Operator Theory: Advances and ApplicationsVol. 205Editor:I. GohbergEditorial Ofce:School of Mathematical SciencesTel Aviv UniversityRamat Aviv IsraelEditorial Board:D. Alpay (Beer Sheva, Israel)J. Arazy (Haifa, Israel)A. Atzmon (Tel Aviv, Israel)J.A. Ball (Blacksburg, VA, USA)H. Bart (Rotterdam, The Netherlands)A. Ben-Artzi (Tel Aviv, Israel)H. Bercovici (Bloomington, IN, USA)A. Bttcher (Chemnitz, Germany)K. Clancey (Athens, GA, USA)R. Curto (Iowa, IA, USA)K. R. Davidson (Waterloo, ON, Canada)M. Demuth (Clausthal-Zellerfeld, Germany)A. Dijksma (Groningen, The Netherlands)R. G. Douglas (College Station, TX, USA)R. Duduchava (Tbilisi, Georgia)A. Ferreira dos Santos (Lisboa, Portugal)A.E. Frazho (West Lafayette, IN, USA)P.A. Fuhrmann (Beer Sheva, Israel)B. Gramsch (Mainz, Germany)H.G. Kaper (Argonne, IL, USA)S.T. Kuroda (Tokyo, Japan)L.E. Lerer (Haifa, Israel)B. Mityagin (Columbus, OH, USA)V. Olshevski (Storrs, CT, USA)M. Putinar (Santa Barbara, CA, USA)A.C.M. Ran (Amsterdam, The Netherlands)L. Rodman (Williamsburg, VA, USA)J. Rovnyak (Charlottesville, VA, USA)B.-W. Schulze (Potsdam, Germany)F. Speck (Lisboa, Portugal)I.M. Spitkovsky (Williamsburg, VA, USA)S. Treil (Providence, RI, USA)C. Tretter (Bern, Switzerland)H. Upmeier (Marburg, Germany)N. Vasilevski (Mexico, D.F., Mexico)S. Verduyn Lunel (Leiden, The Netherlands)D. Voiculescu (Berkeley, CA, USA)D. Xia (Nashville, TN, USA)D. Yafaev (Rennes, France)Honorary and Advisory Editorial Board:L.A. Coburn (Buffalo, NY, USA)H. Dym (Rehovot, Israel)C. Foias (College Station, TX, USA)J.W. Helton (San Diego, CA, USA)T. Kailath (Stanford, CA, USA)M.A. Kaashoek (Amsterdam, The Netherlands)P. Lancaster (Calgary, AB, Canada)H. Langer (Vienna, Austria)P.D. Lax (New York, NY, USA)D. Sarason (Berkeley, CA, USA)B. Silbermann (Chemnitz, Germany)H. Widom (Santa Cruz, CA, USA)Pseudo-Differential Operators: Complex Analysis and Partial Differential EquationsInternational Workshop, York University, Canada,August 48, 2008B.-W. SchulzeM.W. WongEditorsBirkhuserBasel Boston Berlin2000 Mathematics Subject Classication: Primary 30G30, 31A30, 32A25, 33C05, 35, 42B10, 42B15, 42B35, 44A35, 46F12, 47, 58J32, 58J40, 62M15, 65R10, 65T60, 92C55; secondary 30E25, 31A10, 32A07, 32A26, 32A30, 32A40, 42C40, 45E05, 46F05, 65T50Library of Congress Control Number: 2009938998Bibliographic information published by Die Deutsche Bibliothek. Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliograe; detailed bibliographic data is available in the Internet at http://dnb.ddb.deISBN 978-3-0346-0197-9Birkhuser Verlag AG, Basel Boston BerlinThis work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microlms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. 2010 Birkhuser Verlag AG Basel Boston Berlin P.O. Box 133, CH-4010 Basel, SwitzerlandPart of Springer Science+Business MediaPrinted on acid-free paper produced from chlorine-free pulp. TCFPrinted in GermanyISBN 978-3-0346-0197-9e-ISBN 978-3-0346-0198-69 8 7 6 5 4 3 2 1www.birkhauser.chEditors:Bert-Wolfgang SchulzeInstitut fr MathematikUniversitt PotsdamAm Neuen Palais 1014469 PotsdamGermany e-mail: schulze@math.uni-potsdam.deM. W. WongDepartment of Mathematics and StatisticsYork University4700 Keele StreetToronto, Ontario M3J 1P3Canadae-mail: mwwong@mathstat.yorku.caContentsPreface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiB.-W. SchulzeBoundary Value Problems with the Transmission Property. . . . . . . . . . . . . . . . 1A. Dasgupta and M.W. WongSpectral Invariance of SG Pseudo-Dierential Operators onLp(Rn) . . . . . . 51J. Abed and B.-W. SchulzeEdge-Degenerate Families of Pseudo-Dierential Operatorson an Innite Cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59T. Gramchev, S. Pilipovicand L. RodinoGlobal Regularity and Stability inS-Spaces for Classes of DegenerateShubin Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81M. Maslouhi and R. DaherWeyls Lemma and Converse Mean Values for Dunkl Operators . . . . . . . . . . 91H. Begehr, Z. Du and N. WangDirichlet Problems for Inhomogeneous Complex Mixed PartialDierential Equations of Higher Order in the Unit Disc: New View. . . . . 101U. Aksoy and A.O. CelebiDirichlet Problems for Generalizedn-Poisson Equations . . . . . . . . . . . . . . . . 129A. MohammedSchwarz, Riemann, RiemannHilbert Problems and TheirConnections in Polydomains. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143V. Catan a, S. Molahajlooand M.W. WongLp-Boundedness of Multilinear Pseudo-Dierential Operators . . . . . . . . . . . 167J. DelgadoA Trace Formula for Nuclear Operators onLp. . . . . . . . . . . . . . . . . . . . . . . . . . 181V. Catan aProducts of Two-Wavelet Multipliers and Their Traces . . . . . . . . . . . . . . . . . 195S. MolahajlooPseudo-Dierential Operators on Z. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213J. ToftPseudo-Dierential Operators with Symbols in Modulation Spaces . . . . . . 223vi ContentsL. CohenPhase-Space Dierential Equations for Modes . . . . . . . . . . . . . . . . . . . . . . . . . . 235P. Boggiatto, G. De Donno and A. OliaroTwo-Window Spectrograms and Their Integrals . . . . . . . . . . . . . . . . . . . . . . . . 251C.R. Pinnegar, H. Khosravani and P. FedericoTime-Time Distributions for Discrete Wavelet Transforms . . . . . . . . . . . . . . 269C. Liu, W. Gaetz and H. ZhuThe Stockwell Transform in Studying the Dynamics of BrainFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277PrefaceThe International Workshop on Pseudo-Dierential Operators: Complex AnalysisandPartial Dierential Equations washeldatYorkUniversityonAugust48,2008. Therstphaseoftheworkshop onAugust45 consistedofamini-courseon pseudo-dierential operators and boundary value problems given by ProfessorBert-WolfgangSchulzeof Universit atPotsdamforgraduatestudentsandpost-docs. ThiswasfollowedonAugust68byaconferenceemphasizingboundaryvalue problems; explicit formulas in complex analysis and partial dierential equa-tions; pseudo-dierential operators and calculi; analysis on the Heisenberg groupandsub-Riemanniangeometry; andFourieranalysiswithapplicationsintime-frequency analysis and imaging.The role of complex analysis in the development of pseudo-dierential oper-atorscanbestbeseeninthecontextofthewell-knownCauchykernel andtherelated Poisson kernel in, respectively, the Cauchy integral formula and the Pois-sonintegral formulainthecomplexplaneC. Theseformulasareinstrumentalin solving boundary value problems for the Cauchy-Riemann operatorand theLaplacian on specic domains with the unit disk and its biholomorphic compan-ion, i.e.,theupperhalf-plane, as paradigm models. Thecorresponding problemsinseveral complexvariablescanbeformulatedinthecontextof theunitdiskin Cn,whichmay betheunitpolydiskortheunitballin Cn.Analogues oftheCauchy kernel and the Poisson kernel and their ramications to express solutionsofboundaryvalueproblemsinseveral complexvariablescanbelookeduponassingular integral operators, which are de facto equivalent manifestations of pseudo-dierential operators. Itisthevisionthatbringingtogetherexpertsinexplicitformulasforboundaryvalueproblemsincomplexanalysisworkingwithkernelsand specialists in pseudo-dierential operators working with symbolsshould buildsynergy between the two groups. The functional analysis and real-life applicationsof pseudo-dierential operators are always among top priorities in our agenda andthese are well represented in the workshop and in this volume.OperatorTheory:AdvancesandApplications,Vol.205, 150c2009BirkhauserVerlagBasel/SwitzerlandBoundaryValueProblemswiththeTransmissionPropertyB.-W.SchulzeAbstract. Wegiveasurveyonthecalculusof(pseudo-dierential)boundaryvalue problems with the transmisionproperty at the boundary, and ellipticityintheShapiroLopatinskijsense.Apartfromtheoriginal resultsoftheworkofBoutetdeMonvel wepresentanapproachbasedontheideasoftheedgecalculus. Inanal sectionweintroducesymbolswiththeanti-transmissionproperty.MathematicsSubject Classication(2000). 35J40, 58J32, 58J40.Keywords. Pseudo-dierential boundary value problems, transmission andanti-transmissionproperty,boundarysymboliccalculus,ShapiroLopatinskijellipticity,parametrices.1. IntroductionBoundary value problems (BVPs) for elliptic (pseudo-) dierential operators haveattractedmathematiciansandphysicistsduringall periodsof modernanalysis.While the denition of ellipticity of an operator on an open (smooth) manifold isvery simple, such a notion in connection with a (smooth or non-smooth) bound-ary is much less evident. During the past few years the interest in BVPs increasedagainconsiderably, motivatedbynewapplicationsandalsobyunsolvedprob-lems in the frame of the structural understanding of ellipticity in new situations.Several classical periods of the development created deep and beautiful ideas, forinstance, in connection with function theory, potential theory, with boundary op-eratorssatisfyingthecomplementingcondition, cf. Agmon, Douglas, Nirenberg[1], orpseudo-dierentialtheoriesfromVishikandEskin[29], Eskin[7], Boutetde Monvel [4]. Other branches of the development concern ellipticity with globalprojection conditions (analogues of Atiyah, Patodi, Singer conditions, cf. [3]),orelliptic theories on manifolds with geometric singularities, cf. the authors papers[18] or [19].2 B.-W. SchulzeAfter all that it is not easy to imagine how many basic and interesting prob-lems remained open. A part of the new developments is connected with the analysison congurations with singularities that includes boundary value problems. In thatcontext it seems to be desirable to see the pseudo-dierential machinery of Boutetde Monvel and also of Vishik and Eskin from an alternative viewpoint, using theachievements of the cone and edge pseudo-dierential calculus as is pointed out in[16], [21], and in the authors joint paper with Seiler [25], see also the monographs[22], or those jointly with Egorov [6], Kapanadze [12], Harutyunyan [10].Our expositionjust intends toemphasizesuchanapproach, heremainlyfocusedonoperatorswiththetransmissionpropertyattheboundaryfromthework of Boutet de Monvel. We also introduce symbols with the anti-transmissionpropertyattheboundary. Togetherwiththosewiththetransmissionpropertythey span the space of all (classical) symbols that are smooth up to the boundary.A pseudo-dierential calculus for such general symbols needs more tools from theedge algebra than developed here.Thepresentpaperistheelaboratedversionofintroductorylectures, givenduringtheInternational WorkshoponPseudo-Dierential Operators, ComplexAnalysisandPartial Dierential EquationsatYorkUniversityonAugust48,2008, in Toronto.2. InteriorandBoundarySymbolsforDierentialOperatorsLet X be a C manifold with boundary Y= X. Moreover, let 2X be the double,dened by gluing together two copies X of X to a C manifold along the commonboundary Y . Let us x a Riemannian metric on 2X and consider Yin the inducedmetric. There is then a tubular neighbourhood ofYin 2Xthat can be identiedwithY[1, 1],withasplittingofvariablesx=(y, t),wheretisthevariablenormal totheboundaryandy Y .Weassumethat(y, t)belongstoX=:X+for 0 t 1 and toX for 1 t 0 .If MisaCmanifold(withorwithoutboundary), byDi(M)wede-note the set of all dierential operators of order onMwith smooth coecients(smooth up to the boundary whenM ,= ).Local descriptions nearYwill refer to charts : U Rfor openU 2X,U Y ,= , and open Rn1, and induced charts : U Y onYand : U := U X Ron X near the boundary. Concerning the transition maps R R, (y, t) ( y,t), for simplicity we assume that the normal variable remains unchanged nearthe boundary, i.e., t = t for [t[ suciently small. The map y y corresponds to adieomorphism .Boundary Value Problems with the Transmission Property 3LetA Di(X), Bj Dij(V+), V+:=V X, j=1, . . . , N,forsomeN N, and setTu := (Bju[Y )j=1,...,N.Then the equationsAu = fin intX, Tu = g onY (2.1)represent a boundary value problem forA. Consider for the moment functions inC(X); then (2.1) can be regarded as a continuous operator/ =_AT_: C(X) C(X)C(Y, CN). (2.2)IfXis compact, we have the standard Sobolev spacesHs(2X) on 2XandHs(intX) := Hs(2X)[intX,s R. Then (2.2) extends to continuous operators/ : Hs(intX) Hs(intX)Nj=1Hsj12(Y )(2.3)for alls > maxj +12: j = 1, . . . , N.We will give a survey on elliptic boundary value problems (BVPs), startingfrom (2.2), and we ask to what extent we may expect a pseudo-dierential calculus(analgebra) thatcontainstheoperators (2.2)together withtheparametrices ofelliptic elements. First we have to explain what we understand by ellipticity of aboundary value problem.Incontrasttothenotionof ellipticityof adierential operator(ora, say,classical pseudo-dierential operator)A on an openCmanifoldM, in the caseof a manifold with boundary we have from the very beginning a variety of choices.ForanopenCmanifoldMdenotebyLcl(M)thespaceof all classicalpseudo-dierential operatorsof order RonM. Anoperator A Lcl(M)is called elliptic if its homogeneous principal symbol(A)(x, ) of order nevervanishes on TM0 (the cotangent bundle of M minus the zero section). The unionof spacesLcl(M)over R isclosed under theconstruction ofparametrices ofellipticelements, tobemoreprecise,everyellipticA Lcl(M)hasa(properlysupported) parametrixP Lcl(M) such that 1 PA, 1 AP L(M) (hereandinfutureby1 weoftendenoteidentityoperators). ThespaceL(M)canbe identied withC(M M) via a xed Riemannian metric onM.Letusrecallthewell-knownfactthatwhenMiscompactandclosed,theellipticity ofA Lcl(M) is equivalent to the property thatA : Hs(M) Hs(M) (2.4)is a Fredholm operator for some s = s0 R. Moreover, from the Fredholm propertyof(2.4) fors = s0it follows that (2.4) is Fredholm for alls R. In addition it is4 B.-W. Schulzeknown thatLcl(M)for every Rcontains so-called order reducing operators,i.e., elliptic operators Rthat induce isomorphisms,R: Hs(M) Hs(M) (2.5)for alls R; then (R)1 Lcl(M) is again order reducing (of opposite order).Below we shall establish more tools on pseudo-dierential operators.Let us now return to BVPs of the form (2.3), where X is a compact manifoldwith smooth boundaryY .Writing our dierential operator A in local coordinates x R+ near theboundary asA =

