureonS2,thetensorproductofmeasures(X;E1)and(Y;E2),andtheDOITasintegralwithrespecttothisspectralmeasure.4Considerthemappings8>>><>>>:E1():Q7!E1()Q;forˆX;Q2S2;E2(@):Q7!QE2(@);for@ˆY;Q2S2:(1.2.2)EachoperatorE1()isanorthogonalprojectioninS2,themapping7!E1()is˙-additive,andE1(X)=I(theidentitytransformeronS2).SoweseethatE1isaspectralmeasureinS2,andthesameforE2.ThetypesofE1andE2coincidewiththatofE1andE2respectively.Thusforanyboundedmeasurablefunctions˚(x), (y)wehaveZX˚(x)d(E1(x)Q)=ZX˚(x)dE1(x)QandZY (y)d(E2(y)Q)=QZY (y)dE2(y):ThemeasuresE1andE2commute,sinceonecorrespondstothemultiplicationfromtheleftandtheotherfromtheright.ThemappingE(@)=E1()E2(@):Q7!E1()QE2(@)(1.2.3)isanadditiveprojection-valuedfunctiononthesetofall"measurablerectangles"@ˆXY(orthogonalprojectionsonS2).Itturnsout(see[20])thatthisfunctionis˙-additive.The˙-additiveprojection-valuedfunctionEextends,inauniqueway,fromthesetofmeasurablerectangles=@totheminimal˙-algebraA0ofsubsetsinXY,generatedbysuchrectangles,andtheextensionis˙-algebra,soitisaspectralmeasureinS2.WedenoteitbythesamenotationE.ItisconvenienttoaddtoA0allthesubsets0ˆofsets52A0ofE-measurezero,puttingE(0)=0.TheresultingfamilyAisalsoa˙-algebra,andthespectralmeasureEonAisN{full(seeBirmansectionI.3.7).AscalarmeasureoftypeEcanbechosenasthemeasurein(1.1.5).NowwetakebydT=ZXYx;y)dE(x;y);(1.2.4)orTQ=ZXYx;y)d(E(x;y)Q):(1.2.5)So,forboundedthisisaboundedtransformerinS2.ThenwehaveT12=T1+T2;T12=T1T2;(1.2.6)T=T;(1.2.7)kTk=kkL1(XY):(1.2.8)Ifx;y)=˚(x),thenT=RX˚(x)dE1(x),orTQ=RX˚(x)dE1(x)Q.Thesimilarformulaisvalidforx;y)= (y).Fromthisobservationand(1.2.6),weseethatZXY˚(x) (y)d(E(x;y)Q)=ZX˚(x)dE1(x)QZY (y)dE2(y):61.3DOIonS1andB1.3.1ClassMNowweextendthenitionofTtothespaceB=B(H2;H1)ofallboundedoperators.Todothisweneedsomeadditionalassumptionsonthesymbolsinceitisnotalwayspossible.LetS1bethetraceclassofoperators,thenS1ˆS2ˆB:(1.3.1)Moreover,thespaceBisadjointtoS1,withrepecttothedualitygivenby(1.2.1):Q;R=tr(QR);Q2S1;R2B:(1.3.2)Clearly,anytransformerTwithaL1{symbolmapsS1intoS2.SupposethatTisaboundedtransformerfromS1intoS1itselfforagivenfunction.ThenthetransformerTisalsoboundedinS1andhasthesamenorm.TheadjointtransformerTactsinthespaceB.Theequality(1.2.7)showsthatitisnaturaltodTQ=(TjS1)Q;8Q2B:(1.3.3)Theproperties(1.2.6)ofthetransformersTextendtothewholeofB.LetTbeaboundedtransformerwithaL1{symbolthatmapsfromS1intoS1.IfQ2S1(thespaceofallcompactoperators),thenTQ2S1.Indeed,itisttoshowthisforthedenseinS1subsetKofrankoperators.ButifQ2K,then7TQ2S1ˆS1.SoTactsfromS1intoS1andkTkB!B=kTkS1!S1=kTkS1!S1:(1.3.4)Byinterpolation,wegetkTkB!BkTkS2!S2=kkL1:(1.3.5)DenotebyMBthesetofallfunctionsonXY,suchthatthetransformerTisboundedonB.Thisisanormedalgebraoffunction,withrespecttothenormkkMB=kTkB!B:Themapping7!isaninvolutioninMB.Itthenfollowsfrom(1.3.5)thatthealgebraMBiscompleteandhence,isaBanachC{algebra.TheBanachalgebrasMS1andMS1areintroducedinthesameway.ItfollowsfromdualitythatM:=MB=MS1=MS1;includingequalityofthecorrespondingnorms.TheclassMdependsonthechoiceofthespectralmeasuresE1andE2.WeshalluseM(E1;E2)whenitisusefultothisdependenceexplicitly.81.3.2Criterionof2MLet(X;E1)and(Y;E2)betwospectralmeasuresinthespaceH1andH2respectively.Foreachh12H1,thefunctionˆh1()=(E1()h1;h1)isascalarmeasure.Similarly,thefunction˝h2()=(E2()h2;h2)isforeachh22H2.TheclassM(E1;E2)admitsthefollowingdescription.Theorem1.3.1.[17,18,23]Let2L1(E1;E2).Thenthefollowingstatementsareequivalent:(i)2M=M(E1;E2).(ii)Foranyh22H2,h12H1theintegraloperatorKh2;h1:L2(Y;˝h2)!L2(X;ˆh1);(Kh2;h1u)(x)=RYx;y)u(y)d˝h2(y)belongstoS1,andsupkh1k=kh2k=1kKh2;h1kS1=:C<1:Moreover,kkM=C:(iii)Thereexistameasurespace(Z;)andmeasurablefunctionsonXZ,onYZsuchthatx;y)=ZZ(x;z)(y;z)(z)(1.3.6)and8>>><>>>:A2:=(E1)-supxRZj(x;z)j2(z)<1;B2:=(E2)-supyRZj(y;z)j2(z)<1:(1.3.7)ForanysuchfactorizationkkMAB;(1.3.8)9andthereexistsafactorizationsuchthatcABkkM;c>0:(1.3.9)TheconstantcdoesnotdependonthespectralmeasuresE1,E2.Fortheproof,see[17],[18]and[23].Thesetoffunctionsthatadmittherepresentationin(1.3.6)and(1.3.7)iscalledtheintegralprojectivetensorproductofspacesL1(E1)andL1(E2).1.4TransformersonotherclassesLetB=B=(H2;H1),whereH1andH2betwogivenseparableHilbertspaces.ForeachQ2B,thesingularvaluessnisbysn(Q):=n(pQQ),n0.TheSchattenidealsSp,weakSp{idealsSp;w,idealsSp;wandspacesSp;1arebySp=fQ2S1:fsn(Q)g2lpg;0<p<1:(1.4.1)Sp;w=fQ2S1:sn(Q)=O(n1=p)g;0<p<1:(1.4.2)Sp;w=fQ2S1:sn(Q)=o(n1=p)g;0<p<1:(1.4.3)Sp;1=fQ2S1:Xn(n+1)p11sn(Q)<1g;0<p<1:(1.4.4)For1<p<1,certainnormscanbeintroducedsuchthatthesespacesbecomeBanachalgebras.Theywillbecalledthenicesymmetrically-normedideals(See[12]formoredetailsonsymmetrically-normedideals(SNI))andwehavethefollowingdualityrelationsgivenby10(1.2.1)Sp=Sp0;(Sp;w)=Sp0;1;Sp;1=Sp0;w;1=p0=11=p:(1.4.5)GivenaSNIS,thesetofsymbolssuchthatthetransformerTisboundedonS,formacommutativeBanachalgebraoffunctionsonXY,withcomplexconjugationastheinvolution.WedenotethisalgebraasMS.Itfollowsfromthedualityargumentsandinterpolationthatfor1<p<1MSp=MSp0;MSp;w=MSp0;1=MSp;w(1.4.6)andforanyniceSNIS,thefollowingtopologicalimbeddingshold:MˆMSˆMS2=L1(XY):(1.4.7)Furthermore,if2M,thenkkMSkkMforanyniceSNIS.1.5ApplicationsofDOItotheperturbationtheory1.5.1TransformersZ˚Let(R;E1)and(R;E2)betwospectralmeasuresintheseparableHilbertspaceH1andH2respectively.ThespectralmeasureEisasinx1.2.DenotethediagonalofR2bydiag,i.e.diagdef=f(x;x):x2RgˆR2:Thenitisshownin[21]thatEjdiagisanatomicmeasure.LetAandBbeself-adjointoperatorsintheHilbertspacesH1andH2respectively.If˚11isauniformlyLipschitzfunctiononR,thenthefunction˚(x;y)=˚(x)˚(y)xyiswellandcontinuousoutsidethediagonalandbounded.SupposethatitissomehowextendedtodiagandtheextendedfunctionisboundedonR2.NotethatthisfunctionisalwaysE{measuresinceEisN{full.Ifatsomepointx2Rthefunction˚istiable,thenaturalchoiceofextensionis˚(x;x)def=˚0(x).Otherwise,thevalueof˚(x;x)canbechosenarbitrary.Belowwesupposethatsomeextensionof˚tothewholeofR2ischosenandThenthetransformerZA;B˚def=TA;B˚=ZRZR˚(x)˚(y)xydEA1(x)()dEB2(y)(1.5.1)iswellatleastontheclassS2.Wedonotthechoiceofextensioninthenotations,sincetheformulaspresentedinTheorem1.5.1,1.5.2holdtrueindependentlyofit.Moreover,foranySNISthemembership˚2MSdoesnotdependonthischoice.Thisfollowsfromsection7.1of[21].Theorem1.5.1.[18]LetH1=H2beaHilbertspaceandA,Bbeself-adjointoperatorswiththesamedomaininH1,andsupposethatBA2SwhereSisaniceSNI.Supposealsothatthefunction˚issuchthat˚2MS.Then,˚(B)˚(A)=ZA;B˚(BA):(1.5.2)ItallowstheoperatorsA;Btobeunbounded.12Formula(1.5.2)iscalledtheBirman{Solomyakformula.Asimilarformulaalsoholdsforunitaryoperators,inwhichcasewehavetointegrate˚ofafunction˚ontheunitcirclewithwithrespecttothespectralmeasuresofthecorrespondingoperatorintegrals.Theorem1.5.1extendstothequasi{commutatorsJBAJ.HereJisalinearboundedoperatoractingfromH2toH1.TheoperatorsA;Barenotsupposedbounded,andJBAJisunderstoodastheoperatorgeneratedbythesesqui-linearform(JBh2;h1)(Jh2;Ah1)whereh12DomA,h22DomB.Theorem1.5.2.[18]LetAandBbeself-adjointoperatorsinHilbertspaceH1andH2respectivelyandletJ2B(H2;H1).SupposethatJBBA2SwhereSisaniceSNI,andthat˚2MS.Then,independentlyontheway˚isdonthediagonal,J˚(B)˚(A)J=ZA;B˚(JBAJ):(1.5.3)Theorem1.5.2turnsintoTheorem1.5.1ifwetakeH2=H1andJ=I.Boththeoremswereprovedin[18].1.5.2Testsfor˚2MSForpracticalusageofTheorem1.5.1,1.5.2oneneedstoolsforcheckingtheinclusion˚2MSforagivenSNIS.AparticularcaseofTheorem1.5.1saysthatforself-adjointoperatorsAandBj˚(x)˚(y)jLjxyj)k˚(A)˚(B)kS2LkABkS2:13ItiswellknownthataLipschitzfunctiononthereallineisnotnecessarilyoperatorLipschitz,i.e.,theconditionj˚(x)˚(y)jconstjxyjdoesnotimplythatforself-adjointoperatorsAandBk˚(A)˚(B)kconstkABk:DenoteM(R;R)andM(T;T)byM(R)andM(T)respectively.Itwasshownin[23]thatif˚isatrigonometricpolynomialofdegreed,then˚2M(T)andk˚kM(T)constdk˚kL1:(1.5.4)Ontheotherhand,itwasshownin[25]thatif˚isaboundedfunctiononRwhoseFouriertransformissupportedon[˙;˙](inotherwords,˚isanentirefunctionofexponentialtypeatmost˙thatisboundedonR),then˚2M(R)andk˚kM(R)const˙k˚kL1:(1.5.5)Inequalities(1.5.4)and(1.5.5)ledin[23]and[25]tothefactthatfunctionsinfunctionsintheBesovspacesB11;1(T)andB11;1(R)areoperatorLipschitz.ThegegeralBesovspaceBsp;qwillbeinx2.1for1p;q1ands2R.HereweonlygiveanequivalentdescriptionforB11;1(R).Afunction˚onRbelongstoforB11;1(R)ifr0(˚)def=Z10(supx2Rj˚(x+t)2˚(x)+˚(xt)j)dtt2<1:(1.5.6)14Anyfunction˚2B11;1(R)hasuniformlyboundedcontinuousderivative,andwedenoter(˚)def=r0(˚)+supx2Rj˚0(x)j:ThespaceB11;1(T)canbesimilarly.Theorem1.5.3.[25]Let˚2B11;1(R).Then˚2M(R)foranyspectralmeasuresE1,E2,andk˚kM(R)Cr(˚);wheretheconstantCisindependentofthespectralmeasuresE1,E2.Inparticular,foranyniceSNISwehavekJ˚(B)˚(A)JkSCr(˚)kJBBAk(1.5.7)forself{adjointoperatorsA;BandboundedoperatorJasinTheorem1.5.2.Asimilarresultalsoholdswhen˚2B11;1(T).Inchapter2wewillexplaintheresultfoundin[1]thatif˚belongstotheolderclass(R)with0<<1,thenk˚(A)˚(B)kconstkABkforarbitraryself-adjointoperatorsAandB.1.6DOIwithrespecttosemi-spectralmeasuresLetHbeaHilbertspaceandlet(X;A)beameasurablespace.AmapFfromAtothealgebraB(H)ofallboundedoperatorsonHiscalledasemi{spectralmeasureifF0;2A;15F(;)=0andF(X)=I;andforasequencefjgj1ofdisjointsetsinA,F(1[j=1j)=limN!1NXj=1Fj)intheweakoperatortopology:IfKisaHilbertspace,(X;A)isameasurablespace,F:A7!B(K)isaspectralmeasure,andHisasubspaceofK,thenitiseasytoseethatthemapF:A7!B(H)byF=PHFjH;2A(1.6.1)isasemi-spectralmeasure.HerePHstandsfortheorthogonalprojectionontoH.Naimarkprovedin[15]thatallsemi{spectralmeasurescanbeobtainedinthisway,i.e.,asemi{spectralmeasureisalwaysacompressionofaspectralmeasure.AspectralmeasureFsatisfying(1.6.1)iscalledaspectraldilationofthesemi{spectralmeasureF.AspectraldilationFofasemi-spectralmeasureFiscalledminimalifK=closspanfFH:2Ag:Itwasshownin[16]thatifFisaminimalspectraldilationofasemi{spectralmeasureF,thenFandFaremutuallyabsolutelycontinuousandallminimalspectraldilationsofasemi{spectralmeasureareisomorphicinthenaturalsense.If˚isaboundedmeasurablefunctionXandF:A7!B(H)isasemi{spectralmeasure,thentheintegralZX˚(x)dF(x)(1.6.2)16canbeasZX˚(x)dF(x)=PH(ZX˚(x)dF(x))H;(1.6.3)whereFisaspectraldilationofF.Itiseasytoseethattheright{handsideof(1.6.3)doesnotdependonthechoiceofaspectraldilation.Theintegral(1.6.2)canalsobecomputedasthelimitofsumsX˚(x)F);x2;overallmeasurablepartitionsfgofX.IfTisacontractiononaHilbertspaceH,thenbytheSz.