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WEIGHTED COMPOSITION OPERATORS BETWEEN BERS-TYPE SPACES AND BERGMAN SPACES

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WEIGHTED COMPOSITION OPERATORS BETWEEN BERS-TYPE SPACES AND BERGMAN SPACESBERS,TYPE,and,type,Bers,Type

    WEIGHTED COMPOSITION

    OPERATORS BETWEEN BERS-TYPE

    SPACES AND BERGMAN SPACES

    App1.Math.J.ChineseUniv.Ser.B

    2007,22(1):6168

    WEIGHTEDCoMPoSITIoNoPERAToRSBETWEEN

    BERS.TYPESPACESANDBERGMANSPACES

    TangXiaomin

    Abstract.Thispapercharacterizestheboundednessandcompactnessofweightedcomposi

    tionoperatorsbetweenBers-typespace(orlittleBers-typespace)andBergmanspace.Some estimatesforthenormofweightedcompositionoperatorsbetweenthosespacesareobtained. ?1Introduction

    WedenotebyDtheunitdiscinthecomplexplaneC.Let(D)bethefamilyofal1

    holomorphicfunctionsonD.GivenU?H(D)andaholomorphicself-mapofD,theweighted

    compositionoperatorisdefinedasuGly=,.forf?日(D).ItisobviousthatuCvCan

    beregardedasageneralizationofthemultiplicationoperatorandcompositionoperator.The behaviorofthoseoperatorsisstudiedextensivelyonvariousspacesofholomorphicfunction[1-5]

    includingtheHardyandBergmanspaces.

    Recently,HeandJianghavediscussedtheboundednessandcompactnessofthecomposition operatorsonBers

    typespacesin[6.JiangandLihavestudiedthecompositionoperatorsfrom

    BerstypespacetoHardyspaceorotherspaces[7].Zhaohascharacterizedthepropertyof multipliersfromBergmanspaceandHardyspacetoBergmanspacein[8.Inthispaper,we

    obtainthesufficientandnecessaryconditiononuandsuchthatuCvisbounded(orcompact)

betweenBergmanspaceandBers-typespace(orthelittleBers

    typespace).Meanwhile,weget

    someestimatesforthenormofbetweenthoseabovespaces. For0<P<..and1<Q<..,theweightedBergmanspaceisdefinedby =f?日(.);IIfll~,,=(+1)/DIf(z)l(1).dm(z)<..,

    wheredmisanormalLebesgueareameasureonD.When1P<..,isaBanachspace

    underthenormll?lln'p.For0<P<1,isanFspacewithrespecttothetranslation

    invariantmetricdefinedby(,,g)=lIf9ll,p.For0<<..,afunctionf?H(D)issaid

    Received:2006-03-16.

    MRSubjectClassification:47B38.

    Keywords:weightedcompositionoperator,Bers-typespace,Bergmanspace. SupportedbytheNNSFofChina(10471039),theNaturalScienceFoundationofZh~iangPro

    vince(M103

    104)andtheNaturalScienceFoundationofHuzhouCity(2005YZ02).

62App1.Math.J.ChineseUniv.Ser.BVo1.22,No.1

    tobelongtoBerstypespaceifl[fllH~=sup?D(1lzl)l,(z)l<CO,andtothelittle Bers-typespaceHiflimII1(1lzl)l,(z)l=0.ItiswellknownthatisaBanach spaceunderthenormII?llHand%isaclosedsubspaceof.See9,10formoreabout

    Bers-typespace.

    Inwhatfollows,Cwillstandforpositiveconstantswhosevaluemaychangefromlineto

    linebutnotdependonthefunctionsin(D).TheexpressionABmeansCABCA.

    ?2Thecase:

    Lemma2.1.ISupposethat{)isanincreasingsequenceofpositiveintegerswithHadamard gaps,thatis,>1forallk.Let0<P<+?,thenthereisaconstantC>0depending onPandsuchthat

    l

    /N,i

    ,/)(N?ake=1,1/N,/),/)

foranyscalaral,,aNandN=1,2,.

    Theorem2.1.Let?H(D)andbeaholomorphicself-mapofD.Supposethat0<P,<

    COand-1<Ot<CO.Thenthefollowingstatementsareequivalent:

    (1):isbounded;

    (2):野—iscompact;

    Moreover,if:—吕isbounded,then ,

    dm(z)}J

    Proof.(2)(1)isobvious.For(1)(3),takethetestfunction

    Thenf?H(D)and

    +o.

    ,(z)=E2,z?D.=1

    o..

    o.rk+1一十o.