]]a(x)Dx,a C( R+), we dene(A)(x, ) =

]]=a(x), (2.6)(x, ) T( R+) 0, and observe the homogeneity(A)(x, ) = (A)(x, ), R+.Letx = (y, t), = (, ), and set(A)(y, ) =

]]=a(y, 0)(, Dt), (2.7)where(, Dt)=

Dtttfor =(t, tt) Nn, (y, ) T 0, or, equiva-lently, (A)(y, )=(A)(y, 0, , Dt). Theexpression (2.7)represents afamilyof continuous operators(A) : Hs(R+) Hs(R+), s R, (2.8)called the (homogeneous principal) boundary symbol ofA.LetHs(R+) be endowed with the strongly continuous group = R+of isomorphisms : Hs(R+) Hs(R+), (u)(t) = 1/2u(t), R+.Then we obtain the following kind of homogeneity of the boundary symbol,(A)(y, ) = (A)(y, )1, R+. (2.9)Homogeneityinthatsensewill alsoreferredtoastwistedhomogeneity(oforder).It makes sense also to dene the (homogeneous principal) boundary symbolof the trace operatorT=t(T1, . . . , TN), by(Tj)(y, )u := (Bj)(y, 0, , Dt)u[t=0, (2.10)Boundary Value Problems with the Transmission Property 5u Hs(R+),s> maxj +12:j= 1, . . . , N where(Bj)(x, ) is the homoge-neous principal symbol of the operator Bj, and (2.10) is interpreted as a family ofoperators(Tj)(y, ) : Hs(R+) C,(y, ) T 0. Theboundarysymbol (2.10)ishomogeneousinthefollowingsense:(Tj)(y, ) = j+12(Tj)(y, )1, R+. (2.11)It is often convenient to compose (2.2) from the left by an operatordiag(1, R1, . . . , RN) (2.12)whereRj L(j+12)cl(Y ) is an order reducing operator on the boundary in theabove-mentioned sense and to pass to a modied operator_At(R1T1, . . . , RNTN)_: Hs(intX) Hs(intX)Hs(Y, CN),related to the former one by a trivial pseudo-dierential reduction of orders on theboundary. This is formally a little easier (later on we admit such trace operatorsanyway). Instead of(2.11) we then obtain(RjTj)(y, ) = (RjTj)(y, )1, R+, (2.13)where(RjTj)(y, ) = (Rj)(y, )(Tj)(y, )with(Rj)(y, )beingthehomogeneousprincipal symbol of Rjof order (j +12) as a classical pseudo-dierential operator on the boundary.Letusnowexplaintheroleof thetraceoperatorsinconnectionwiththeellipticity of a boundary value problem. We call the pair(/) = ((/), (/))the principal symbol of /, consisting of the (principal) interior symbol(/) :=(A) and the (principal) boundary symbol (/):=t((A), (T1), . . . , (TN))of /,(/) : Hs(R+) Hs(R+)CN.Ellipticity of / requires the bijectivity of both components on TX0 and TY 0,respectively, the latter as an operator function for s > 12. Since the operators(Tj)(y, ) are of nite rank, (A)(y, ) has to be a family of Fredholm operators.The following lemma shows that this is an automatic consequence of the ellipticityofA with respect to.6 B.-W. SchulzeLemma2.1. LetA be an ellipticdierential operator; then(A)(y, ) : Hs(R+) Hs(R+)is asurjectivefamilyof Fredholmoperators foreveryreal s > 12, andthekernel ker (A)(y, ) is a nite-dimensional subspace of o(R+) = o(R)[R+whichisindependent of s. Moreover, dimker (A)(y, )=dimker (A)(y, /[[)forall (y, ) TY 0.Proof. Setforthemomenta():=(A)(y, 0, , )withfrozenvariables(y, ), ,=0. Then(/) =a(Dt)canbewrittenasop+(a) :=r+opt(a)e+fortheoperator of extension e+of functions by 0 to t < 0 and r+the restriction to t > 0,andopt() is thepseudo-dierential operatoronRwiththesymbol a(), i.e.,opt(a)u(t) = __ei(tt