{Nagydilationtheorem(see[8]),Thasaunitarydilation,i.e.,thereexistaHilbertspaceKsuchthatHˆKandaunitaryoperatorUonKsuchthatTn=PHUnjH;n0;(1.6.4)wherePHistheorthogonalprojectionontoH.LetFUbethespectralmeasureofU.ConsidertheoperatorsetfuntionFontheBorelsubsetsoftheunitcircleTbyF=PHFUjH;ˆT:ThenFisasemi{spectralmeasure.Itfollowsfrom(1.6.4)thatTn=ZTndF()=PHZTndFU()H;n0:(1.6.5)Suchasemi{spectralmeasureFiscalledasemi{spectralmeasureofT.Notethatitisnotunique.Tohaveuniqueness,weconsideraminimalunitarydilationUofT,whichisunique17uptoanisomorphism(see[8]).Itfollowseasilyfrom(1.6.5)that˚(T)=PHZT˚()dFU()Hforanarbitraryfunction˚inthedisk{algebraCA.In[24]and[27]DOIwithrespecttosemi{spectralmeasureswereintroduced.Supposethat(X1;A1)and(X2;A2)aremeasurablespaces,andF1:A17!B(H1)andF2:A27!B(H2)aresemi{spectralmeasures.ThendoubleoperatorintegralZX1X2x1;x2)dF1(x1)QdF2(x2)werein[27]inthecasewhenQ2S2andisaboundedsymbol.DOIwerealsoin[27]inthecasewhenQisaboundedlinearoperatorandbelongstotheintegralprojectivetensorproductofthespacesL1(F1)andL1(F2).Inparticular,thefollowingBirman{Solomyakformulaholds:˚(R)˚(Q)=ZTT˚(;˝)dFR()(RQ)dFQ(˝):(1.6.6)HereRandQarecontractionsonHilbertspace.FRandFQaretheirsemi{spectralmea-sures,and˚isananalyticinDofclass(B11;1)+(Forseesectionx2.2.1).18Chapter2OperatorolderFunctionsandarbitrarymoduliofcontinuity2.1IntroductionLet0<<1.Inwasprovedin[1]thatthefunctionsintheolderclassarealsooperatoroldercontinuousforarbitraryself{adjointoperators,unitaryoperatorsandcontractions,andtheirsharpestimatesarealsoobtained.Similarresultsformaximaldissipativeoperators,normaloperatorsandn{tuplesofself{adjointopratorsarealsoobtainedin[1],[4],[5]and[10].Inthosepapers,theauthorsalsoextendedtheresultstoclass!.Wewillshowtheproofforself{adjointoperatorsandunitaryoperatorsandgiveashortdiscussionforothertypesofoperators.Anintroductiontothefunctionspacesand!aregivenbelowinx2.2.1andx2.2.2.2.2Functionspaces2.2.1BesovclassesInthissubsectionwegiveabriefintroductiontotheBesovspacesthatplayanimportantroleintheproblemsofperturbationtheory.WestartwithBesovspacesontheunitcircle.19Let1p;q1ands2R:TheBesovclassBsp;qoffunctions(ordistributions)onTcanbedinthefollowingway.LetwbeannitelytiablefunctiononRsuchthatw0;suppwˆ[12;2];andw(x)=1w(x2)forx2[1;2]:(2.2.1)aC1functionvonRbyv(x)=1forx2[1;1]andv(x)=w(jxj)ifjxj1:(2.2.2)trigonometricpolynomialsWn,W]nandVnbyWn(z)=Xk2Zw(k2n)zk;n1;W0(z)=z+1+z;andW]n(z)=Wn(z);n0andVn(z)=Xk2Zv(k2n)zk;n1:VniscalleddelaValleePoussintypekernel.IffisadistributiononT,wedefn,n0byfn=fWn+fW]n;n1;andf0=fW0;Thenf=Pn0fnandffVn=P1k=n+1fn.TheBesovclassBsp;qconsistsoffunctions(inthecases>0)ordistributionsfonTsuchthatfk2nsfWnkLpgn12`qandfk2nsfW]nkLpgn12`q(2.2.3)20Besovclassesadmitmanyotherdescriptions.Inparticular,fors>0,thespaceBsp;qadmitsthefollowingcharacterization.Afunctionf2LpbelongstoBsp;q,s>0,ifandonlyif8>>>><>>>>:RTkn˝fkqLpj1˝j1+sqdm(˝);forq<1;sup˝6=1kn˝fkLpj1˝js<1;forq=1:(2.2.4)HeremisthenormalizedLebesguemeasureonT,nisanintegergreaterthans,and˝,˝2T,istheoperator:˝f)()=f(˝)f();2T:WeusethenotationBspforBsp;p:Thespacesdef=B1formtheolder{Zygmundclass.If0<<1,thenf2ifandonlyifjf()f(˝)jconstj˝j;;˝2T:Thesespacesarecalledtheolderspaces.Afunctionf21ifandonlyiffiscontinuousandjf(˝)2f()+f(˝)jconstj1˝j;;˝2T:By(2.2.4),for>0,f2ifandonlyiffiscontinuousandjn˝f)()jconstj1˝j;wherenisapositiveintegersuchthatn>.21Notethatthe(semi)normofafunctionfinisequivalenttosupn12(kfWnkL1+kfW]nkL1):ItiseasytoseefromtheofBesovclassesthattheRieszprojectionP+,P+f=Xn0^f(n)zn;isboundedonBsp;q.Functionsin(Bsp;q)+def=P+Bsp;qadmitanaturalextensiontoanalyticfunctionsintheunitdiskD.Itiswellknownthatthefunctionsin(Bsp;q)+admitthefollowingdescription:f2(Bsp;q)+,Z10(1r)q(ns)1kf(n)rkqpdr<1;q<1;andf2(Bsp;1)+,sup0<r<1(1r)(ns)kf(n)rkp<1;wherefr()def=f(r)andnisanonnegativeintegergreaterthans.LetusproceednowtoBesovspacesontherealline.WeconsiderhomogeneousBesovspacesBsp;q(R)offunctions(distributions)onR.Weusethesamefunctionsw,vasin(2.2.1),(2.2.2),andfunctionsWn,W]nandVnonRbyFWn(x)=w(x2n);FW]n(x)=FWn(x);n2Z22andFVn(x)=v(x2n);n2Z;whereFistheFouriertransform:(Ff)(t)=ZRf(x)eixtdx;f2L1:VnisalsocalleddelaValleePoussintypekernel.IffbelongstoS0(R),thespaceoftempereddistributiononR,wedefnbyfn=fWn+fW]n;n2Z:Initiallywethe(homogeneous)Besovclass_Bsp;q(R)asthesetofallf2S0(R)suchthatf2nskfnkLpgn2Z2`q(Z):(2.2.5)Accordingtothisthespace_Bsp;q(R)containsallpolynomials.Moreover,thedistributionfisbythesequenceffngn2Zuniquelyuptoapolynomial.ItiseasytoseethattheseriesPn0fnconvergesinS0(R).However,theseriesPn<0fncandivergeingeneral.ItiseasytoprovethattheseriesPn<0f(r)nconvergesuniformlyonRforeachnonnegativeintegerr>s1=p.Notethatinthecaseq=1theseriesPn<0f(r)nconvergesuniformly,wheneverrs1=p.Nowwethemo(homogeneous)BesovclassBsp;q(R).Wesaythatadistri-butionfbelongstoBsp;q(R)iff2nskfnkLpgn2Z2`q(Z)andf(r)=Pn2Zf(r)ninthespaceS0(R),whereristheminimalnonnegativeintegersuchthatr>s1=p(rs1=p23ifq=1).Nowthefunctionfisdetermineduniquelybythesequenceffngn2Zuptoapolynomialofdegreelessthanr,andapolynomial˚belongstoBsp;q(R)ifandonlyifdeg˚<r.BesovspacesBsp;q(R)admitequivalentthataresimilartothosediscussedaboveincaseofBesovspacesoffunctionsonT.Inparticular,theolder{Zygmundclasses(R)def=B1(R);>0,canbedescribedastheclassesofcontinuousfunctionsfonRsuchthatjmt)(x)jconstjtj;t2R;wheretheenceoperatortisbytf)(x)=f(x+t)f(x);x2R;andmisanintegergreaterthan.Asinthecaseoffunctionsontheunitcircle,wecanintroducethefollowingequivalent(semi)normon(R):supn2Z2(kfWnkL1+kfW]nkL1);f2(R):Thefollowingresultwillbeusedinx2.3.Theorem2.2.1.