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manuscriptNo. (willbeinsertedbytheeditor) The Maslov index in weak symplectic functional analysis BernhelmBooß-Bavnbek ChaofengZhu · 3 1 thedateofreceiptandacceptanceshouldbeinsertedlater 0 2 Abstract WerecalltheChernoff-Marsdendefinitionofweaksymplecticstructureandgive n a a rigorous treatment of the functional analysis and geometry of weak symplectic Banach J spaces.WedefinetheMaslovindexofacontinuous pathofFredholmpairsofLagrangian 0 subspacesincontinuouslyvaryingBanachspaces.WederivebasicpropertiesofthisMaslov 3 indexandemphasizethenewfeaturesappearing. ] Keywords Closedrelations,FredholmpairsofLagrangians,Maslovindex,spectralflow, G symplecticsplitting,weaksymplecticstructure. D 2010MathematicsSubjectClassification Primary53D12;Secondary58J30 . h t a m 1 Introduction [ 1.1 Oursettingandgoals 1 v First,werecallthemainfeaturesoffinite-dimensionalandinfinite-dimensionalstrongsym- 8 plectic analysis and geometry and argue for the need to generalize from strong to weak 4 assumptions. 2 7 . 1.1.1 Thefinite-dimensionalcase 1 0 3 ThestudyofdynamicalsystemsandthevariationalcalculusofN-particleclassicalmechan- 1 icsautomaticallyleadtoasymplecticstructureinthephasespaceX =R6N ofpositionand : v ThesecondauthorwaspartiallysupportedbyKPCMENo.106047andNNSFNo.10621101. i X B.Booß-Bavnbek r DepartmentofScience,SystemsandModels/IMFUFA a RoskildeUniversity,DK-4000Roskilde,Denmark E-mail:[email protected] C.Zhu ChernInstituteofMathematicsandLPMC NankaiUniversity,Tianjin300071,thePeople’sRepublicofChina E-mail:[email protected] 2 impulsevariables:whenwetracethemotionofNparticlesin3-dimensionalspace,wedeal withabilinear(inthecomplexcasesesquilinear)anti-symmetric(inthecomplexcaseskew- symmetric)andnon-degenerateformw : X X R.Thereasonfortheskew-symmetryis × → theasymmetrybetweenpositionandimpulsevariablescorresponding totheasymmetryof differentiation. To carry out the often quite delicate calculations of mechanics, the usual trickistoreplacetheskew-symmetricform w byaskew-symmetricmatrixJwithJ2= I − suchthat w (x,y) = Jx,y forallx,y X, (1) h i ∈ where , denotestheinnerproductinX. h· ·i For geometric investigations, the key concept is a Lagrangian subspace of the phase space.FortwocontinuouspathsofLagrangiansubspaces,anintersectionindex,theMaslov indexiswell-defined.Itcanbeconsideredasare-formulationorgeneralizationofcounting conjugate points on a geodesic. In Morse Theory, this number equals the classical Morse index, i.e.,thenumber ofnegativeeigenvalues oftheHessian(the secondvariation ofthe action/energyfunctional).ThisMorseIndexTheorem(cf.M.Morse[30])forgeodesicson RiemannianmanifoldswasextendedbyW.Ambrose[1],J.J.Duistermaat[22],P.Piccione andD.V.Tausk[34,35],andthesecondauthor[43,44].SeealsotheworkofM.Musso,J. Pejsachowicz,andA.PortalurionaMorseindextheoremforperturbedgeodesicsonsemi- Riemannianmanifolds in[31]whichhasinparticularleadN. Waterstraattoa K-theoretic proofoftheMorseIndexTheoremin[39]. Forasystematicreview ofthebasicvectoranalysisand geometry andforthephysics background,werefertoV.I.Arnold[2]andM.deGosson[25]. 