)a()u(tt)dttd, d= (2)1d.Then op+(a1) is a rightinverse of op+(a), since op+(a)op+(a1) = op+(aa1) + r+op(a)eop+(a1) = 1,becauseofr+op(a)e=0.Thisshowsthesurjectivityofop+(a). Thefactthatsolutionsuof thehomogeneousequationa(Dt)u=0formanite-dimensionalsubspaceof o(R+)isstandard. However,wewill showthosethingsbelowonceagain independently, cf. Theorem 3.29 below. The last assertion follows from thehomogeneity (2.9). Example. LetA=betheLaplacian,= nj=12x2jinlocalvariables.Then() = [[2, and()() = [[2+D2t: Hs(R+) Hs2(R+).We haveker ()() = ce]]t: c C,i.e., dimker ()() = 1 for all ,= 0 and alls > 3/2.Remark2.2. The operatorsTk:Hs(X) Hsk1/2(Y ), locally nearY denedbyTku := Dktu[t=0, k N,have the boundary symbols(Tk)u = Dktu[t=0, (Tk)() : Hs(R+) Cand are (although they are independent of) of homogeneityk +12, i.e.,(Tk)()u = k+12(Tk)()1u, R+.Moreover, asweseefromLemma2.1togetherwithLemma2.3below,thecolumn matrix_()()(Tk)()_: Hs(R+) Hs2(R+)C(2.14)is an isomorphism for every ,= 0, s > max32, k +12; this is true for every k N.Observe thatT0 represents Dirichlet andT1Neumann conditions.Boundary Value Problems with the Transmission Property 7In other words, the boundary symbol (Tk) lls up the Fredholm operators()() : Hs(R+) Hs2(R+),s > 3/2, to a family of isomorphisms (2.14). Inthis way we have examples of so-called elliptic BVPs, namely,/k :=_Tk_: Hs(X) Hs2(X)Hsk12(Y )(2.15)foreveryk N. Inconnectionwithsuchconstructionsitisuseful torecallthefollowing simple algebraic result.Lemma2.3. Let H, H,LbeHilbert spacesanda: H H, b: H Llinearcontinuousoperators.Thenthecolumnmatrixoperatora := _ab_ :H HLisanisomorphism if and only ifa : H His surjective, andb : H L restricts to anisomorphismb[ker a : ker a L.Proof. Leta : H Hbe surjective, andb0 := b[ker a : ker a L an isomorphism.Thena is obviously surjective. Moreover,au = 0 impliesu ker a andb0u = 0;then, since b0 is an isomorphism it follows that u = 0. Thus a is injective and henceanisomorphism.Conversely,assumethataisanisomorphism.Thesurjectivityof aimpliesthat a: H H, b: H Larebothsurjective. Inparticular, ifH1 denotes the orthogonal complement of ker a1 inH, we obtain an isomorphisma1 := a[H1: H1 H, anda can be written as a block matrixa =_a10b1b0_:H1ker aHLforb1:=b[H1. Itremainstoshowthatb0: ker a Lisanisomorphism. Theoperator_a110b1a111_:HLH1Lis an isomorphism, and we have_a110b1a111__a10b1b0_=_1 00 b0_:H1ker aH1L.Therefore, sincebothfactorsontheleft-handsideareisomorphisms, itfollowsthat alsob0 : ker a ker a is an isomorphism. This shows us the meaning of the above-mentionedN, the number of traceoperators which turns the boundary symbol(/)(y, ) : Hs(R+) Hs(R+)CN(2.16)8 B.-W. Schulzetoafamilyof isomorphisms. AccordingtoLemmas2.1and2.3, foranellipticdierential operatorA we haveN= dimker (A)(y, ); this number is requiredtobeindependentof yand ,=0. Asiswell-known, if Aisof order2mandadmits boundary operators B1, . . . , Bm satisfying the so-called complementingconditionwithrespecttoA,then(form =N)thatproperty holds (cf.Agmon,Douglis, Nirenberg [1], Lions, Magenes [13]).Denition2.4. The block matrix operator / =t(A T) is said to be a (ShapiroLopatinskij) ellipticboundary value problem fortheellipticdierential operatorAiftheboundary symbol (2.16) isa familyofisomorphisms forany sucientlylarges, for all (y, ) TY 0. We also talk about ShapiroLopatinskij trace (orboundary) conditions for the operator A.Remark 2.5.Observe that not every elliptic dierential operator A admits ShapiroLopatinskij elliptic trace conditions. The simplest example is the Cauchy-Riemannoperatorinthecomplexplane. MoregeneralexamplesareDiracoperators ineven dimensions, and other important geometric operators. We will return later onto this discussion in the context of the AtiyahBott obstruction for the existenceof ShapiroLopatinskij elliptic conditions.If we ask for an algebra of BVPs a rst essential formal problem is that columnmatrices cannot be composed with each other in a reasonable manner. However, weextend the notion algebra and talk about block matrix operators where the al-gebraic operations are carried out only under natural conditions, namely, additionwhen the matrices have the same number of rows and columns and multiplicationwhen the number of rows and columns in the middle t together. For instance, ifwe consider the Dirichlet problem /0for the Laplacian, cf. the formula (2.15) fork = 0, we have invertibility of/0 : C(X) C(X)C(Y ).Denotingby T:=(P0K0) theinverseof /0(whichbelongstothepseudo-dierential operatorcalculustobediscussedhere), thenwehavetwokindsofcompositions, namely, forA = ,/0T0 =_AT0_(P0K0) =_AP0AK0T0P0T0K0_=_1 00 1_, (2.17)and(P0K0)_AT0_= P0A+K0T0 = 1. (2.18)It also makes sense to consider/kT0 =_1 0TkP0TkK0_(2.19)foreveryk N.Thelower rightcorner ofthelattermatrix hasthemeaningofthereductionof theboundaryconditionTktotheboundary(bymeansof theBoundary Value Problems with the Transmission Property 9Dirichlet problem). It turns out that TkK0 is a classical elliptic pseudo-dierentialoperator of order k on the boundary. Its symbol will be computed in the followingsection, cf. the formula (3.28).3. InversesofBoundarySymbolsLet us rst recall that the construction of a parametrix of an elliptic operator A Lcl(M) on an open C manifold Mcan be started by inverting the homogeneousprincipalsymbolandformingaB Lcl(M)suchthat(B)=1(A)(Bisobtained via an operator convention). In a second step we form1 BA = C L1cl(M)(everything in the frame of properly supported pseudo-dierential operators), thenwe pass to aD L1cl(M) such that (1 + D)(1 C) = 1 modL(M). Such aDcanbefoundasanasymptoticsum j=1 Cj, andP=(1 + D)Bisthenaleft parametrix ofA. (For future references we call the latter procedure a formalNeumannseries argument.) Inan analogous manner we nda right parametrix,and then a simple algebraic consideration shows that Pis a two-sided parametrix.These arguments are based on the following properties of (classical) pseudo-dierential operators:1. everypseudo-dierential operatorhasaproperlysupportedrepresentativemodulo a smoothing operator;2. anysequenceof operatorsof order j, j N, hasanasymptoticsum,uniquely determined modulo a smoothing operator;3. there is a symbolic map that assigns the unique principal symbol of an opera-tor; the algebraic operations between operators are compatible with those forassociated principal symbols (in particular, the principal symbol of a compo-sition is equal to the composition (product) of the principal symbols);4. every smooth homogeneous function of order onTM 0 is the principalsymbol of an associated pseudo-dierential operator of order (i.e., there isan operator convention that is right inverse of the principal symbolic map of3.);5. an operator of order with vanishing principal symbol is of order 1.It turns out that boundary value problems as in Section 2 can be completed toa graded algebra of 2 2 block matrix operators with a two-component principalsymbolichierarchy=(, ), whereanaloguesof theproperties1.5. hold.Such an algebra has been introduced by Boutet de Monvel [4], and we discuss here(among other things) some elements of that calculus.The rst essential point is to analyse the nature of inverses of bijective bound-ary symbols. Since such inverses are computed (y, )-wise for (y, ) TY 0 werst freeze those variables and look at operators on R+. Let us consider a classicalsymbola() Scl(R), R. Examples of such symbols arel() := (1 i). (3.1)10 B.-W. SchulzeLet us setop+(a)u(t) = r+op(a)(e+u)(t), (3.2)for every u Hs(R+) (= Hs(R)[R+),s > 12, where e+u ot(R) is the distribu-tion obtained by extendingu by zero to R, i.e.,e+u(t) = u(t) fort > 0, e+u(t) = 0 fort < 0.Moreover r+is the operator of restriction from R to R+, andop(a)v(t) =__ei(tt