[1]Let>0.Thenforeach>0andeachfunctionf2(R)thereexistsafunctiong2(R)withcompactsupportsuchthatf(t)=g(t)fort2[0;1]andkgkkfk+24wheretheconstantcandependonlyon.ToproveTheorem2.2.1,weusethewell{knownfactthatif˚andfarefunctionsin(R)and˚hascompactsupport,then˚f2(R).Wereferthereaderto[11],Section4:5:2fortheproof.DenotebyS0+(R)thesetofallf2S0(R)suchthatsuppFfˆ[0;1).WetheanalyticBesovspaceBsp;q(R)+asBsp;q(R)[S0+(R).Put(R)+def=(R)[S0+(R).Forf2S0+(R),wehavefW]n=0,n2Z.Wereferthereaderto[1],[13]and[26]formoredetailedinformationonBesovspaces.2.2.2Spaces!Let!beamodulusofcontinuity,i.e.,!isanondecreasingcontinuousfunctionon[0;1)suchthat!(0)=0,!(x)>0forx>0,and!(x+y)!(x)+!(y);x;y2[0;1):Wedenoteby!(R)thespaceoffunctionsonRsuchthatkfk!(R)def=supx6=yjf(x)f(y)j!(jxyj):Thespace!(T)ontheunitcirclecanbeinasimilarway.Theorem2.2.2.[1]Thereexistsaconstantcsuchthatforanarbitrarymodulusofcon-tinuity!andforanarbitraryfunctionfin!(R),thefollowinginequalitiesholdforalln2Z:kffVnkL1c!(2n)kfk!(R):(2.2.6)25Proof.Wehavejf(x)(fVn)(x)j=2nZR(f(x)f(xy))V(2ny)dy2nkfk!(R)ZR!(jyj)jV(2ny)jdy2nkfk!(R)Z2n2n!(jyj)jV(2ny)jdy+2n+1kfk!(R)Z12n!(y)jV(2ny)jdy:Clearly,2nZ2n2n!(jyj)jV(2ny)jdy!(2n)kVkL1:Ontheotherhand,keepinginmindtheobviousinequality2n!(y)2y!(2n)fory2n,weobtain2n+1Z12n!(y)jV(2ny)jdy422n!(2n)Z12nyjV(2ny)jdy=4!(2n)Z11yjV(y)jdyconst!(2n)Thisproves(2.2.6).Remark2.2.3.[1]AsimilarinequalityholdsforfunctionsonTofclass!:kffVnkL1c!(2n)kfk!;n>0:Toproveit,ittoextendfasa2ˇ{periodicfunctiononRandapplyTheorem2.2.2.26Corollary2.2.4.[1]Thereexistsaconstantcsuchthatforanarbitrarymodulusofconti-nuity!andforanarbitraryfunctionfin!,thefollowinginequalitiesholdforalln2Z,inRcase,orforalln0,inTcase:kfWnkL1c!(2n)kfk!;kfW]nkL1c!(2n)kfk!:(2.2.7)Put!(R)+def=!(R)\S0+(R)andC+def=fz2C:Imz>0g.Thenafunctionin!(R)belongstothespace!(R)+ifandonlyifithasa(unique)continuousextensiontotheclosedupperhalf-planceclosC+thatisanalyticintheopenupperhalf-planeC+withatmostapolynomialgrowthrateaty.2.3olderestimatesforself{adjointoperatorsInthissectionweshowthatolderfunctionsonRoforder,0<<1;mustalsobeoperatorolderoforder.NotethatifAandBareself{adjointoperators,wesaythatoperatorABisboundedifB=A+Kforsomeboundedself{adjointoperatorK.Inparticular,thisimpliesthatDomA=DomB.WesaythatkABk=1ifthereisnosuchaboundedoperatorKthatB=A+K.Lemma2.3.1.[3]LetAandBbeself{adjointoperatorsandletRbeanoperatorofnorm1.ThenthereexistasequenceofoperatorsfRngn1andsequencesofboundedself{adjointoperatorsfAngn1andfBngn1suchthat(i)thesequencefkRnkgn1isnondecreasingandlimn!1kRnk=1;(ii)limn!1Rn=Rinthestrongoperatortopology;27(iii)foreverycontinuousfunctionfonR,thesequencefkf(An)RnRnf(Bn)kgn1isnondecreasingandlimn!1kf(An)RnRnf(Bn)k=kf(A)RRf(B)k;(iv)iffisacontinuousfunctiononRsuchthatkf(A)RRf(B)k<1,thenlimn!1f(An)RnRnf(Bn)=f(A)RRf(B)inthestrongoperatortopology;(v)iffisacontinuousfunctiononRsuchthatkf(A)RRf(B)k<1,thenthesequencefsj(f(An)RnRnf(Bn))gn1isnondecreasingforeveryj0andlimn!1sj(f(An)RnRnf(Bn))=sj(f(A)RRf(B)):Proof.PutPndef=EA([n;n])andQndef=EB([n;n])whereEAandEBarethespectralmeasuresofAandB.PutAndef=PnA=APnandBndef=QnB=BQn.Clearly,Pn(f(A)RRf(B))Qn=f(An)PnRQnPnRQnf(Bn);n1:(2.3.1)28ItremainstoputRndef=PnRQn.Theorem2.3.2.[1]Let0<<1.Thenthereisaconstantc>0suchthatforeveryf2(R)andforarbitraryself{adjointoperatorsAandBonHilbertspacethefollowinginequalityholds:kf(A)f(B)kckfk(R)kABk:(2.3.2)Proof.DuetoLemma2.3.1,wecanassumethatAandBareboundedoperators.ItthenfollowsfromTheorem2.2.1thatwemayassumethatf2L1(R)andwehavetoobtainanestimateforkf(A)f(B)kthatdoesnotdependonkfkL1.Putfn=fWn+fW]n:Letusshowthatf(A)f(B)=1Xn=(fn(A)fn(B))(2.3.3)andtheseriesontherightconvergesabsolutelyintheoperatornorm.ForN2Z,weputgNdef=fVN.Clearly,f=fVN+Xn>NfnandtheseriesontherightconvergesabsolutelyintheL1norm.Thusf(A)=(fVN)(A)+Xn>Nfn(A)andf(B)=(fVN)(B)+Xn>Nfn(B)29andtheseriesconvergeabsolutelyintheoperatornorm.Wehavef(A)f(B)Xn>N(fn(A)fn(B))=f(A)Xn>Nfn(A)f(B)Xn>Nfn(B)gN(A)gN(B):SincegN2L1(R)andgNisanentirefunctionofexponentialtypeatmost2N+1,itfollowsfrom(1.5.2)and(1.5.5)thatkgN(A)gN(B)kconst2NkfVNkL1kABkconst2NkfkL1kABk!0asN!.Thisproves(2.3.3).LetNbetheintegersuchthat2N<kABk2N+1:(2.3.4)Wehavef(A)f(B)=XnN(fn(A)fn(B))+Xn>N(fn(A)fn(B)):30Itfollowsfrom(2.2.5)and(2.3.4)thatXnN(fn(A)fn(B))XnNk(fn(A)fn(B))kconstXnN2nkfnkL1kABkconstXnN2n2kfk(R)kABkconst2N(1)kfk(R)kABkconstkfk(R)kABk:Ontheotherhand,Xn>N(fn(A)fn(B))Xn>N(kfn(A)k+kfn(B)k)2Xn>NkfnkL1constXn>N2Nkfk(R)const2Nkfk(R)constkfk(R)kABkby(2.3.4).Thiscompletestheproof.2.4olderestimatesforotherclassesofoperatorsInthissectionweobtainanalogsoftheresultoftheprevioussectionforfunctionsofunitaryoperators,contractions,maximaldissipativeoperators,normaloperatorsandn{Tuplesofcommutingself{adjointoperators.312.4.1ThecaseofunitaryoperatorsTheorem2.4.1.