    )l2J9?2(V~)2=dx--/12()d1k==l (2?kg)e(2g))

    wherez=pen.

    Thusf?,letg0(z)=f(ez),theng0(z)?

    SetD1={z;l(z)l<1)andD2:{z;l(z)l).Since:

    get?fromthefactthat=(1).So (2.1)

    (2.2)

    (2.3)

    andllgollH~=llfllH~?

    isbounded,we

    dm(z)(1IzI).I(z)ldm(z)II,<oo _,一一Z一??卜

    

    Tan9XiaominWEIGHTEDCOMPOSITIONOPERATORSBETWEEN63 Thus,toprove(2.1)weneedonlyshowthat .1)p/3<....,D(I(z)I……'

    Letr=I(z)I.Since:isbounded,thereexistsaconstantCsuchthat

    CullIII(1IIII))lPd)

    ?ei2.(2.4)

    IntegratingInequality(2.4)withrespectto0andapplyingFubiniTheoremandLemma2.1,

    weget

    "c""9"

    2

    (1--Izl2)~lu(z)lp[I2czkei2kIddmcz

    .

    c?z?a?cz?2?cz?k+.dmcz.

    ?2r?.=2-2?2+1)rH.2-2/3?/.22/3xr2x+loooork1d=+?+++ k=lk=lk=lJ

    2

    +oo

    22/3Xr2X+~dx>

    _

    2-4/3

    J1(1og/J4log2-1e?u5-\

    (1r)'

    <...5

    (3)(1).LetM=dm(z)<.o.For,?节,itisclearthat rDI(c,)(z)I(1IzI)dm(z)=J(1IzI)I(z)I-[!_={{dm(z) lldm(MIIfll~.(2.6)

    SoIluC.IICll$11H~.Thatis,:isbounded?

    (3)(2).Suppose(2.1)holds,then:卵一isboundedfrom"(3)(1)". Let{}cHybesuchthatII/.1l-r1,then{}isanormalfamilyonD?ByMontel

Theoremthereisasubsequenceof{}whichconvergesuni~rmlyonanycompactsubsetof

    D

    toaholomorphicfunction{.Bypassingtothissubsequencewemayassumethatthesequence

    App1.Math.ChineseUniv.Ser.BVo1.22,No?1 {)itselfconvergesto,.Since厶?andlIAlIj=f1,Weknowthat?爿and

    I1?Thus,weobtain

    I()(z)(,))Ip=lu(z)lpI((z)),((z))I

    2Plu(z)lP(I,(())lp+I,((z))Ip)

    (1lY.1l~;+IlfllH$~)?

    Therefore,byLebesguedominatedc.nvergencetheorem,lIu厶一u,Il0?Tha meai18:iscompact?

    By(2.6)wehave

    =

    .?.{

    7

    Therefore,(2.5)and(2.7)give(2.2). The0rem2.2.Let?H(D)andbeaholomorphicself-mapofD.Supposethat0<P,<

    .od1<Q<OO,thenthefollowingstatementsareequivalent:

    (1):Hooisbounded;

    (2)u:Hooiscompact;

    .

    '(

    (

    1-

    -)lzl2)'~lu(z)ff'din(z)<

    .

    OO.

    21isobviousFor(1)(3),take,(z)justaB(2.3)andletrn:.1

,Pr00f.()=:().()===}(3),(zJIz?Jndt,

    (z):y(rnz),thenA?oo,0.Hence,theproofof"(1)(3)"issimilartothatofTheoem

    2.1,andisomittedhere.

    F0(3):(2),supposethat厶高dm(z)<?,thenbyThe.rem2?1,:

    iscompact.SinceHoo,

    0isclosedsubspaceof

    ,Weget:is

    compact.

    ?3Thecase:

    Lemma3.1.12ILet0<p<o.,1<Q<oo,thenforallf?,I,(z)I(1-1zl2)-p,z?D,

    whereCisapositiveconstantdependingOilPandOtonly. The0rem3.1.Let?日()andbeaholomorphicself-mapofD.Supposethat0<P,<

    ?and1<Ot<(30,then:isboundedifandonlyif ?

    Moreover,if:isbounded,then

    

    sup?

    (3.1)

    (3.2)

    TangXiaominWEIGHTEDCOMPOSITIONOPERATORSBETWEEN...65 Proof.Supposeu:A_+ISbounded?ForA?'set

    =

    [

    then?andIIpC.SothereexistsaconstantCsuchthat IIull<IluIpCllu~ll?