)a()u(tt)dttd,d= (2)1d. In an analogous manner we dene the extension eby zero fromRto R and the restriction r from R to R. The operator (3.2) denes a linearmapop+(a) : Hs(R+) ot(R+)foreverys> 12, ot(R+) := ot(R)[R+.Asiswellknown(cf.[7]),insomecasesop+() induces a continuous operatorop+(a) : Hs(R+) Hs(R+) (3.3)for everys > 1/2, namely, whena() is a so-called minus-symbol.Let /(U), UCopen, denotethespaceof all holomorphicfunctionsinU, andsetC:= z=+ i: 0. Thena() Scl(R)issaidtobeaminus-symbol if a()hasanextensiontoafunctionin /(C+) C(C+)suchthat[a(z)[ c(1 +[z[2)/2(3.4)for all z C+, for some constant c > 0. By a plus-symbol of order we understandan element a() Scl(R) that extends to a function in /(C)C(C) such thatthe estimates (3.4) hold for all z C. For s R we have a relation similar to (3.3)whenwereplacee+byacontinuousextensionoperatore+s: Hs(R+) Hs(R)withe+s u[R+=u; thenintheminus-casethelattermapisindependentof thechoice of e+s .If a()isa plus-symbol oforderandHs0(R+) := u Hs(R): u = 0 onR, thenop+(a) : Hs0(R+) Hs0(R+) (3.5)iscontinuous foreverys R.Concerning aproof ofthecontinuity of (3.3)and(3.5), see [7, Lemma 4.6 and Theorem 4.4], (cf. also [10, Section 4.1.2]). Moreover,for an arbitrary p() Scl(R), R, we haveop+(ap) = op+(a)op+(p) (3.6)whena() Scl(R) is a minus-symbol (since r+op(a)e = 0) andop+(pa) = op+(p)op+(a) (3.7)whena() Scl(R) is a plus-symbol (since rop(a)e+= 0).Example. A polynomial inis both a minus- and a plus-symbol.Boundary Value Problems with the Transmission Property 11Remark 3.1. Thefunctionl()= (1 i)isaminus-symbol oforder R,andop+(l) : Hs(R+) Hs(R+)isanisomorphismforeverys R, s>max12, 12,where(op+(l))1=op+(l). Moreover, l+() = (1 +i)is a plus-symbol of order R, andop+(l+) : Hs0(R+) Hs0(R+)is an isomorphism for everys R where (op+(l+))1= op+(l+).A classical symbola() Scl(R) has an asymptotic expansiona()

j=0aj (i)jfor (3.8)foruniquecoecientsaj C(theimaginaryuniti= 1istakenforconve-nience; powersaredenedas(i)=e log(i)withtheprincipal branchofthelogarithm).If () C(R)isanexcisionfunctionin (i.e., ()=0for [[ c1, for some 0 < c0< c1), then we havea()

j=0()a(j)() (3.9)fora(j)() = a+j+() +aj()(i)j, (3.10)with+being the characteristic function of the half-axis in, where (3.9) hasthe meaning of an asymptotic expansion of symbols,()a(j)() Sjcl(R).Denition 3.2. A symbola() Scl(R) for Z has the transmission property ifa+j= ajfor allj N. (3.11)Let Str(R) denote the space of all symbols in Scl(R) with the transmission property.Remark3.3. A symbola() Scl(R) has the transmission property exactly whena(j)() = (1)ja(j)() (3.12)for all R 0 and allj N.In fact, the transmission property means that a(j)() = cj(i)jfor cj :=a+j= ajfor allj N, and this shows the relation (3.12). Conversely from (3.12)we deducea+j+() +aj()(i)j= (1)ja+j+() +aj()(i)j= a+j+() +aj()(i)jfor all ,= 0, which implies a+j (+() +()) = aj (() ()). For > 0we have+() = () = 0 and+() = () = 1 which yieldsa+j= aj .12 B.-W. SchulzeRemark 3.4. ThespaceScl(R)isanuclearFrechetspaceinanaturalway,andStr(R) is a closed subspace in the induced topology.Example. 1. Every polynomial inhas the transmission property;2. the-wise product of two symbols with the transmission property has againthe transmission property;3. If a Str(R) and a+0= a0,= 0, then it follows that ()a1() Str(R) fora suitable excision function(). If in additiona() ,= 0 for all R, thena1() Str(R).In particular, the symbols (3.1) for Z have the transmission property.Remark3.5. The multiplication of symbols byl+() (orl()) induces an iso-morphismStr(R) S0tr(R).Remark 3.6. Leta() S0cl(R), andformtheboundedsetL(a):= a() C: R which is a smooth curve (with admitted self-intersections) and end pointsa0=a(). Thenwehavea() S0tr(R)ifandonlyif L(a)isaclosedcurvewhich is smooth includinga+0= a0 .Remark3.7. Every symbola() Str(R) can be written in the forma() = p() + b()wherep()is apolynomial inoforder(onlyrelevant for 0)andb() S1tr(R).In fact, this is an evident consequence of Denition 3.2.Proposition 3.8. Leta() Str(R); then for everyN N there is a minus-symbolmN() Scl(R)andaplus-symbol pN() Scl(R) suchthat a() mN() S(N+1)cl(R) anda() pN() S(N+1)cl(R).Proof. Since a polynomial in is a plus- and a minus-symbol it suces to assume= 1. Bydenitionthereareconstants ajsuchthat foranyxedexcisionfunction()a() = ()N

j=1aj(i)j+rN() (3.13)where rN() S(N+1)cl(R). The relation1i= 11i +1i11ican be iterated, andwe obtain1i= 11i+ 11i+1i11i11i= 11i 1(1i)2+1i1(1i)2=. . . =

Nk=11(1i)k+1i1(1i)N .Thisyields(i)j= _

Nk=1(1 i)k_j+rj,N()for everyj N 0 where()rj,N() S(N+1)cl(R) for every excisionfunction(). Thus, settingmj,N() := aj_

Nk=1(1 i)k_jwe obtain()aj(i)j= mj,N() +()ajrj,N() (3.14)Boundary Value Problems with the Transmission Property 13modulo a symbol inS(R), wheremj,N() is a minus-symbol, cf. Remark 3.1.Then from (3.13) we obtain the rst assertion, for mN() =

Nj=1 mj,N(). More-over, writing1i=11+i+1i11+i= Nk=11(1+i)k+1i1(1+i)Nweobtain aplus-symbolpN() :=

Nj=1 pj,N(),pj,N() := aj_Nk=1(1 +i)k_jwith the desiredproperty. Corollary3.9. Leta() Str(R); then op+(a) induces a continuous operatorop+(a) : Hs(R+) Hs(R+)for every real s > 12.Proof. Letuswriteop(a)=op(mN) + op(cN)where, accordingtoProposition3.8,mNis a minus-symbol of order, andcN S(N+1)cl(R). Then we haveop+(a) = op+(mN) + op+(cN). (3.15)We observedbefore that op+(mN) has the desiredmappingproperty. Let usnowassumes (12, 0]. Weemploytheknownfactthatforthoseswehavee+Hs(R+) = Hs0(R+). As noted before we have an isomorphismop(ls+) : Hs0(R+) L2(R+) = H00(R+)with the inverse op+(ls+). Moreover, using the relation (3.7) we haveop+(cN) = op+(cNls+)op+(ls+)where(cNls+)() S(N+1)scl(R). Thusitremainstoverifythatop+(cNls+):L2(R+) Hs(R+)iscontinuous. However, whenNislarge enough, wehavethe continuityop+(cNls+) : L2(R+) HN+1+s(R+).Thus forNso large thatN + 1 we obviously obtain the desired continuity.Finallyfors 0itsucestoemploythecontinuousembeddinge+Hs(R+)L2(R), i.e., we can argue similarly as before and obtain the continuity op+(cN) :L2(R+) Hs(R+) forN + 1 > s . Proposition 3.10. Every symbol a() S1tr(R) can be written in the forma() = a+() +a() (3.16)for uniquely determineda+() Ft_e+o(R+)_, a() Ft_eo(R)_which are plus/minus symbols inS1tr(R).Concerning a proof of Proposition 3.10, see [15, Section 2.1.1.1].Corollary3.11. Leta() Str(R); then op+(a) induces a continuous operatorop+(a) : o(R+) o(R+).14 B.-W. SchulzeProof. For N the symbol a() is equal to a polynomial in of order , moduloasymbol inS1tr(R). Thuswithoutlossof generalityweassume= 1. TheFourier transformF= Ftinduces a continuous operatorF: e+o(R+) o1tr(R). (3.17)Moreover, the multiplication between symbols with the transmission propertyis bilinear continuous. In particular, the composition of (3.17) with the multipli-cation by the symbol (3.16) gives us a continuous operatora()F: e+o(R+) S1tr(R).FinallyF1:S1tr(R) e+o(R+) + eo(R)is a topological isomorphism (thesum on the right-hand side is direct) andr+: e+o(R+) + eo(R) o(R+)is obviously continuous. Thus op+(a) = r+F1a()Fe+is a composition of con-tinuous operators. Proposition 3.12. Leta() S0tr(R); then the adjointofop+(a) : L2(R+) L2(R+)withrespect tothe L2(R+)-scalarproduct hastheformop+(a)forthecomplexconjugatea() S0tr(R).Proof. The computation is completely elementary. Proposition3.13. Let a() S0cl(R)beasymbol withthetransmissionproperty,let : RRbe denedby (t) =t, and : L2(R)L2(R) thecorresponding function pull back. Thenr+op(a)e, rop(a)e+: L2(R+) L2(R+) (3.18)induce continuous operatorsL2(R+) o(R+).Proof. If a() is a constant, both operators are zero. Therefore, it suces to assumea() S1cl(R). By virtue of the identity,r+op(a) = op+((lN)(lN))r+op(a) = op+(lN)r+op(lNa)for anyN Z (cf. the relation (3.6) taking into account thatlN()are minus-symbols); we may even consider the symbol lN()a() S(N+1)cl(R) rather thana(), for anyN> 1, since op+(lN) : o(R+) o(R+) is continuous, cf. Corollary3.11. In other words, leta() S2cl(R); thenr+op(a)ev(t) = r+_R_0ei(t+t