1.1.2 Thestrongsymplecticinfinite-dimensionalcase AsshownbyK.Furutaniandthefirstauthorin[7],thefinite-dimensional approachofthe MorseIndexTheoremcanbegeneralizedtoaseparableHilbertspacewhenweassumethat theformw isboundedandcanbeexpressedasin(1)withabounded operatorJ,whichis skew-self-adjoint (i.e., J = J) and not only injective but invertible. The invertibility of ∗ − J isthewholepointofastrongsymplecticstructure.Then,withoutlossofgenerality,one can assume J2 = I like in the finite-dimensional case (see Lemma 1 below), and many − calculationsofthefinite-dimensionalcasecanbepreservedwithonlyslightmodifications. Themodel spaceforstrongsymplecticHilbert spacesisthevonNeumann spaceb (A):= dom(A )/dom(A)ofnaturalboundaryvaluesofaclosedsymmetricoperatorAinaHilbert ∗ spaceX withsymplecticformgivenbyGreen’sform w (g (u),g (v)):= A∗u,v u,A∗v forallu,v dom(A∗), (2) h i−h i ∈ where , denotes the inner product in X and g : dom(A ) b (A) is the trace map. A ∗ h· ·i → typicalexampleisprovidedbyalinearsymmetricdifferentialoperatorAoffirstorderover amanifoldM withboundary S .Herewehavetheminimaldomaindom(A)=H1(M)and 0 the maximal domain dom(A ) H1(M). Note that the inclusion is strict for dimM >1. ∗ RecallthatH1(M)denotesthec⊃losureofC¥ (M S )inH1(M).Forbetterreadingwedonot 0 0 \ mention the corresponding vectorbundles in the notation of the Sobolev spaces of vector bundlesections. As in the finite-dimensional case, the basic geometric concept in infinite-dimensional strong symplectic analysis is the Lagrangian subspace, i.e., a subspace which is isotropic and co-isotropic at the same time. Contrary to the finite-dimensional case, however, the 3 common definition of a Lagrangian as a maximal isotropic space or an isotropic space of halfdimensionbecomesinappropriate. InordertodefinetheMaslovindexintheinfinite-dimensionalcaseasintersectionnum- ber of two continuous paths of Lagrangian subspaces, one has to make the additional as- sumption that corresponding Lagrangians make a Fredholm pair so that, in particular, we havefiniteintersectiondimensions. In[23],A.Floersuggestedtoexpressthespectralflowofacurveofself-adjointopera- torsbytheMaslovindexofcorrespondingcurvesofLagrangians.Followinghissuggestion, a multitude of formulae was achieved by T. Yoshida [41], L. Nicolaescu [32], S. E. Cap- pell,R.Lee,andE.Y.Miller[18],thefirstauthor, jointlywithK.Furutani andN.Otsuki [8,9] and P. Kirk and M. Lesch [27]. The formulae are of varying generality: Some deal withafixed(elliptic)differentialoperatorwithvaryingself-adjointextensions(i.e.,varying boundary conditions); others keep the boundary condition fixed and let the operator vary. Anexampleforapathofoperators isacurveofDiracoperators onamanifoldwithfixed RiemannianmetricandCliffordmultiplicationbutvaryingdefiningconnection(background field). Seealso theresults by the present authors in[13] for varying operator and varying boundaryconditionsbutfixedmaximaldomainandin[14](inpreparation)alsoforvarying maximal domain. Recently, M. Prokhorova [36] considered a path of Dirac operators on a two-dimensional disk with a finite number of holes subjected to local elliptic boundary conditions andobtainedabeautiful explicitformulaforthespectralflow(respectively,the Maslovindex). 