[1]Let0<<1.Thenthereexistsaconstantc>0suchthatforeveryf2(T)andforarbitraryunitaryoperatorsUandVonHilbertspacethefollowinginequalityholds:kf(U)f(V)kckfk(T)kUVk:(2.4.1)Proof.Letf2(T).Wehavef=P+f+Pf=f++f:Weestimatekf+(U)f+(V)k.Thenormofkf(U)f(V)kcanbeobtainedinthesameway.Thusweassumethatf=f+.Letfndef=fWn:Thenf=Xn0fn:(2.4.2)Clearly,wemayassumeU6=V.LetNbethenonnegativeintegersuchthat2N<kUVk2N+1:(2.4.3)Wehavef(U)f(V)=XnN(fn(U)fn(V))+Xn>N(fn(U)fn(V)):32ItfollowsfromtheBirman{Solomyakformulaforunitaryoperatorsand(1.5.4)thatXnN(fn(U)fn(V))XnNk(fn(U)fn(V))kconstXnN2nkfnkL1kUVkconstXnN2n2kfk(T)kUVkconst2N(1)kfk(T)kUVkconstkfk(T)kUVk:Ontheotherhand,Xn>N(fn(U)fn(V))Xn>N(kfn(U)k+kfn(V)k)2Xn>NkfnkL1constXn>N2Nkfk(T)const2Nkfk(T)constkfk(T)kUVkby(2.4.3).Thiscompletestheproof.2.4.2ThecaseofcontractionsRecallthatifTisacontractiononHilbertspace,itfollowsfromvonNeumann'sinequalitythatthepolynomialfunctionalcalculusf7!f(T)extendstothedisk{algebraCAandkf(T)kkfkCA,f2CA.Theorem2.4.2.[1]Let0<<1.Thenthereexistsaconstantc>0suchthatforeveryf2)+andforarbitrarycontractionsTandRonHilbertspacethefollowinginequality33holds:kf(T)f(R)kckfkkTRk:(2.4.4)Proof.TheproofofTheorem2.4.2isalmostthesameastheproofofTheorem2.4.1.Forf2)+,weuseexpansion(2.4.2)andchooseNsuchthat2N<kTRk2N+1:ThusasintheproofofTheorem2.4.2,fornN,weestimatekfn(T)fn(R)kintermsofconst2nkTRk(see(1.6.6)and(1.5.4)),whileforn>NweusevonNeumann'sinequalitytoestimatekfn(T)fn(R)kintermsof2kfnkL1.Therestoftheproofisthesame.Corollary2.4.3.[1]Letfbeafunctioninthedisk{algebraand0<<1.Thenthefollowingtwostatementsareequivalent:(i)kf(T)f(R)kconstkTRkforallcontractionsTandR,(ii)kf(U)f(V)kconstkUVkforallunitaryoperatorsUandVRemark2.4.4.[1,9]Thiscorollaryisalsotruefor=1.ThiswasprovedbyKissinandShulman(see[9]).2.4.3Thecaseofmaximaldissipativeoperators2.4.3.1DissipativeoperatorsInthissectionwegivenecessaryinformationofdissipativeoperatorsinordertointerprettheconstructionofthesemi{spectralmeasureofamaximaldissipativeoperator.Wereferthereaderto[4],[8]and[7]formoreinformation.342.4.5.LetHbeaHilbertspace.AnoperatorL(notnecessarilybounded)withdensedomainDLinHiscalleddissipativeifIm(Lu;u)0;u2DL:Adissipativeoperatoriscalledmaximaldissipativeifithasnoproperdissipativeexten-sion.NotethatifLisasymmetricoperator(i.e.,(Lu;u)2Rforeveryu2DL),thenLisdissipative.However,itcanhappenthatLismaximalsymmetric,butnotmaximaldissipative.TheCayleytransformofadissipativeoperatorLisbyTdef=(LiI)(L+iI)1withdomainDT=(L+iI)DLandrangeRangeT=(LiI)DL(theoperatorTisnotdenselyingeneral).Tisacontraction,i.e.,kT(u)kkuk,u2DT,1isnotaneigenvalueofT,andRange(IT)def=fuTu:u2DTgisdense.Conversely,ifTisacontractiononitsdomainDT,1isnotaneigenvalueofT,andRange(IT)isdense,thenitistheCayleytransformofadissipativeoperatorLandListheinverseCayleytransformofT:L=i(I+T)(IT)1;DL=Range(IT):AdissipativeoperatorismaximalifandonlyifthedomainofitsCayleytransformisthewholeHilbertspace.35Everydissipativeoperatorhasamaximaldissipativeextension.Everymaximaldissipa-tiveoperatorisnecessarilyclosed.IfLisamaximaldissipativeoperator,thenLisalsomaximaldissipative.IfLisamaximaldissipativeoperator,thenitsspectrum˙(L)iscontainedintheclosedupperhalf{planeclosC+andk(L)1k1jImj;Im<0:(2.4.5)IfLandMaremaximaldissipativeoperators,wesaythattheoperatorLMisboundedifthereexistsaboundedoperatorKsuchthatL=M+K.Anelementaryfactis(see[4]fortheproof)thatifLisamaximaldissipativeoperatorandMisadissipativeoperatorsuchthatLMisbounded,thenMisalsomaximaldissipative.Theconstructionofthefunctionalcalculusfordissipativeoperatorswasgivenin[4].LetLbeamaximaldissipativeoperatorandletTbeitsCayleytransform.ConsideritsminimalunitarydilationU,i.e.,UisaunitaryoperatoronaHilbertspaceKthatcontainsHsuchthatTn=PHUnjH;n0;andK=closspanfUnh:h2Hg.Since1isnotaneigenvalueofT,itfollowsthat1isnotaneigenvalueofU(see[8],Ch.II,x6).TheSz{Nagy{Foia˘sfunctionalcalculusallowsusafunctionalcalculusforTontheBanachalgebraCA;1def=g2H1:giscontinuousonTnf1g:36Ifg2CA;1,weputg(T)def=PHg(U)jH:Thisfunctionalcalculusislinearandmultiplicativeandkg(T)kkgkH1;g2CA;1:AfunctionalcalculusforthedissipativeoperatorontheBanachalgebraCA;1def=ff2H1(C+):fiscontinuousonRgbyf(L)def=(f!)(T);f2CA;1;where!istheconformalmapofDontoC+by!