)a()v(tt)dttd= r+_0__ei(t+t

)a()d_v(tt)dtt.Boundary Value Problems with the Transmission Property 15By virtue of Proposition 3.10 we have_eira()d e+o(R+) + eo(R)withr R being the variable on the right-hand side. Sincer has the meaning oft +ttfort > 0,tt> 0, we obtainr+op(a)ev(t) = r+_0f(t +tt)v(tt)dtt(3.19)for some f(r) e+o(R+). It remains to observe that the right-hand side of (3.19)represents a continuous operator L2(R+) o(R+). The second operator in (3.18)can be treated in an analogous manner. Corollary 3.14. Let g denote one of the operators in (3.18), and let g be its adjointinL2(R+). Thengandginduce continuous operatorsg, g : L2(R+) o(R+). (3.20)Proof. The assertion forg is contained in Proposition 3.13. Moreover, because of(r+op(a)e)=rop(a)e+by Proposition 3.13 we also obtain theresult forg. Remark 3.15. It canbeprovedthat anoperator g L(L2(R+)) that denescontinuous operators (3.20) can be represented in the formgu(t) =_0c(t, tt)u(tt)dttfor somec(t, tt) o(R+R+) (= o(R R)R+R+), see [10, Theorem 2.4.87].Denition3.16. 1. Anoperatorg L(L2(R+))whichinducescontinuousop-erators (3.20)iscalledaGreenoperatoroftype0.Let0(R+)denotethespace of those operators.2. An operator of the form

dj=0 gjjtfor gj 0(R+), d N, is called a Greenoperator of typed. Let d(R+) denote the space of those operators.Remark3.17. Anyg d(R+) induces a compact operatorg : Hs(R+) Hs(R+)for everys R,s > d 12. Moreover g induces a continuous operatorg : Hs(R+) o(R+) (3.21)for thoses.Infact, jt:Hs(R+) Hsj(R+)is continuous for everyj Nas well asg0 : Hsj(R+) o(R+) whens j> 12,g0 0(R+).Lemma3.18. Letg 0(R+),and let1 + g :L2(R+) L2(R+)bean invertibleoperator. Then there is anh 0(R+) such that (1 +g)1= 1 +h.16 B.-W. SchulzeProof. Let b:=(1 + g)1whichbelongsto L(L2(R+)). Writingb=1 + hforh := b1 we obtain (1+g)(1+h) = 1, i.e., h+g+gh = 0. This yields h = g(1+h),andhenceh: L2(R+) o(R+)iscontinuous, cf. Denition3.16(i). Moreover(1 +g)(1 +h) = 1 yieldsg +h +gh = 0, i.e.h = g(1 +h) which showsagain the continuityh:L2(R+) o(R+). In other words,h 0(R+). Corollary3.19. Leta() Str(R), b() Str(R); thenop+(a)op+(b) = op+(ab) +g (3.22)for someg 0(R+).Proof. For = = 0 we haveop+(a)op+(b) = r+op(a)e+r+op(b)e+= r+op(a)op(b)e++ r+op(a)op(b)e+for the characteristic functionof R. Sincer+op(a)op(b)e+= (r+op(a)e)(rop(b)e+) =: gandthefactorsinthemiddleareGreenoperatorsoftypezero, cf. Proposition3.13, we obtaing 0(R+), since 0(R+) is closed under compositions.It remains to consider N or N. In this case we writea() = a0() +p(), b() = b0() +q()for a0, b0 S1tr(R) and polynomials p and q of degree and , respectively. Sincepolynomials are minus- and plus-symbols at the same time we haveop+(p)op+(b0) = op+(pb0), op+(a0)op+(q) = op+(a0q),i.e., when we denegby op+(a0)op+(b0) = op+(a0b0) + g(according to the rstpart of the proof) we obtainop+(a)op+(b) = (op+(a0)+op+(p))(op+(b0)+op+(q)) = op+((a0+p)(b0+q))+g.

More generally we have the following composition property.Theorem3.20. Let a() Str(R), b() Str(R),andg d(R+), h e(R+).Then(op+(a) +g)(op+(b) +h) = op+(ab) +kfor a certaink max+d,e](R+).Proof. By virtue of Corollary 3.19 it remains to discuss the compositionsop+(a)h, gop+(b), andgh.Itisevidentthatop+(a)h e(R+)andgh e(R+). Fortheoperatorinthemiddle we writegop+(b) = g(op+(b0) + op+(p))Boundary Value Problems with the Transmission Property 17whereb0 S1tr(R)andpisapolynomial in of order (whichvanishesfor 1). It is clear thatgop+(p) +d(R+). What concernsgop+(b0) it sucesto assumeg = g0Djtfor any 0 j d. Sincejis a minus-symbol we havegop+(b0) = g0op+(bj) forbj() = jb0() Sj1tr(R).Thus, writing bj() = cj() +qj1() for a polynomial qj1() in of degree j 1(whenj 1 0) and somecj S1tr(R) it follows thatgop+(b0) = g0op+(cj) +g0op+(qj1).The secondsummandonthe right obviouslybelongs toj1(R+) for j 1while the rst one belongs to 0(R+) which follows from the continuity op+(cj) :L2(R+) L2(R+) and g0 : L2(R+) o(R+) and an analogous conclusion for theadjoints. Remark3.21. As a special case of Theorem 3.20 fora() Str(R), g d(R+),weobtainthat(op+(a) + g)op+(lN)=op+(alN) + kfor k 0(R+)whenN +d 0.Let us now turn to 2 2 block matrices of operators with upper left cornersof the form_op+(a) +g11g12g21g22_:Hs(R+)CHs(R+)C(3.23)for arbitrary a() Str(R), Z,g11 d(R+),d N,s > d 12,g22 C,g21u(t) =d

l=0g21,lltu(t) u Hs(R+) (3.24)forg21,lv(t) :=_0f21,l(t)v(t)dt,f21,l o(R+),l = 0, . . . , d, andg12c := cf(t), c C, (3.25)forsomef o(R+). Anoperatorof theform(3.24)iscalledatraceoperatorof typed, and (3.25) a potential operator (for the boundary symbolic calculus ofoperators with the transmission property at the boundary).Inasimilarmannerwedeneanaloguesof (3.23)whereContheleftisreplacedbyCjandontherightbyCj+forcertainj, j+ N(if oneof thedimensions iszero, thenwe have row or columnmatrices whichare admittedaswell). Let B,d(R+; j, j+) denote the space of such block matrices. Moreover, letBdG(R+; j, j+) be the subspace of operators (3.23) dened bya 0.Thus BdG(R+; 0, 0)=d(R+); infuturewealsowrite BdG(R+)ratherthand(R+).Remark 3.22. Moregenerally wehave B,d(R+; v)forv=(k, l; j, j+),denedtobethespaceof22blockmatriceswheretheupperleftcorneritself isanl k matrix of operators as in the upper left corner of (3.23), while g12 is a jkmatrix of potential operators, etc. For every xed Z,d N, the space of such18 B.-W. Schulzematricesisa(nuclear)Frechet spaceinanaturalway. Thefuturehomogeneousboundary symbols of BVPs are symbols in (y, ) with values in such spaces.Remark3.23. It can easily be proved, cf. [22, Proposition 4.1.46], that everyg d(R+) ford > 0 has a unique representationg = g0 +d1

j=0kj rtDjtfor g0 0(R+), potential operators kj, and rtu := u(0). Similarly, a trace operatorb of typed > 0 can uniquely be written asb = b0 +d1

j=0cj rtDjtfor a trace operatorb0 of type 0 and constantscj.Example. The operator_op+([[22)rtDkt_: Hs(R+) Hs2(R+)C, (3.26)rtv =v[t=0, belongs to B2,k+1(R+; 0, 1),k N, and (3.26) is an isomorphism forevery Rn1 0, s > max32,k +12. The operator family (3.26) fork = 0 isjust the boundary symbol of the Dirichlet problem for the Laplace equation andfork = 1 of the Neumann problem.The inverse of(3.26) fork = 0 is explicitly computed in [12, Section 3.3.4].Settinga(, ) := [[2[[2the result is_op+(a)()rt_1= (op+(l1+)()op+(l1)() d())forl() := [[ iand a potential operatord()denedbyd() :c ce]]t,c C. By virtue of Corollary 3.19 we haveop+(l1+)()op+(l1)() = op+(a1)() +g()for a Green operator family g() of type 0. Note that g() is just the homogeneousboundary symbol of the well-known Greens function of the Dirichlet problem forthe Laplacian (twisted homogeneous of order 2).Itisnoweasyalsotocomputetheinversesof (3.26)forarbitraryk N,especially, of the boundary symbol of the Neumann problem. In fact, similarly as(2.19), now on the level of boundary symbols, we have_op+(a)()rtDkt_(p() d()) =_1 0b() qk()_(3.27)Boundary Value Problems with the Transmission Property 19for p() :=op+(a1)() +g(), b() :=rtDkt (op+(a1)() +g()), gk() :=rtDktd(). We havertDktd() = Dkte]]tt=0 = (i[[)k(3.28)whichisjustthehomogeneous principalsymboloftheellipticoperatorTkK0 Lkcl(Y ) occurring in the lower right corner of the operator (3.27). Thus_op+(a)()rtDkt_1= (p() d())_1 0b() qk()_1= (p() d()q1k()b() d()q1()).General compositions of boundary symbols are studied in Theorem 3.26 be-low.Remark 3.24. Itisinterestingtoconsiderellipticboundaryvalueproblemsforthe elliptic operatorTkK0on a smooth submanifold ofYwith boundaryZ. Thismakes sense, for instance, when we reduce the Zaremba problem for (dened byjumping conditions from Dirichlet to Neumann along Z) toY . Then a basic di-culty is that TkK0 fails to have the transmission property at Z, cf. Denition 4.11below, unlessk is even. Mixed problems (i.e., with jumping boundary conditions)belongtothemotivationtostudyBVPsforoperators withoutthetransmissionproperty. Another(possiblyevenstronger)motivationisthesimilaritybetweenmixed and (specic) edge problems.Theorem 3.25. We havea B,d(R+; j0, j+), b B,e(R+; j, j0) ab B+,max+d,e](R+; j, j+),and (a, b) ab denes a bilinear continuous mapB,d(R+; j0, j+) B,e(R+; j, j0) B+,max+d,e](R+; j, j+)between the respective Frechet spaces.Proof. The result for the composition of upper left corners is contained in Theorem3.20. The proof for the remaining entries is straightforward and left to the reader.