1.1.3 Beyondthelimitsofthestrongsymplecticassumption Weak(i.e.,notnecessarilystrong)symplecticstructures ariseonthewaytoaspectralflow formula in the full generality wanted: for continuous curves of, say linear formally self- adjoint elliptic differential operators of first order over a compact manifold of dimension 2withboundary andwithvaryingmaximaldomain(i.e.,admittingarbitrarycontinuous ≥ variation of thecoefficients of first order) and with continuously varying regular(elliptic) boundary conditions, see[14]. An interesting new feature for the comprehensive general- ization is the following “technical” problem: For regular (elliptic) boundary value problems(sayforalinearformallyself-adjointellipticdifferentialoperatorAoffirstorderona compactsmoothmanifoldMwithboundaryS ),therearethreecanonicalspacesofbound- ary values: the above mentioned von Neumann space b (A)=dom(A )/dom(A), which ∗ isasubspace ofthe distributional Sobolev space H 1/2(S );thespaceof boundary values − H1/2(S ) H1(M)/H1(M)oftheoperatordomainH1(M);andthemostfamiliarandbasic L2(S ).1A≃sin(2),Gre0en’sforminducessymplecticformsonallthreesectionspaceswhich aremutuallycompatible. Moreprecisely,Green’sformyieldsastrongsymplecticstructurenotonlyonb (A),but alsoonL2(S )by w (x,y):= −hJx,yiL2(S ). Here J denotes the principal symbol of the operator A over the boundary in innernormal direction.Themultiplicativeoperatorinducedby J isinvertible(=injectiveandsurjective, 1 Inthetraditionofgeometricallyinspiredanalysis,wethinkmostlyofhomogeneoussystemswhentalk- ingofellipticboundaryvalueproblems.Ourkeyreferenceisthemonograph[11]byK.P.Wojciechowski andthefirstauthorandthesupplementaryelaborationsbyJ.Bru¨ningandM.Leschin[16].Foramorecom- prehensivetreatment,emphasizingnon-homogeneousboundaryvalueproblemsandassemblingallrelevant sectionspacesinahugealgebra,werefertothemorerecentarticle[38]byB.-W.Schulze. 4 i.e.,with bounded inverse) since A is elliptic. Forthe induced symplectic structure on the Sobolev space H1/2(S ) the corresponding operator J is not invertible for dimS 1, see ′ Remark2binSection2.1below.So,fordimS 1thespaceH1/2(S )becomesonly≥aweak ≥ symplecticHilbertspace,touseanotionintroducedbyP.R.ChernoffandJ.E.Marsden[19, Section1.2,pp.4-5]. Anadditionalincitementtoinvestigateweaksymplecticstructurescomesfromastun- ning observation of E. Witten (explained by M.F. Atiyah in [3] in a heuristic way). He considered a weak (and degenerate) symplectic form on the loop space Map(S1,M) of a finite-dimensional closed orientable Riemannian manifold M and noticed that a (future) thoroughunderstandingoftheinfinite-dimensionalsymplecticgeometryofthatloopspace “shouldleadratherdirectlytotheindextheoremforDiracoperators”(l.c.,p.43).Ofcourse, restrictingourselvestothelinearcase,i.e.,tothegeometryofLagrangiansubspacesinstead ofLagrangianmanifolds,wecanonlymarginallycontributetothatprograminthispaper. 1.2 Mainresultsandplanofthepaper Inthispaperweshalldealwiththeprecedingtechnicalproblem.Todothat,wegeneralize the results of J. Robbin and D. Salamon [37], S.E. Cappell, R. Lee, and E.Y. Miller[17], K.Furutani,N.Otsukiandthefirstauthorin[8,9]andofP.KirkandM.Leschin[27].We givearigorous definition oftheMaslovindexforcontinuous curves ofFredholm pairs of LagrangiansubspacesinafixedBanachspacewithvaryingweaksymplecticstructuresand continuouslyvaryingsymplecticsplittingsandderiveitsbasicproperties.Partofourresults willbeformulatedandprovedforrelationsinsteadofoperatorstoadmitwiderapplication. Throughout,weaimforacleanpresentationinthesensethatresultsareprovedinsuit- ablegenerality.Wewishtoshowclearlytheminimalassumptionsneededinordertoprove the various properties. We shall, e.g., prove purely algebraic results algebraically in sym- plecticvectorspacesandpurelytopologicalresultsinBanachspaceswheneverpossible-in