()def=i(1+)(1)1,2D.LetLbeamaximaldissipativeoperator,TbeitsCayleytransformandletETbethesemi{spectralmeasureofTontheunitcircle.Theng(T)=ZTg()dET();g2CA;1:(2.4.6)Thesemi{spectralmeasureELofLcanbebyELdef=ET(!1;isaBorelsubsetofR:37Itfolllowsfrom(2.4.6)thatf(L)=ZRf(x)dEL(x);f2CA;1:(2.4.7)2.4.3.2olderEstimatesItwasshownin[4]thatiffisaboundedfunctiononRwhoseFouriertransformhascompactsupportin(0;1),andifLandMaremaximaldissipativeoperatorssuchthatLMisbounded,thentheBirman{SolomyakformulaholdsforLandMwithrespecttotheirsemi{spectralmeasuresandkf(L)f(M)k8˙kfkL1(R)kLMk:(2.4.8)Itthenfollowsthatiff2B11;1(R)+,wecanassociatewithfthesequenceffngn2Zbyfndef=fWn,whichgivesf=1Xn=fn:Theseriesconvergesuniformly.ThentheBirman{Solomyakformulaalsoholdsforf.Notethatfisnotnecessarilybounded,andwhenitisnotbounded,theoperatorf(L)f(M)isbyf(L)f(M)def=1Xn=(fn(L)fn(M)):(2.4.9)Asinthecaseofself{adjointoperators,theseriesontherightconvergesabsolutelyandthedoesnotdependonthechoiceofthefunctionsWn.Furthermore,thefunctionsin38B11;1(R)+areoperatorLipschitzontheclassofmaximaldissipativeoperators.Theorem2.4.6.[4]Thereisaconstantc>0suchthatforevery2(0;1),forarbitraryf2(R)+,andforarbitrarymaximaldissipativeoperatorsLandMwithboundedLM,thefollowinginequalityholds:kf(L)f(M)kc(1)1kfk(R)kLMk;(2.4.10)wheref(L)f(M)isdby(2.4.9).Proof.UsingthesameargumentsasintheproofofTheorem2.3.2,wegetkf(L)f(M)kconstkfk(R)kLMk:Thefactthattheconstantinthisinequalitycanbeestimatedintermsofc(1)1followsimmediatelyformTheorem2.5.5below.2.4.4ThecaseofnormaloperatorsIn[5]itwasshownthattheBirman{Solomyakformulaholdsforarbitrarynormaloperatorsonlyforlinearfunctionsandanewformulaforthef(A)f(B)wasestablishedforfunctionsintheBesovspaceB11;1(R2)andnormaloperatorsN1,N2intermsofDOI.Readersarereferredto[5]fortheofB11;1(R2)andtheconstructionofthetheoryofDOIfornormaloperators.AlsodenotebyFtheFouriertransformonL1(Rn),n1by:(Ff)(t)=ZRnf(x)ei(x;t)dx;where39x=(x1;:::;xn);t=(t1;:::;tn);(x;t)def=x1t1+:::+xntn:Thefollowingimportantresultwasprovedin[5]:LetfbeaboundedcontinuousfunctiononR2suchthatsuppFfˆf2C:jj˙g;˙>0:Thereexistsaconstantc>0suchthatforarbitrarynormaloperatorsN1andN2,k(f(N1)f(N2)kc˙kfkL1kN1N2k:(2.4.11)Theclass(R2)ofolderfunctionsoforder,0<<1,isby:(R2)def=(f:kfk(R2)=supz16=z2jf(z1)f(z2)jjz1z2j<1:)Theclass(Rn),n>2isinthesameway.Using(2.4.11),itwasprovedin[5]thatthefunctionsinB11;1(R2)areoperatorLipschitzfornormaloperatorsandthereexistsaconstantc>0suchthatkf(N1)f(N2)kckfkkN1N2kforeveryfunctionin(R2)andarbitrarynormaloperatorsN1andN2.2.4.5Thecaseofn{Tuplesofcommutingself{adjointoperatorsIn[10],anotherformulaforthef(A1;:::;An)f(B1;:::;Bn)wasestablishedforfunctionsintheBesovspaceB11;1(Rn)andn{tuplesofcommutingself{adjointoperators(A1;:::;An),(B1;:::;Bn)intermsofDOI.Readersarereferredto[10]fortheof40B11;1(Rn)andtheconstructionofthetheoryofDOIforn{Tuplesofcommutingself{adjointoperators.Thefollowingimportantresultwasprovedin[10]:LetfbeaboundedcontinuousfunctiononRnsuchthatsuppFfˆf˘2Rn:j˘j˙g;˙>0:Thereexistsaconstantcn>0suchthatforarbitraryn{tuplesofcommutingself-adjointoperators(A1;:::;An)and(B1;:::;Bn),kf(A1;:::;An)f(B1;:::;Bn)kcn˙kfkL1max1jnkAjBjk:(2.4.12)Using(2.4.12),itwasprovedin[10]thatthefunctionsinB11;1(Rn)areoperatorLips-chitzfornormaloperatorsandthereexistsaconstantcn>0suchthatkf(A1;:::;An)f(B1;:::;Bn)kcn(1)1kfkmax1jnkAjBjkforeveryfunctionin(Rn)andn{tuplesofcommutingself-adjointoperators(A1;:::;An)and(B1;:::;Bn).2.5ArbitrarymoduliofcontinuityInthissectionweconsidertheproblemofestimatingkf(A)f(B)kforself{adjointoperatorsAandBandfunctionsinthespace!,where!isanarbitrarymodulusofcontinuity.Wealsoshowsimilarresultsforunitaryoperators,contractions,maximaldissipativeoperators,andn{tuplesofcommutingself{adjointoperators.41Givenamodulusofcontinuity!,wethefunction!and!]by!(x)=xZ1x!(t)t2dt;x>0and!](x)=xZ1x!(t)t2dt+Zx0!(t)tdt;x>0:Inthispaper,weassumethat!]isvaluedwheneveritisused.Forexample,ifwe!by!(x)=x;x>0;0<<1;then!](x)const!(x).Itiswellknown(see[6],Ch.3,Theorem13.30)thatif!isamodulusofcontinuity,thentheHilberttransformmaps!intoitselfifandonlyif!](x)const!(x).Theorem2.5.1.[1]Thereexistsaconstantc>0suchthatforeverymodulusofcontinuity!,everyfin!(R)andforarbitraryself-adjointoperatorsAandB,thefollowinginequalityholds:kf(A)f(B)kckfk!(R)!(kABk):(2.5.1)Proof.DuetoLemma2.3.1,wecanassumethatAandBareboundedoperatorsandtheirspectraarecontainedin[a;b].Wereplacethefunctionf2!(R)withtheboundedfunction42f[byf[(x)=8>>>>>>>><>>>>>>>>:f(b);x>b;f(x);x2[a;b];f(a);x<a:(2.5.2)Clearly,kf[k!(R)kfk!(R).Thuswemayassumethatfisbounded.LetNbeaninteger,weclaimthatf(A)f(B)=NXn=fn(A)fn(B)+(ffVn)(A)(ffVn)(B);(2.5.3)andtheseriesconvergesabsolutelyintheoperatornorm.Herefn=fWn+fW]nandthedelaValleePoussintypekernelVnisdasinx2.2.1.SupposethatM<N,itiseasytoseethatf(A)f(B) NXn=M+1fn(A)fn(B)+(ffVN)(A)(ffVN)(B!=(ffVM)(A)(ffVM)(B):Clearly,ffVMisanentirefunctionofexponentialtypeatmost2M+1.Thusitfollowsfrom(1.5.5)thatk(ffVM)(A)(ffVM)(B)kconst2MkfkL1kABk!0asM!:43SupposenowthatNistheintegersatisfying(2.3.4).ItfollowsfromTheorem2.2.2thatk(ffVN)(A)(ffVN)(B)k2kffVnkL1constkfk!(R)!(2N)constkfk!(R)!(kABk):Ontheotherhand,itfollowsfromCorollary2.2.4andfrom(1.5.5)thatNXn=kfn(A)fn(B)kconstNXn=2nkfnkL1kABkconstNXn=2nkfk!(R)!(2N)kABk=constXk02Nkkfk!(R)!(2(Nk))kABkconst Z12N!(t)t2dt!kfk!(R)kABk=const2N!(2N)kfk!(R)kABkconstkfk!(R)!(kABk)Theresultnowfollowsfromtheobviousinequality!