Theorem 3.26. Leta B0,0(R+; j, j+), and dene the adjointaby(au, v)L2(R+)Cj+= (u, av)L2(R+)Cjfor allu L2(R+) Cj, v L2(R+) Cj+. Then we havea B0,0(R+; j+, j),anda adenes an(antilinear), continuous mapB0,0(R+; j, j+) B0,0(R+; j+, j).Proof. The result for the upper left corner follows from Proposition 3.12, togetherwith Corollary 3.14. The proof for the remaining entries is straightforward and leftto the reader. 20 B.-W. SchulzeDenition3.27. A symbola() Str(R) is called elliptic (of order) ifa() ,= 0forall R,andif a0(=a0=a+0 )doesnotvanish(cf.thenotationin(3.8)).Moreover, we call ana B,d(R+; j, j+) elliptic if the symbol a() Str(R) inthe upper left corner of(3.23) is elliptic.Theorem 3.28. Leta() Str(R) be elliptic, andg d(R+); thena := op+(a) +g : Hs(R+) Hs(R+) (3.29)is aFredholmoperatorforevery s >max, d 12, andp:=op+(a1)is aparametrix ofa.Proof. Because of the assumption ons the operatorop+(a1) : Hs(R+) Hs(R+)is continuous. From Corollary 3.19 and Theorem 3.20 we haveop+(a1)op+(a) +g = op+(aa1) +k = 1 +k (3.30)where k = h+op+(a1)g for an h 0(R+) and op+(a1)g d(R+). Thus sincek : Hs(R+) Hs(R+)is compact for s >d 12, cf. Remark3.17, the operator op+(a1) is aleftparametrix. Ina similar mannerwe obtainthatop+(a1)is aright parametrix.In fact, we have to computeop+(a) +gop+(a1) = op+(aa1) +k = 1 +kwhere k = h +gop+(a1), for ah 0(R+), and gop+(a1) max+d,0](R+).This can be applied to functions in Hs(R+) when s satises the conditionss > 12and s > max +d, 0 12. In the case max +d, 0 = 0 thelatter is the same as the rst condition whilefor max + d, 0 = + d 0theconditioniss > + d 12,i.e., s>d 12.Forsitfollows altogethers > max, d 12, and we can apply again Remark 3.17. Theorem3.29. Let a() Str(R) be elliptic, and g d(R+). Then V :=ker(op+(a) + g)isanite-dimensional subspaceof o(R+),andthereisanite-dimensional subspaceW o(R+) such thatim(op+(a) +g) +W= Hs(R+). (3.31)Thisistrueforall real s>max, d 12withthesamespacesVandW. Itfollowsthat ind(op+(a) +g) is independent ofs.Proof. Let us seta := op+(a) + gand assumeu Hs(R+), au = 0. Then fromtherelation(3.30)it followsthat (1 +k)u=0, i.e., u= ku, whichimpliesu o(R+), cf. theformula(3.21). Inotherwords, V =ker a ker(1 + k)isanite-dimensional subspace of o(R+),independentofs = max, d 12.In thecased = 0, 0 we can do the same for the formal adjointa, and we may setW= ker awhich is a nite-dimensional subspace of o(R+) independent ofs.Boundary Value Problems with the Transmission Property 21To ndWin general we seta := op+(a) + g which we check as an operatora : Hs(R+) Hs(R+) for s > max, d 12. Let us set lN:= op+(lN) for anyN Z. In particular, forN:= max, d we havea = alN= a0lN: Hs(R+) Hs(R+) (3.32)wherea0=op+(alN) + kforsomek 0(R+), i.e., a0 B,0(R+)for = N 0. Thena can be regarded as a chain of operatorsa : Hs(R+) HsN(R+) HsN(R+) = Hs(R+)wheretherstone,namely, lNisanisomorphismwheres N 12, andthesecond one a0 is elliptic of order . For the latter we apply the rst part of the proof,i.e., we nd a nite-dimensionalW o(R+) such that ima0 + W=Hs(R+).This entails ima +W= Hs(R+), sincea = a0lN. Proposition3.30. Leta = op+(a) + g B,d(R+)wherea() Str(R)isellipticof order. Moreover, let W o(R+)beanite-dimensional subspace, andk:Cj Wa linear map. Then_uc_Hs(R+)Cj, au +kc = 0 (3.33)foranys> max, d 12impliesu o(R+),andthespaceofall solutionsof(3.33) is a nite-dimensional subspace ofHs(R+) Cj, independent ofs.Proof. First observe that (a k) is a Fredholm operator(a k) :Hs(R+)Cj Hs(R+). (3.34)Then, analogously as in the proof of Theorem 3.29 we pass to the operator_p0_(a k) =_pa pk0 0_:Hs(R+)CjHs(R+)Cjforp := op+(a1). The compositionl := pkis a potential operator, and we havepa=1 + hforanoperator h d(R+). Thekernel of (3.34)iscontainedinthe kernel of (1 + h l). The kernel of (1 +h l) consists of allt(u c) such that(1 +h)u +lc = 0, i.e.,u = hu +lc o(R+). Proposition3.31. Let op+(a) + g B,d(R+)beellipticof order. Thenthereexists a 2 2 block matrix operatora =_op+(a) +g kb q_:Hs(R+)CjHs(R+)Cj+(3.35)22 B.-W. Schulzeforatraceoperatorb,apotential operatorkandaj+jmatrixq,suchthat(3.35) is an isomorphismfor all s > max, d 12, and we haveind(op+(a) +g) = ind op+(a) = j+j. (3.36)The operator(3.35) is an isomorphismif and only ifa :o(R+)Cjo(R+)Cj+(3.37)is an isomorphism.Proof. Applying Theorem 3.29 we nd a nite-dimensional subspaceW o(R+)such that (3.31) holds for all s > max, d 12. Choose any j N, j dimW,and a linear surjective mapk : Cj W. Then(op+(a) +g k) :Hs(R+)Cj Hs(R+)isobvisouslysurjectiveforall s. ByvirtueofProposition3.30itskernel V isasubspace oft(o(R+) Cj) of nite dimensionj+. Choosing an isomorphism(b q) : V Cj+it suces to extend b to a trace operator b : Hs(R+) Cj+(for simplicity denotedbythesameletter). Then, accordingtoLemma2.3weobtainanisomorphism(3.35). Theorem3.32. Let a B,d(R+; j, j+)begivenasin(3.35),lettheupperleftcornerbeellipticinthesenseof Denition3.27,andassumethat adenesanisomorphism(3.37). Thenwehave a1 B,(d)+(R+; j+, j)where+:=max, 0.Proof. By virtue of Theorem 3.28 the operator (3.29) is Fredholm where op+(a1)is a parametrix. According to Proposition 3.31 there is a 22 block matrix iso-morphism of the formp :=_op+(a1) hc r_:o(R+)Cg+o(R+)Cgfor a suitable trace operator c of type 0 and a potential operator h. Since op+(a1)is a parametrix of op+(a), cf. Theorem 3.28, we have ind op+(a1) = indop+(a)= jj+and from (3.36)ind op+(a1) = gg+ = jj+.InthecaseN:=g j Nwhichimpliesg+ j+=Nwepassfromatoa idCNwhichisagainanisomorphismwith(j, j+)replaced by(g, g+).Onthe other hand, whenN:=j g N wherej+ g+ =N, fromp we pass topidCNwhich is an isomorphism with (g, g+) replaced by (j, j+). In any case,Boundary Value Problems with the Transmission Property 23to nda1it suces to assume thatj = g,j+ = g+. Now the compositionapis of the formap =_1 +g11g12g21g22_:o(R+)Cj+o(R+)Cj+forag= (gij)i,j=1,2 B(d)+G(R+; j+, j+).ByvirtueofLemma3.33belowwehave__1 00 0_+g_1=_1 00 0_+l (3.38)for anl B(d)+G(R+; j+, j+). Then Theorem 3.25 gives usa1= p__1 00 0_+l_ B,(d)+(R+; j+, j). Lemma3.33. Letg BdG(R+; j, j), and assume that_1 00 0_+g :Hs(R+)CjHs(R+)Cj(3.39)is invertible for any s > d12. Then the inverse of (3.39) has the form_1 00 0_+lfor somel BdG(R+; j, j).Proof. Forconveniencewesetg=_G KT Q_. Then, inparticular, Qisajjmatrix. Since isomorphisms in a Hilbert space form an open set, a small pertur-bation ofQ allows us to pass to an invertible operator_1 +G KT R_ whereRisan invertible j j matrix. Assume that we have computed_1 +G KT R_1. Thenwe have_1 +G KT R_1_1 +G KT Q_ =_1 0D J_ which is again invertible; thisentails the invertibility ofJ. We obtain_1 +G KT Q_1=_1 0J1D J1__1 +G KT R_1. (3.40)Thus it remains tocharacterisethe secondfactor onthe right of (3.40). Theidentity_1 KR10 1__1 +G KT R__1 0R1T R1_=_1 +C 00 1_24 B.-W. SchulzeforC:=G KR1Tshows that the operator 1 + Cis invertible, and it followsthat_1 +G KT R_1=_1 0R1T R1__(1 +C)100 1__1 KR10 1_.This reduces the task to the computation of (1 +C)1.The operator C BdG(R+) can be written in the form C = C0 +