spiteofthefactthatweshalldealwithsymplecticHilbertspacesinmostapplications. Theroutesof[8,9]and[27]arebarredtousbecausetheyrely ontheconceptofstrong symplecticHilbertspace.Consequently,wehavetoreplacesomeofthefamiliarreasoning of symplectic analysis by new arguments. A few of the most elegant lemmata of strong symplectic analysis cannot beretained, but, luckily, thenew weak symplecticset-up will showaconsiderablestrengththatisillustrativeandapplicablealsointheconventionalstrong case. InSection2,wegive athorough presentation of weaksymplecticfunctional analysis. Basic concepts are defined in Subsection 2.1. A new feature of weak symplectic analysis isthelackofacanonicalsymplecticsplitting:forstrongsymplecticHilbertspace,wecan assume J2 = I by smooth deformation of the metric, and obtain the canonical splitting − X=X+ X intomutuallyorthogonalclosedsubspacesX :=ker(J iI)whichareboth − ± ⊕ ∓ invariantunderJ.ThatpermitstherepresentationofallLagrangiansubspacesasgraphsof unitaryoperators from X+ toX (seeLemma2),whichyieldsatransferofcontractibility − fromtheunitarygrouptothespaceofLagrangiansubspaces.Moreover,thatrepresentation isthebasisforafunctionalanalyticaldefinitionoftheMaslovindex.Forweaksymplectic HilbertorBanachspaces,theprecedingconstructiondoesnotworkanylongerandwemust assumethatasymplectic splittingisgiven andfixed(itsexistencefollows,however, from Zorn’s Lemma). Given an elliptic differential operator A of first order over a manifold M withboundary S ,however,wehaveanaturalsymplecticsplittingofthesymplecticspaces 5 ofsectionsoverS ,bothinthestrongandweaksymplecticcase,seeRemark3a,Equation 11. In Subsection 2.2, we turn to Fredholm pairs of Lagrangian subspaces to prepare for thecountingofintersectiondimensionsinthedefinitionoftheMaslovindex.Hereanother newfeatureofweaksymplecticanalysis isthat theFredholm indexofaFredholmpairof Lagrangiansubspacesdoesnotneedtovanish.Ontheonehand,thisopensthegatetonew interestingtheorems. Ontheotherhand, there-formulation ofwell-known definitions and lemmata in the weak symplectic setting becomes rather heavy since we have to add the vanishingoftheFredholmindexasanexplicitassumption. As a sideeffect of our weak symplectic investigation, we hope to enrich the classical literature with our new purely algebraic conditions for isotropic subspaces becoming La- grangians,inLemma4andPropositions1and2. Atpresent, thehomotopy types ofthefull Lagrangian Grassmannianand oftheFred- holmLagrangianGrassmannianremainunknown forweaksymplecticstructures.Wegive alistofrelated open problems inSubsection 2.3below. Tous, however, it seemsremark- ablethat awiderange offamiliargeometric features canbere-gainedinweaksymplectic functionalanalysis—inspiteoftheincomprehensibilityofthebasictopology. InSubsection2.4,welaythenextfoundationforarigorousdefinitionoftheMaslovin- dexbyinvestigatingcontinuouscurvesofoperatorsandrelationsthatgenerateLagrangians inthenewwidersetting.ReferringtotheconceptsofourAppendix,wedefinethespectral flowofsuchcurves. InSection3wefinallycometotheintersectiongeometry.InSubsection3.1,weshow howtotreatvaryingweaksymplecticstructuresinafixedBanachspacewithcontinuously varyingsymplecticsplittingsanddefinetheMaslovindexforcontinuouscurvesofFredholm pairsofLagrangiansubspacesinthissetting.Weobtainthefulllistofbasicpropertiesofthe MaslovindexaslistedbyS.E.Cappell,R.Lee,andE.Y.Millerin[17].Wecannot claim thatthisnewMaslovindexisalwaysindependentofthesplittingprojections.However,for