    Hence,foreachz?D,(1Izl)Iu(z)ll((z))IClluC~l1.Inparticular,whenz=weget

    viiu

    (1I()I2).

upIu<...(3.3)

    zED(1I(z)I)".,

    c.nverselyjsupp.sethat(3?1)h0lds?LetM=s

    ?

    u

    .

    p

    (

    (

    1

    l

    

    -

    l

    l

    

    z

    (

    12)

    )l2)

    ~lu

    (z)J<.o?F0r,?,by

    Lemma3.1,

    (1Izl)I(u),(z)I=(1Izl)~lu(z)llf(cp(z))I

    IlfllMCIIfll(3.4)

    (1l(z)I)p

    foranyz?D.Consequently,IIu,IlCllfll~,p?Henceu:isbounded?

    By(3.4)wehave

    <1lC~ofllHr<_ll,sllaulp<pIflla,p.1Ip<?(3.5)l1l1(l(z)l)

    Thus,(3.3)and(3.5)imply(3.2).

Tocharacterizethecompactnessofubetweenand,wewillneedthefollowing

    result,whoseproofisaneasymodificationofthatofProposition3.11in13].

    Lemma3.2.LetXandYbeor.Thenu:XYiscompactifandonlyiffor anyboundedsequence{}inXwhichconvergesto0uniformlyonanycompactsubsetofD,

    wehaveu-_+0inY.

    Theorem3.2.Letu?H(D)andbeaholomorphicself-mapofD.Supposethat0<P,<

    .oand1<oL<.o,thenu:iscompactifandonlyifu?and

    ..1

    -o.(3.6)l

    (z)l(I(z)I)

    Proof.Supposeu:iscompact.Takef(z)1,thenu=u,??Let

    {}beasequenceinDsuchthatI(zt1)I-_+1as-_+.o.Set

    App1.Math.J.ChineseUniv.Set.BVo1.22,No.1 Itiseasytocheckthat{}isaboundedsequenceinand__+0uniformlyonanycompact subsetofD.ByLemma3.2,limlJ~ll-r:0.Becausen---~o.r J

    

    sup()JJ?,

    wegetu-Thismealim

    1)12l1-1~o(z)l-0?(l();l(z)l'()

    c.nversely1supp0sethat?and(3?6)h0lds,the?s

    ?

    u

    .

    p

    (

    (

    1

    t

    -

    lzlU)a

    l

    1

    )

    ~(z)l<??s.:

    一卵isboundedfromTheorem3.1.Foranyg>0,by(3.6)thereisan7-?(0,1)such thatr<J(z)J<1implies

    <E.(3.7)

    (1Iv(z),

    Supposethat{)isaboundedsequenceinwhichconvergesto0uniformlyonanycompact

    subsetofD.ThenfortheaboveE>0,thereexistsannosuchthatforJ(z)lrandnno,

    wegetI,())I<E.Thus,forI)Irandnno,weobtain (1)I(z)IIA(z))I<IlulIH;e.(3.8) NowforI()I>7-andalln,by?7)weget (1JzJ)pJ(z)JJ_n(())JlJ_nlJa,plJ_nIJa,pg?(3?9) Combining(3?8)and(3?9)1wehave

    lim

    ..

    lJ0?Hence,:墨一c.mp

    fromLeInina3.2.

    ?4Thecase:HOO,0

    Lemma4.1.Let?H(D)andbeaholomorphicself-mapofD.Supposethat0<P,<? and-1<Q<?.then

    lim

    Izl-~1

    o(4_)

    (l(z)l)

ifandonlyif?HOO0and

    Iv(1

    o.(4.2)

    z)l(l(z)l)

    Pr.0f.supp.sethat(4?1)h.,then.(1)plu(z)l(

    1

    1

    

    -

    1zlU)

    )l2)

    Ol~(z)l.as__+1.

    SouEHOO

    ,..

    Ifl(z)l1,weknowthat1?Thl(1ir)Ial(x-lzlZ)Ol~(z)l=.? Conversely,forany>0,by(4.2)thereexistsanr?(0,1)suchthatr<J(z)J<1implies

    12fIfz)I)

Tan9XiaominWEIGHTEDCOMPOSITIONOPERATORSBETWEEN..67

    Fortheabove>0,sinceu?Hoo

    ,

    0,thereexistsome?(0,1)suchthatif<II<1,then (1[zl~)/glu()I<(r).Thus,for<lzl<1,ifr<Iv()I<1,then

    12(I()I)

    andifI()Ir,then

    Iv(=(1)I.)(1r.).

    So(4.1)holds.

    TocharacterizethecompactnessofuCvfromtoHOO.0'wegivethefollowingresultwhose

    proofissimilartothatofLemma1in14].

    Lemma4.2.AclosedsetEinHpoo

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