(x)!(x);x>0.Corollary2.5.2.[1]Let!beamodulusofcontinuitysuchthat!xconst!(x);x>0.Thenforanarbitraryfunctionf2!(R)andforarbitraryself{adjointoperatorsAandBonHilbertspacethefollowinginequalityholds:kf(A)f(B)kckfk!(R)!(kABk):(2.5.4)Belowwegivesimilarresultsforothertypesofoperators.Theirproofsaresimilartothe44proofofTheorem2.5.1.Theorem2.5.3.[1]Thereexistsaconstantc>0suchthatforeverymodulusofcontinuity!,everyfin!andforarbitraryunitaryoperatorsUandV,thefollowinginequalityholds:kf(U)f(V)kckfk!!(kUVk):(2.5.5)Theorem2.5.4.[1]Thereexistsaconstantc>0suchthatforeverymodulusofcontinuity!,everyfin!)+andforarbitrarycontractionsTandR,thefollowinginequalityholds:kf(T)f(R)kckfk!!(kTRk):(2.5.6)Theorem2.5.5.[4]Thereexistsaconstantc>0suchthatforeverymodulusofcontinuity!,everyfin!)+andformaximaldissipativeoperatorsLandMwithboundedence,thefollowinginequalityholds:kf(L)f(M)kckfk!!(kLMk):(2.5.7)Let!beamodulusofcontinuity,theclass!(R2)isby:!(R2)def=(f:kfk!(R2)=supz16=z2jf(z1)f(z2)j!(jz1z2j)<1:)Theclass!(Rn),n>2isinthesameway.Theorem2.5.6.[5]Thereexistsaconstantc>0suchthatforeverymodulusofcontinuity!,everyfin!(R2)andforarbitrarynormaloperatorsN1andN2,thefollowinginequality45holds:kf(N1)f(N2)kckfk!(R2)!(kN1N2k):(2.5.8)Theorem2.5.7.[10]Letnbeapositiveinteger.Thereexistsaconstantcn>0suchthatforeverymodulusofcontinuity!,everyfin!(Rn)andforarbitraryn{tuplesofcommutingself{adjointoperators(A1;:::;An)and(B1;:::;Bn),thefollowinginequalityholds:kf(A1;:::;An)f(B1;:::;Bn)kcnkfk!max1jn!(kAjBjk):(2.5.9)46Chapter3Estimatesonsingularvalues3.1ResultsforperturbationofclassSpLetl0beanintegerandp1.DenotebySlpthenormedidealthatconsistsofallboundedlinearoperatorsequippedwithnormkTkSlpdef=lXj=0(sj(T))p1p:(3.1.1)ClassesSlpandSparebothniceSNI.ThusTheorem1.5.1and(1.5.5)canbeappliedtothem,i.e.,iffisanexponentialfunctionoftypeatmost˙thatisboundedonR,thenforarbitraryself{adjointoperatorsAandB,wehave:kf(A)f(B)kSlpconst˙kfkL1kABkSlp;(3.1.2)andkf(A)f(B)kSpconst˙kfkL1kABkSp:(3.1.3)Similarresultsalsoholdformaximaldissipativeoperators,normaloperatorsandn{tuplesofself{adjointoperators.Wealsohaveiffisatrigonometricpolynomialofdegreed,thenforarbitraryunitary47operatorsUandV,kf(U)f(V)kSlpconstdkfkL1kUVkSlp;andkf(U)f(V)kSpconstdkfkL1kUVkSp:Similarresultsalsoholdforcontractions.Theorem3.1.1.Let0<<1.Thenthereexistsaconstantc>0suchthatforeveryl0,p2[1;1),f2(R),andforarbitraryself{adjointoperatorsAandBonHilbertspacewithboundedAB,thefollowinginequalityholdsforeveryeveryjl:sjf(A)f(B)ckfk(R)(1+j)=pkABkSlp:(3.1.4)Proof.Putfndef=fWn+fW]n;n2Z,andanintegerN.WehaveNXn=fn(A)fn(B)SlpNXn=kfn(A)fn(B)kSlpconstNXn=2nkfnkL1kABkSlpconstkfk(R)NXn=2n(1)kABkSlpconst2N(1)kfk(R)kABkSlp:48Ontheotherhand,Xn>Nfn(A)fn(B)2Xn>NkfnkL1constkfk(R)Xn>N2const2Nkfk(R):PutRNdef=NPn=fn(A)fn(B)andQNdef=Pn>Nfn(A)fn(B).Clearly,forjl,sjf(A)f(B)sj(RN)+kQNk(1+j)1=pkRNkSlp+kQNkconst(1+j)1p2N(1)kfk(R)kABkSlp+2Nkfk(R):Toobtainthedesiredestimate,ittochoosethenumberNsuchthat2N<(1+j)1=pkABkSlp2N+1:Usingthesametypeofarguments,wecangetsimilarestimatesforunitaryoperators,contractions,maximaldissipativeoperators,normaloperatorsandn{tuplesofself{adjointoperators.493.2Estimatesonsingularvaluesoffunctionsofper-turbedself-adjointandunitaryoperatorsInthissection,wegeneralizetheestimateinx3.1totheclass!andalsoobtainsomelower{boundestimatesforrankoneperturbationswhichalsoextendtheresultsin[2].Insectionx3.3,similarestimatesaregivenwithoutproofsincaseofcontractions,maximaldissipativeoperators,normaloperatorsandn{tuplesofcommutingself{adjointoperators.Theorem3.2.1.Thereexistsaconstantc>0suchthatforeverymodulusofcontinuity!,everyfin!(R)andforarbitraryself-adjointoperatorsAandB,thefollowinginequalityholdsforalllandforallj,1jl:sj(f(A)f(B))c!(1+j)1pkABkSlpkfk!:(3.2.1)Proof.DuetoLemma2.3.1,AandBcanbetakenasboundedoperators,thenwemayfurtherassumefisbounded.LetRN=PNn=(fn(A)fn(B)),QN=(ffVN)(A)(ffVN)(B).HerefnandthedelaValleePousstypekernelVNareasinx2.2.1.Thenf(A)f(B)=RN+QN,withconvergenceintheuniformoperatortopology.Notethatforanyintegerm2Z,functionsfmandffVmareentirefunctionsofexponentialtypeatmost2m+1:Thusitfollowsfrom(3.1.2),(2.2.6)and(2.2.7)thatkQNkc!(2N)kfk!50andkRNkSlpNXn=kfn(A)fn(B)kSlpcNXn=2nkfnkL1kABkSlpc2N!(2N)kABkSlpkfk!:Thensj(f(A)f(B))sj(RN)+kQNk(1+j)1kRNkSlp+kQNkc(1+j)1p2N!(2N)kABkSlp+!(2N)kfk!:TakeNsuchthat1(1+j)1p2NkABkSlp2andusethefactthat!(t)!(t)foranyt>0,weget(3.2.1).Theorem3.2.2.Thereexistsaconstantc>0suchthatforeverymodulusofcontinuity!,everyfin!(T)andforarbitraryunitaryoperatorsUandV,thefollowinginequalityholdsforalllandforallj,1jl:sj(f(U)f(V))c!(1+j)1pkUVkSlpkfk!:(3.2.2)Proof.If(1+j)1pkUVkSlp2,theproofissimilartoTheorem3.2.1withRN=PNn=0(fn(U)fn(U));if(1+j)1pkUVkSlp>2,thensj(f(U)f(V))kf(U)f(V)kc!(kUVk)kfk!c!(2)kfk!:51Corollary3.2.3.Let!beamodulusofcontinuitysuchthat!(x)const!(x);x0:Thenforanarbitraryfunctionf2!(R)andforarbitraryself-adjointoperatorsAandB,thefollowinginequalityholdsforalllandforallj,1jl:sj(f(A)f(B))const!