strong symplectic Banach space the independence will be proved in Proposition 6. That establishesthecoincidencewiththecommondefinitionoftheMaslovindex. In Subsection 3.2, in our general context, we establishthe relation between real sym- plecticanalysis(inthetraditionofclassicalmechanics)ontheoneside,andthemoreelegant complexsymplecticanalysis(asfoundedbyJ.Lerayin[28])ontheotherside. In Subsection 3.3, we pay special attention to questions related to the embedding of symplecticspaces,Lagrangiansubspacesandcurvesintolargersymplecticspaces.Ourin- vestigations are inspired by the extremely delicate embedding questions between the two strongsymplecticHilbertspacesb (A)andL2(S )asstudiedbyK.Furutani,N.Otsukiand thefirstauthorin[9].Oneadditionalreasonforourinterestinembeddingproblemsisour observationofRemark2c,thateachweaksymplecticHilbertspacecannaturallybeembed- dedinastrongsymplecticHilbertspace,imitatingtheembeddingofH1/2(S )intoL2(S ). InAppendixA.1andA.2,werecallthebasicknowledgeandfixournotationsregarding gapsbetweenclosedsubspacesinBanachspace,uniformproperties,closedlinearrelations andtheirspectralprojections. Then, inAppendix A.3,wegivearigorous definitionofthe spectralflowforadmissiblefamiliesofclosedrelations.Ourdiscussionofcontinuousoper- atorfamiliesinSubsection2.4andthewholeofSection3isbasedonthatdefinition. Themainresultsofthispaperwereachievedmanyyearsagoby theauthorsandinfor- mallydisseminatedin[12].Throughalltheyears,ourgoalwastoestablishatrulygeneral spectralflowformulabyapplyingtheweaksymplecticfunctionalanalysis.Butherewemet atechnical gapintheargumentation: Onlyrecentlywefound thecorrect sufficient condi- tionsforcontinuousvariationoftheCauchydataspaces(or,alternativelystated,thecontin- 6 uousvariationofthepseudo-differentialCalderońprojection)forcurvesofellipticoperators injointworkwithG.ChenandM.Lesch[6].Nowthatgapisbridged,afullgeneralspectral flow formula isobtained in[14]and therelevance ofweaksymplecticfunctional analysis hasbecomesufficientlyclearforaregularpublicationofourresults. 2 Weaksymplecticfunctionalanalysis 2.1 Basicsymplecticfunctionalanalysis Wefixournotation.Tokeeptrackoftherequiredassumptions,weshallnotalwaysassume that the underlying space is a Hilbert space but permit Banach spaces and — for some concepts—evenjustvectorspaces.Foreasierpresentationandgreatergenerality,webegin withcomplexsymplecticspaces. Definition1 LetX beacomplexBanachspace.Amapping w : X X C × −→ iscalleda(weak)symplecticformonX,ifitissesquilinear,bounded,skew-symmetric,and non-degenerate,i.e., (i)w (x,y)islinearinxandconjugatelineariny; (ii) w (x,y) C x y forallx,y X; (iii)|w (y,x)|=≤ kw k(kx,yk); ∈ (iv)Xw := x−X w (x,y) = 0forally X = 0 . Thenwecall{(X∈,w )|a(weak)symplecticBa∈nac}hspa{ce}. Thereisapurelyalgebraicconcept,aswell. Definition2 LetX beacomplexvectorspaceandw aformwhichsatisfiesalltheassump- tionsofDefinition1except(ii).Thenwecall(X,w )acomplexsymplecticvectorspace. Definition3 Let(X,w )beacomplexsymplecticvectorspace. (a)Theannihilatorofasubspacel ofX isdefinedby l w := y X w (x,y) = 0 forallx l . { ∈ | ∈ } (b)Asubspacel iscalledsymplectic,isotropic,co-isotropic,orLagrangianif l l w = 0 , l l w , l l w , l = l w , ∩ { } ⊂ ⊃ respectively. (c)TheLagrangianGrassmannianL(X,w )consistsofallLagrangiansubspacesof(X,w ). Definition4 Let(X,w )beasymplecticvectorspaceandX+,X belinearsubspaces.We − call (X,X+,X ) a symplectic splitting of X, if