d1j=0 KjTjforaC0 B0G(R+), potential operatorsKjandtraceoperatorsTj:=rtDjt, cf.Remark3.23.SinceC0iscompactinSobolevspaces,wehaveind(1 + C0)=0.Because of the nature of V:= ker(1+C0) and W= coker(1+C0) (which are of thesame dimensionl) there is a trace operatorBof type 0 and a potential operatorD which induces isomorphismsB =t(B1, . . . , Bl) : V Cl, D = (D1, . . . , Dl) : Cl Wsuch that1 +C0 +DB : Hs(R+) Hs(R+)is an isomorphism. Note thatC1 := C0 +DB B0G(R+). We obtain1 +C = 1 +C1 d+l

k=1DkBk(3.41)forDl+j+1= Kj, Bl+j+1=Tjforj=0, . . . , d 1. Nowweemploythefactthat there is aC2 B0G(R+) such that 1 + C2= (1 + C1)1, say, as an operatorL2(R+) L2(R+), cf. Lemma 3.18. In order to characterise (1 + C)1we form(1 +C2)(1 +C) = 1 + (1 +C2)d+l

k=1DkBk = 1 +d+l

k=1MkBk = 1 +/BforMk= (1 + C2)Dk, / := (M1, . . . , Md+l), B :=t(B1, . . . , Bd+l). This reducesthe task to invert the operator 1 +Cto the inversion of1 +/B : Hs(R+) Hs(R+).With the operators / and B we can also associate the operator1 +B/: Cl+d Cl+d, 1 := idCl+d.Nowweverify that1 + /Bisinvertibleifandonlyif1 + B/ isinvertible.Infact, settingM:=_1 /0 1_, B :=_1 0B 1_, F :=_1 /B 1_,it follows thatMFB =_1 + /B 00 1_, BFM=_1 00 1 +B/_which gives us the desired equivalence. At the same time we see that(1 +/B)1= 1 /(1 +B/)1BBoundary Value Problems with the Transmission Property 25which is of the form 1 +G1for aG1 BdG(R+). Thus(1 +C)1= (1 +C1)(1 +G1) = 1 +C1 +G1 +C1G1whereC1 +G1 +C1G1 BdG(R+). Remark3.34. Theorem 3.32 easily extends to B,d(R+; (k, k; j, j+)) for arbitraryk, j, j+ N (cf. Remark 3.22). The technique for the proof which mainly employscompositions of some operators also shows that the inverse continuously dependson the given operatora.4. Pseudo-Dierential BoundaryValueProblemsWe develop basics on pseudo-dierential BVPs with the transmission property atthe boundary. Other material may be found in the authors joint monographs withRempel [15], with Kapanadze [12], or with Harutyunyan [10], and in the monographof Grubb [8]. The ideas here are related to the calculus on manifolds with edges.Letus rst consider operators inlocal coordinatesx = (y, t) Rn1 R+.TheoperatorconventionreferstotheembeddingofRn+intotheambientspace Rn.Therefore, we rst look at operatorsOpx(p)u(x) =__ei(xx

)p(x, )u(xt)dxtd. (4.1)HerepbelongstoH ormanderssymbol classes.LetS(URn)for RandU Rmopen denote the set of allp C(U Rn) such that[DxDp(x, )[ c)]]for all (x, ) KRn, KU, andall Nm, Nn, for constantsc=c(, , K) >0. Wewill freeleyemployvariousstandardpropertiessuchasasymptoticexpansions, etc., developedintextbooksonpseudo-dierentialoper-ators. Thesubspace Scl(URn)of classical symbolsisdenedbyasymptoticexpansionsp(x, )

j=0()p(j)(x, )wherep(j)(x, ) C(U(Rn 0)), p(j)(x, ) =jp(j)(x, )forall R+, andis anyexcisionfunction. If someassertionis validfor theclassical and the general case we also writeS(cl)(URn). Recall that the spacesS(cl)(URn) are Frechet in a natural way. It is then obvious thatS(cl)(Rn) (thespace ofx-independent elements) is closed inS(cl)(U Rn), and thatS(cl)(U Rn) = C(U, S(cl)(Rn)). (4.2)In order to illustrate some consequences of the presence of a boundary, heret=0,werephrase(4.1)inanisotropicform,bycarryingouttheactionrstintandtheniny.Itwill benotessentialthatyvariesin Rn1;weoftenassumey foranopenset Rn1. Moreover, forsimplicity, werstconsidera26 B.-W. Schulzet-independent symbol, i.e., p(y, , ) S(Rn1R). We form Opt(p)(y, ) :Hs(R) Hs(R) as an operator family parametrised by (y, ) Rn1andthenOpx(p) = Opy(Opt(p))where Opt(p)(y, ) is regarded as an operator-valued symbol in the variables andcovariables (y, ).In order to formulate the latter aspect in a more precise manner we x a groupR+of isomorphisms : Hs(R) Hs(R) by setting (u)(t) = 1/2u(t), R+. Then a simple computation shows the identity1)Opt(p)(y, )) = Opt(p)(y, ) (4.3)forp(y, , ) = p(y, ), ). (4.4)Usingthesymbolicestimatesforp, especially, [p(y, , )[ c, )forall(y, , ) K Rn,K, and constantsc(K) > 0, it follows that[p(y, ), )[ c)), (4.5)taking into account the relation ), ) = )).Lemma4.1. Under the above assumptions we have|1)DyDOpt(p)(y, ))|1(Hs(R),Hs+||(R)) c)]](4.6)forall (y, ) KRn1, K, andall , Nn1, andeverys R, forconstantsc = c(, , K, s) > 0.Proof. Let rst = = 0, and set a(y, ) := Opt(p)(y, ). Then the relation (4.3)together with the estimate (4.5) yields|1)a(y, ))u|2Hs(R) =_ )2(s)[p(y, ), ) u()[2d supR,yK)2[p(y, ), )[2_ )2s[ u()[2d c)2|u|2Hs(R).This implies (4.6) for = = 0. The assertion for arbitrary, follows inan analogous manner, usingDyDp(y, , ) S]](U Rn). Remark 4.2. Lemma4.1remainstrueinanalogousformundertheassumptionp(y, t, , ) S(RRn,)whenpisindependentof tfor [t[ >constforaconstant > 0 (and also under certain weaker assumptions with respect to [t[ ).Denition4.3. 1. ByagroupactiononaHilbertspace Hweunderstandastronglycontinuousgroup= R+of isomorphisms : HH,suchthat = forall , t R+(strongly continuousmeansthath C(R+, H) for everyh H).Boundary Value Problems with the Transmission Property 272. Let Hand HbeHilbertspaceswithgroupactions and , respectively.ThenS(Rq; H, H) for Rpopen, R, is dened to be the set ofalla(y, ) C( Rq, L(H, H)) such that| 1)DyDa(y, ))|1(H,H) c)]]forall (y, ) KRq, K, andall Np, Nq, forconstantsc = c(, , K) > 0.3. ThespaceScl(Rq; H, H)ofclassical elementsisthesetofall a(y, ) S(Rq; H, H) such that there are functionsa(j)(y, ) C((Rq0), L(H, H)), j N, witha(j)(y, )=j a(j)(y, )1forall R+, (y, ) (Rq 0), witha(y, ) N