(1+j)1pkABkSlpkfk!:LetH,HbetheHankeloperatorsin[2].Theorem3.2.4.Let!beamodulusofcontinuityonT.ThereexistunitaryoperatorsU,Vandarealfunctionhin!]((T))suchthatrank(UV)=1andsm(h(U)h(V))!(1+m)1:Proof.ConsidertheoperatorsUandVonspaceL2(T)withrespecttothenormalizedLebesguemeasureonTby(see[2])Uf=zfandVf=zf2(f;1)z;f2L2:52Forf2C(T),wehave(f(U)f(V))zj;zk=28>>>>>>>><>>>>>>>>:^f(jk);ifj0;k<0;^f(jk);ifj<0;k0;0;otherwise:functiongbyg()=1Xn=1!(4n)(4n+4n);2T:ThenwehavekgWnkL1const!(2n);kgW]nkL1const!(2n);n0:Let˘,betwoarbitrarilytpointsonT,chooseN0suchthat122Nj˘j1,then53jg(˘)g()jNXn=0jgn(˘)gn()j+j(ggVN)(˘)(ggVN)()jNXn=0jgn(˘)gn()j+21Xn=N+1kgnkL1constNXn=02nkgnkL1j˘j+21Xn=N+1kgnkL1constNXn=02n!(2n)j˘j+const1Xn=N+1!(2n)const!(j˘j)+constZ2N0!(t)tdtconst!](j˘j):Considerthematrixg=f^g(jk)gj1;k0=f^g(j+k)gj1;k0.Letn1.matrixTn=f^g(j+k+4n1+1)g0j;k34n1,thenTn=26666666664!(4n)!(4n):::!(4n)37777777775:IfRisanymatrixwiththesamesizeofTnsuchthatrank(R)<34n1,thenkTnRk!(4n).Itfollowsthatsj(Tn)!(4n)forj<34n1.ForeachTn,thereissomeorthogonalprojectionPnsuchthatTn=PngPn,hencesjg)sj(Tn)!(4n)forall54nandforallj,j<34n1.Thusforallj0,wehavesjg)!316(j+1)1332!(j+1)1:Tocompletetheproof,ittotakeh=323g.Corollary3.2.5.Let!beamodulusofcontinuitysuchthat!](x)const!(x);0x2:ThereexistunitaryoperatorsU,Vandarealfunctionhin!(T)suchthatrank(UV)=1andsm(h(U)h(V))!(1+m)1:Theorem3.2.6.Let!beamodulusofcontinuityonTandfbeacontinuousfunctiononT.IfforallunitaryoperatorsUandV,wehavesn(f(U)f(V))const!(1+n)1pkUVkSp;foralln0;thenf2!(T):Proof.Let;2T,wecanselectcommutingunitaryoperatorsUandVsuchthats0(UV)=s1(UV)=:::=sn(UV)=jjandsk(UV)=0;kn+1.Thensn(f(U)f(V))=jf()f()j,kUVkSp=(1+n)1pjj.Theorem3.2.7.Let!beamodulusofcontinuityonRandfbeacontinuousfunctionon55R.Ifforallself-adjointoperatorsAandB,wehavesn(f(A)f(B))const!(1+n)1pkABkSp;foralln0;thenf2!(R):Proof.SimilartoTheorem3.2.6.Theorem3.2.8.Let!beamodulusofcontinuityoverR.Thereexistself-adjointoperatorsA,B,andarealfunctionfin!](R)suchthatrank(AB)=1andsm(f(A)f(B))!(1+m)1,forallm0:Proof.WLOG,weassume!(t)=!(2),forallt2,thatis,!canberegardedasamodulusofcontinuityonT:Wethenchooseafunction(see[2],Lemma9.6)ˆ2C1(T)suchthatˆ()+ˆ()=1,ˆ()=ˆ()forall2T,andˆvanishesinaneighborhoodoftheset1;1g:Notethatˆ2!(T);since!(st)s2!(t);forallt0ands,0<s<1:functiong1byg1()=1Xn=1!(4n)(4n+4n);2T:Theng12!](T):Ifg0def=Cˆg1foratlargenumberC,theng02!](T);vanishesinaneighborhoodoftheset1;1gandg0()=g0()forall2T,andsm(Hg0)!(1+m)1forallm0:'(x)=(x2+1)1(asin[2],Theorem9.9),thenthereexistsacompactlysupported56realboundedfunctionfsuchthatf('(x))=g0(xix+i)andasimplecalculationshowsthatfbelongsto!](R):DenoteL2e(R)thesubspaceofevenfunctionsinL2(R).ConsideroperatorsAandBonL2e(R)byA(g)=H1M'H(g)andB(g)='g;hereHistheHilberttransformonL2(R)(see[2])andM'isthemultiplicationby'.Thenrank(AB)=1;andwehavesm(f(B)f(A))p2sm(Hf')=p2sm(Hg0)p2!(1+m)1:3.3EstimatesforothertypesofoperatorsThefollowingestimatesaregivenwithoutproofsincaseofcontractions,maximaldissipativeoperators,normaloperatorsandn-tuplesofcommutingself-adjointoperators.Theorem3.3.1.Thereexistsaconstantc>0suchthatforeverymodulusofcontinuity!,everyfin!(R)+andforarbitrarycontractionsTandRonHilbertspace,thefollowinginequalityholdsforalllandforallj,1jl:sj(f(T)f(R))c!(1+j)1pkTRkSlpkfk!:Toprovethisresult,thefollowingresultisimportant(see[1],[2]and[24]):ThereexistsaconstantcsuchthatforarbitrarytrigonometricpolynomialfofdegreenandforarbitrarycontractionsTandRonHilbertspace,k(f(T)f(R)kSpcnkfkL1kTRkSp:57Theorem3.3.2.Thereexistsaconstantc>0suchthatforeverymodulusofcontinuity!,everyfin!(R)+andforarbitrarymaximaldissipativeoperatorsLandMwithboundedence,thefollowinginequalityholdsforalllandforallj,1jl:sj(f(L)f(M))c!(1+j)1pkLMkSlpkfk!:Toprovethisresult,thefollowingresultisimportant(see[4]):Thereexistsaconstantc>0suchthatforeveryfunctionfinH1(C+)withsuppFfˆ[0;˙];˙>0;andforarbitrarymaximaldissipativeoperatorsLandMwithboundedk(f(L)f(M)kSpc˙kfkL1kLMkSp:Theorem3.3.3.Thereexistsaconstantc>0suchthatforeverymodulusofcontinuity!,everyfin!(R2)andforarbitrarynormaloperatorsN1andN2,thefollowinginequalityholdsforalllandforallj,1jl:sj(f(N1)f(N2))c!(1+j)1pkN1N2kSlpkfk!:Toprovethisresult,thefollowingresultisimportant(see[5]):Thereexistsaconstantc>0suchthatforeveryboundedcontinuousfunctionfonR258withsuppFfˆf2C:jj˙g;˙>0;andforarbitrarynormaloperatorsN1andN2,kf(N1)f(N2)kSpc˙kfkL1kN1N2kSp:Theorem3.3.4.Letnbeapositiveintegerandp1.Thereexistsapositivenumbercnsuchthatforeverymodulusofcontinuity!,everyfin!(Rn)andforarbitraryn-tuplesofcommutingself-adjointoperators(A1;:::;An)and(B1;:::;Bn),thefollowinginequalityholdsforalllandforallj,1jl:sj(f(A1;:::;An)f(B1;:::;Bn))cnmax1jn!(1+j)1pkAjBjkSlpkfk!:Toprovethisresult,thefollowingresultisimportant(see[10]):Thereexistsaconstantcn>0suchthatforeveryboundedcontinuousfunctionfonRnwithsuppFfˆf˘2Rn:j˘j˙g;˙>0;andforarbitraryn-tuplesofcommutingself-adjointoperators(A1;:::;An)and(B1;:::;Bn),kf(A1;:::;An)f(B1;:::;Bn)kSpcn˙kfkL1max1jnkAjBjkSp:59BIBLIOGRAPHY60BIBLIOGRAPHY[1]A.B.Aleksandrov,V.V.Peller,Operatorolder-Zygmundfunctio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