X =X+ X , the quadratic form iw is − − ⊕ − positivedefiniteonX+andnegativedefiniteonX ,and − w (x,y) = 0 forallx X+andy X−. (3) ∈ ∈ 7 Remark1 (a) By definition, each one-dimensional subspace in real symplectic space is isotropic,andtherealwaysexistsaLagrangiansubspace.However,therearecomplexsym- plectic Hilbert spaces without any Lagrangian subspace. That is, in particular, the case if dimX+=dimX inN ¥ forasingle(andhenceforall)symplecticsplittings. − (b)Ifdim6 X isfinite,as∪u{bsp}acel isLagrangianifandonlyifitisisotropicwithdiml = 1dimX. 2 (c)InsymplecticBanachspaces,theannihilatorl w isclosedforanysubspacel .Inpartic- ular,allLagrangiansubspacesareclosed,andwehaveforanysubspacel theinclusion l ww l . (4) ⊃ (d)LetXbeavectorspaceanddenoteits(algebraic)dualspacebyX .Theneachsymplectic ′ formw inducesauniquelydefinedinjectivemappingJ: X X suchthat ′ → w (x,y) = (Jx,y) forallx,y X, (5) ∈ whereweset(Jx,y):=(Jx)(y). If(X,w )isasymplecticBanach space,then theinduced mapping J isabounded, injective mapping J: X X where X denotes the (topological) dual space. If J is also ∗ ∗ surjective (so, invertibl→e), the pair (X,w ) is called a strong symplectic Banach space. As mentionedintheIntroduction,wehavetakenthedistinctionbetweenweakandstrongsym- plecticstructuresfromChernoffandMarsden[19,Section1.2,pp.4-5]. IfX isaHilbertspacewithsymplecticformw ,weidentifyX andX .Thentheinduced ∗ mapping J isabounded, skew-self-adjoint operator(i.e., J = J)onX withkerJ= 0 ∗ − { } iA 0 andcanbewrittenintheformJ= + withA >0boundedself-adjoint(butnot 0 iA ± necessarilyinvertible,i.e.,A 1 not(cid:18)necess−aril−y(cid:19)bounded).Asinthestrongsymplecticcase, − wethenhavethatl X isL±agrangianifandonlyifl =Jl . ⊥ ⊂ Theproofofthefollowinglemmaisstraightforwardandisomitted. Lemma1 AnystrongsymplecticHilbertspace(X, , ,w )(i.e.,withinvertibleJ)canbe madeintoastrongsymplecticHilbertspace(X, , h,·w·i)withJ2= Ibysmoothdeforma- ′ ′ h· ·i − tionoftheinnerproductofX into hx,yi′ := h√J∗Jx,yi withoutchangingw . Remark2 (a) In a strong symplectic Hilbert space many calculations become quite easy. E.g.,theinclusion(4)becomesanequality,andallFredholmpairsofLagrangiansubspaces havevanishingindex,seebelowDefinition5,Equations(12)-(14). (b) From the Introduction, we recall an important example of a weak symplectic Hilbert space: Let A be a formally self-adjoint linear elliptic differential operators of first order over a smooth compact Riemannian manifold M with boundary S . As mentioned in the Introduction,wehave(wesuppressmentioningthevectorbundle) H1/2(S ) H1(M)/H1(M) (6) ≃ 0 with uniformly equivalent norms. Green’s form yields a strong symplectic structure on L2(S )by {x,y} := −hJx,yiL2(S ). (7) 8 Here J denotes the principal symbol of the operator A over the boundary in innernormal direction.ItisinvertiblesinceAiselliptic.FortheinducedsymplecticstructureonH1/2(S ) wedefineJ by ′ {x,y} = −hJ′x,yiH1/2(S ) forx,y∈H1/2(S ). LetBbeaformallyself-adjointellipticoperatorBoffirstorderonS .ByGa˚rding’sinequal- ity,theH1/2 norm isequivalent totheinduced graph norm. This yields J =(I+ B) 1J. ′ − SinceBiselliptic,ithascompactresolvent.So,(I+ B) 1 iscompactinL2(S );a|nd|sois − J.HenceJ isnotinvertible.Inthesameway,anyden|se|subspaceofL2(S )inheritsaweak ′ ′ symplecticstructurefromL2(S ). (c)EachweaksymplecticHilbertspace(X, , ,w )withinducedinjectiveskew-self-adjoint Jcannaturallybeembeddedinastrongsymhp·l·eicticHilbertspace X , , ,w withinvert- ′ ′ ′ h· ·i ibleinducedJ bysetting x,y := J x,y asinLemma1andthencompletingthespace. ′ ′ Thisimitatesthesituationhofthieemhb|ed|dinigofH1/2(S )intoL2(S(cid:0)).Itshowst(cid:1)hattheweak symplectic Hilbert space H1/2(S ) with its embedding into L2(S ) yields a model for all weaksymplecticHilbertspaces.InSection3.3,weshallelaborateontheembeddingweak ֒ strongalittlefurther. → Thefollowing lemmaisakeyresult insymplectic analysis.Therepresentation ofLa- grangiansubspacesasgraphsofunitarymappingsfromonecomponentX+ tothecomple- mentarycomponent X oftheunderlyingsymplecticvectorspace(tobeconsidered asthe − induced complex spaceinclassicalrealsymplecticanalysis,see,e.g.,K.Furutani andthe firstauthor[7,Section1.1])goesbacktoJ.Leray[28].Wegiveasimplificationforcomplex vectorspaces,firstannouncedin[43].Ofcourse,themainideaswerealreadycontainedin therealcase.TheLemmaisessentiallywell-knownandwillbeobtainedinthemoregeneral settingbelow:(i)isclear;(ii)willfollowfromLemma3;and(iii)fromProposition2. Lemma2 Let(X,w )beastrongsymplecticHilbertspacewithJ2= I.Then − (i) thespaceX splitsintothedirectsumofmutuallyorthogonalclosedsubspaces X = ker(J iI) ker(J+iI), − ⊕ whicharebothinvariantunderJ; (ii) thereisa1-1correspondencebetweenthespaceUJ ofunitaryoperatorsfromker(J iI)toker(J+iI)andL(X,w )underthemappingU l :=G(U)(=graphofU); − (iii) ifU,V UJ andl :=G(U),m :=G(V),then(l ,m )7→isaFredholmpair(seeDefinition 5b)ifa∈ndonlyifU V,or,equivalently,UV 1 I isFredholm.Moreover,we − − − ker(J+iI) haveanaturalisomorphism ker(UV−1−Iker(J+iI))≃l ∩m . (8) The preceding method to characterize Lagrangian subspaces and to determine the di- mensionoftheintersectionofaFredholmpairofLagrangian subspacesprovidesthebasis for defining the Maslov index in strong symplectic spaces of infinite dimensions (see, in different formulations anddifferent settings,thequoted references [7],[9], [24],[27],and ZhuandLong[45]). Surprisingly, itcanbegeneralized toweaksymplecticBanachspacesinthefollowing way. 9 Lemma3 Let(X,w )beasymplecticvectorspacewithasymplecticsplitting(X,X+,X ). − (a)Eachisotropicsubspacel canbewrittenasthegraph l = G(U) ofauniquelydeterminedinjectiveoperator U: dom(U) X− −→ withdom(U) X+.Moreover,wehave ⊂ w (x,y) = w (Ux,Uy) forallx,y dom(U). (9) − ∈ (b) If X is a Banach space, then X are always closed and the operator U defined by a ± Lagrangian subspace l is closed as an operator from X+ toX (not necessarilydensely − defined). (c) For a closed isotropic subspace l in a strong symplectic Banach space X, we have dom(U)andimUareclosed.Moreover,ifl isLagrangian,thendom(U)=X+andimU= X ;i.e.,thegeneratingU isboundedandsurjectivewithboundedinverse. − Proof a. Let l X be isotropic and v +v ,w +w l with v ,w X . By the + + ± isotropicpropert⊂yofl andourassumptionab−outthespl−itti∈ngX=X+± X± ∈wehave − ⊕ 0 = w (v +v ,w +w ) = w (v ,w )+w (v ,w ). (10) + + + + − − − − Inparticular,wehave w (v +v ,v +v ) = w (v ,v )+w (v ,v ) = 0 + + + + − − − − and sov =0 ifand only if v =0. So, ifthe first(respectively thesecond) components + oftwop−oints v +v ,w +w l coincide, thenalsothesecond (respectivelythefirst) + + componentsmustcoi−ncide. −∈ Nowweset dom(U) := x X+ y X−suchthatx+y l . { ∈ |∃ ∈ ∈ } Bytheprecedingargument, yisuniquelydetermined,andwecandefineUx:=y.Bycon- struction,theoperatorU isaninjectivelinearmapping,andproperty(9)followsfrom(10). b. By Definition 4 of a symplectic splitting, Equation (3) we have X (X+)w . Now − let x +x (X+)w withx X . Then w (x +x ,x )=w (x ,x )=⊂0 x =0 + ± + + + + + since iw −is∈positivedefinite±o∈nX+.ThatprovesX −=(X+)w ,andcorrespon⇐din⇒glyX+= − (X )w−.AsnoticedinRemark1c,annihilatorsarealwaysclosed.Thisprovesthefirstpart − of(b).Nowletl beaLagrangiansubspaceandletU betheuniquelydeterminedinjective operatorU: dom(U) X withdom(U) X+ andG(U)=l .ByDefinition3bwehave − l =l w ,hencel iscl→osedasanannihilator⊂andsoisthegraphofU,i.e.,U isclosed. c. Let l =G(U). Let x be a sequence in dom(U) convergent to x X+. Since X is n { } ∈ strong,weseefrom(9)thatthesequence Ux isaCauchysequenceandthereforeisalso n convergent. Denote by y the limit of U{x . S}ince l is closed, we have x domU and n y=Ux.Thusdom(U)isclosed.Weap{plyth}esameargumenttodom(U 1) ∈X ,relative − − totheinnerproductiw andobtainthatimU isclosed.Thisprovesthefirstpa⊂rtof(c). Nowassumethatl isaLagrangiansubspace.FirstlyweshowthatU isdenselydefined in X+. Indeed, if dom(U)=X+, there would be a v V, v=0, where V denotes the 6 ∈ 6 orthogonal complement ofdom(U)inX+ withrespecttotheinnerproduct on X+ defined 10 by iw .Clearly(dom(U))w =V+X .So,V =(dom(U))w X+.Thenv+0 l w l . − Tha−tcontradictstheLagrangianpropertyofl .So,wehavedom∩(U)=X+. ∈ \ We have shown that dom(U) is closed and dense. Hence dom(U) =X+. Now the boundedness ofU follows fromtheclosednessof G(U).Applying thesamearguments to dom(U 1) X relativetotheinnerproduct iw yieldsimU =dom(U 1)=X andU 1 − − − − − ⊂ isbounded. Remark3 (a)Notethatthesymplecticsplittingisnotunique.Itsexistencecanbeprovedby Zorn’s Lemma. In our applications, the geometric background provides natural splittings. Let A be an elliptic differential operator of first order, acting on sections of a Hermitian vector bundle E over the Riemannian manifold M with boundary S . Then the symplectic Hilbertspacestructuresof L2(S ;E S )andH1/2(S ;E S )of(7)and(6)arecompatibleand | | theirsymplecticsplittingisdefinedbythebundleendomorphism(theprincipalsymbolofA ininnernormaldirection)J: E S E S inthefollowingway: | → | H± := H1/2(S ;E±S ) and L± := L2(S ;E±S ) | | positive withE±|S := lin.spanof negative eigenspacesofiJ. (11) (cid:26) (cid:27) NotethatL+,L changecontinuouslyifJchangescontinuously.Forvaryingsplittingssee − alsothediscussionbelowinSection3. (b)ThesymplecticsplittingandthecorrespondinggraphrepresentationofisotropicandLa- grangiansubspacesmustbedistinguishedfromthesplittingincomplementaryLagrangian subspaces which yields thecommon representation of Lagrangian subspaces as images in therealcategory(seeLemma11below). 2.2 FredholmpairsofLagrangiansubspaces AmainfeatureofsymplecticanalysisisthestudyoftheMaslovindex.Itisanintersection index between apath of Lagrangian subspaces with the Maslovcycle, or, more generally, withanotherpathofLagrangiansubspaces. Beforegivingarigorous definitionoftheMaslovindexinweaksymplecticfunctional analysis(seebelowSection3)wefixtheterminologyandgiveseveralsimplecriteriafora pairofisotropicsubspacestobeLagrangian. Werecall: Definition5 (a) The space of (algebraic) Fredholm pairs of linear subspaces of a vector spaceX isdefinedby F2 (X) := (l ,m ) diml m <+¥ anddimX/(l +m )<+¥ (12) alg { | ∩ } with index(l ,m ) := diml m dimX/(l +m ). (13) ∩ − (b)InaBanachspaceX,thespaceof(topological)Fredholmpairsisdefinedby F2(X) := (l ,m ) F2 (X) l ,m andl +m X closed . (14) { ∈ alg | ⊂ }