j=0()a(j)(y, ) S(N+1)( Rq; H, H)for everyN N and any excision function().Example. 1. Forp(y, , ) S(Rn)anda(y, )=Opt(p)(y, )wehavea(y, ) Scl( Rn1; Hs(R), Hs(R)) for everys R.2. Forp(y, t, , ) S(RRn)undertheassumption ofRemark 4.2 wehavea(y, ) S( Rn1; Hs(R), Hs(R)) for everys R.Remark 4.4. Observethat inthelatter Examplewedidnot exhaust the fullinformation of(4.6) with respect tos. In fact, dierentiation ingives us bettersmoothnessintheimagespaces. ForourpurposesitsucestoxtheHilbertspaces H and H; in applications it will be clear anyway to what extent we can saymore when those spaces run over scales of spaces, parametrised bys.Parallel to the spaces of operator-valued symbols we have vector-valued ana-logues of Sobolev spaces.Denition4.5. LetHbe a Hilbert space with group action = R+. ThenJs(Rq, H) fors R is dened to be the completion of o(Rq, H) with respect tothe norm |)s1) u()|L2(Rq,H).Thespace Js(Rq, H)iscontainedin ot(Rq, H)= L(o(Rq), H). Foreveryopen Rqwe dene Jscomp(, H) to be the set of all u Js(Rq, H) with com-pact support and Jsloc(, H) Tt(, H) = L(C0(), H) byu Jscomp(, H)for every C0().Example. 1. LetH:=Hs(Rm),(u)(x) :=m/2u(x)for R+.Then foreverys R we haveJs(Rq, Hs(Rm)) = Hs(RqRm).2. LetH:=Hs(R+), (u)(t) =1/2u(t), R+. Then for everys R wehaveJs(Rq, Hs(R+)) = Hs(RqR+).28 B.-W. SchulzeRemark4.6. The notion of group actions also makes sense for Frechet spaces thatare written as projective limits of Hilbert spaces. An example is the Schwartz spaceo(Rm) = limjNx)jHj(Rm)with being dened as in the above example. Then there are natural extensionsof Denitions 4.3 and 4.5 as well as comp/loc spaces to the case of Frechet spaceswith group action (for more details cf. also [20], [22]).Theorem4.7. Let Hand HbeHilbert (Frechet )spaces withgroupactionanda(y, ) S(Rq; H, H).ThenOpy(a) :C0(, H) C(, H) extendsto acontinuous operatorOpy(a) : Jscomp(, H) Jsloc(, H) (4.7)foreverys R. If a(y, ) S(Rq Rq; H, H)isindependent of yfor [y[ const > 0, then we obtain a continuous operatorOpy(a) : Js(Rq, H) Js(Rq, H) (4.8)for everys R.Remark 4.8. Thecontinuityof (4.8) canbeprovedunder muchmoregeneralassumptions ona(y, ) than in Theorem 4.7, see, for instance, [20] or [28].Let us now turn to what we did at the beginning of this section.Forp(y, t, , ) S(Rn1R Rn) we haveOpt(p)(y, ) S(Rn1Rn1; Hs(R), Hs(R))whenpsatisestheassumptionof Remark4.2. Forourpurposesitsucestoassume thatp is a classical symbol oforder Z,and independentof (y, t) for[y, t[ const for some constant> 0.Ina theoryof elliptic boundaryvalue problems that relies onstandardSobolev spacesHs(Rn+) = Hs(Rn)[Rn+we should possess the continuity ofOp+(p) = r+Op(p)e+: Hs(Rn+) Hs(Rn+) (4.9)fors> 1/2, similarlyas inCorollary 3.9;heree+istheoperator ofextensionbyzerofrom Rn+to Rn, andr+therestrictionto Rn+(analogously wehavetheextension and restriction operators e and r, respectively). It turns out that thecontinuity of (4.9) requires certain very restrictive assumptions on the symbolp.For instance, forp(x, ) = ()[[ where is some excision function, the operator(4.9) will not be continuous for alls > 12.According to Theorem 4.7 for the continuity in Sobolev spaces it suces toknow thatop+(p)(y, ) S(Rn1Rn1; Hs(R+), Hs(R+)) (4.10)fors > 12, i.e.,|1)DyDop+(p)(y, ))|1(Hs(R+),Hs(R+)) c)]](4.11)Boundary Value Problems with the Transmission Property 29for all (y, ) Rn1Rn1, and all, , forc = c(, , K, s) > 0.Moreover, it is desirable to haveop+(p)(y, ) S(Rn1 Rn1; o(R+), o(R+)). (4.12)Inordertoillustratetheeectforthemomentweconsiderthecasethat pisindependent ofy andt. To obtain (4.11) we assume p(, ) := p(), ) S(Rq, Str(R)). (4.13)ThenotationS(Rq, E)foraFrechetspaceEwiththesemi-normsystem(k)kNmeans the set of alla() C(Rq, E) such thatk(Da()) c)]]for all Rq, Nq,k N, for constantsc = c(, k) > 0.Lemma4.9. LetEandFbeFrechetspaceswiththesemi-normsystems(j)jNand (j)jN, respectively, and letB : E Fbe a continuous operator. ThenTB : C(Rq, E) C(Rq, F) (4.14)denedbythecompositiona: RqEandB: E FinducesacontinuousoperatorTB : S(Rq, E) S(Rq, F) (4.15)for every R.Proof. Withoutloss of generality we assumej+1() j()andj+1() j()for allj. Then continuity ofBmeans that for everyk N there is aj N suchthatk(Bu) cj(u) for allu E, for somec> 0. Analogously, the continuityof(4.15) means that for everyk N, Nq, there arej, N N such thatsupRq)+]]k(DTBa()) c supRq]]N)+]]j(Da()) (4.16)for some c>0. Since TBa()=(Ba)() with pointwise composition and D(Ba)()= B(Da)() it follows thatsupRq)+]]k(DTBa()) c supRq)+]]j(Da())which implies (4.16). Lemma4.10. Letp(y, , ) C(Rn1, Scl(Rn)) and p(y, , ) C(Rn1, S(Rn1, Str(R))).Then we have the relations(4.10) fors > 12, and(4.12).Proof. For (4.10) we have to verify the estimates (4.11). Let rst = = 0. Forsimplicity letp be independent ofy. An analogue of the relations (4.4) gives us1)op+(p)()) = op+( p)(). (4.17)30 B.-W. SchulzeThe operation op+() induces a continuous operatorop+() : Str(R) L(Hs(R+), Hs(R+))for every s > 12. That means, for every s there is a semi-norm j from the Frechettopology ofStr(R) such that|op+(a)|1(Hs(R+),Hs(R+)) cj(a)foreverya Str(R). Thus, forE=Str(R), F= L(Hs(R+), Hs(R+)), fromLemma 4.9 it follows thatsup)|op+( p)()|1(Hs(R+),Hs(R+)) c sup)j( p(, )) < ,i.e., using (4.17), that|1)op+(p)())|1(Hs(R+),Hs(R+)) c).In a similar manner we can proceed with the derivativesDp(, ) for every Nn1.The proof(4.12) is straightforward as well and left to the reader. Denition4.11. Asymbol p(y, t, , ) Scl(yRRn,)for Zissaidtohave the transmission property att = 0 if the homogeneous components p(j)ofp satisfy the conditionsDy,tD,p(j)(y, t, , ) (1)jp(j)(y, t, , ) = 0 (4.18)ontheset (y, t, , ) RRn: y , t =0, =0, R 0ofnon-vanishing conormal vectors over theboundary, for all, Nn, j N.LetStr( R Rn) denote the space of all symbols of that kind. Moreover, setStr( R Rn) := p[RRn : p Str( R Rn).Sincethetransmissionpropertyisalocalconditionneart=0itcaneas-ilybeextendedtosymbolsinanarbitraryopensetURnintersecting t=0. (It isclearthatit sucestoask(4.18)onlyforall =(0, . . . , n), =(1, . . . , n1, 0)).Operators withsymbols withthetransmission property inconnection withboundary value problems (and also transmission problems) have been studied bymany outhors, rst of all Boutet de Monvel [5], [4], Eskin [7], and later on Myshkis[14], Rempel and Schulze [15], Grubb [8], [9], and many others. One of the main mo-tivations was to nd a framework to express parametrices of elliptic boundary valueproblems for dierential operators and to prove an analogue of the AtiyahSingerindex theorem. In this connection it appeared not too perturbing that genericallysymbols (that are smooth up to the boundary) have not the transmission propertyat the boundary. We will return to more general symbols below.Therstimportantaspectisthatapseudo-dierential theory ofboundaryvalue problems concerns continuous operators (4.9) (and analogously on manifoldswithsmoothboundary).Anotheressentialpointistounderstandthebehaviourof such operators under compositions.Boundary Value Problems with the Transmission Property 31Proposition 4.12. For everyp(y, t, , ) Str( R+Rn) we have p(y, t, , ) := p(y, t, ), ) C( R+, S(Rn1, Str(R))).The simple proof is left to the reader.In the local analysis of BVPs it suces to assume that the involved symbolsare independent oft for larget.Proposition 4.13. For everyp(y, t, , ) Str( R+Rn) which is independentoft for larget we haveop+(p)(y, ) S( Rq; Hs(R+), Hs(R+)) (4.19)for everys > 12, andop+(p)(y, ) S( Rq; o(R+), o(R+)). (4.20)Thet-independent case is contained in Lemma 4.10. After that the proof ingeneral is straightforward.Theorem 4.7 together with (4.19) entails the continuity ofOp+(p) = Opy(op+(p)) : Hs[comp)( R+) Hs[loc)( R+); (4.21)here Hs[comp)/[loc)(R+) = Jscomp/loc(, Hs(R+)), cf. also Example 4 (ii). Let usnow give a motivation of the conditions (4.18) in Denition 4.11. First it is evidentthatwhenpisapolynomialin, thehomogeneouscomponentsp(j)oforder j,j = 0, . . . , , satisfy the relations (4.18). For instance, we have in this casep()(y, t, , ) = p()(y, t, , ) (4.22)for every R, not only for R+, and hence,p()(y, t, , ) = (1)p()(y, t, , ),even for all (y, t, , ).If p(x, ) isellipticof order , thentheLeibnizinversewhichbelongstoScl( R Rn) satises those conditions as well with respect to the order .Thebehaviourof operatorsundercompositionslocallyneartheboundarycanbereducedtothecompositionof operatorswithoperator-valuedsymbols,modulo smoothing operators. In general, ifH, H, and Hare Hilbert spaces withgroup actions = R+, = R+, and = R+, respectively, anda(y, ) S( Rq, H, H), a(y, ) S ( Rq; H, H),for simplicity, with compact support with respect toy, then we can formOpy(a)Opy( a) = Op(a# a)with the Leibniz producta# a(y, ) S+ ( Rq; H, H) that can be computedby an operator-valued analogue of the respective oscillatory integral expression in32 B.-W. SchulzeKumano-gos formalism.This entails an asymptotic expansiona# a(y, )

Nq1!(a(y, ))Dy a(y, ),:= 1/y11. . . q/yqq.If we apply this to the casea(y, ) = op+(p)(y, ), a(y, ) = op+( p)(y, )for symbolsp(x, ) Str(RRn), p(x, ) S tr(RRn) (say, under thesimplifyingconditionofcompactsupportin(y, t)), thenwehavetounderstandthe compositions( op+(p)(y, ))Dyop+( p)(y, ) = op+(p)(y, )op+(Dy p)(y, ).Since, Z are arbitrary, andp S]]tr,Dy p S tr, we may consider, forinstance, the case = 0. From the information of Section 3 we know thatop+(p)(y, )op+( p)(y, ) = op+(p#t p)(y, ) +g(y, )where p#t p is the Leibniz product between p and p with respect to thet-variable,andg(y, ) is a family of operators in 0(R+).More precisely, the operator families g(y, ) are Green symbols in the follow-ing sense.Denition4.14. 1. An operator-valued symbol g(y, ) belongs to 1,0G(Rn1)ifg(y, ), g(y, ) Scl( Rn1; L2(R+), o(R+)).Hereg(y, ) is the (y, )-wiseL2(R+)-adjoint. Elements of 1,0G( Rn1)are called Green symbols of type 0.2. An operator familyg(y, ) belongs to 1,dG(Rn1), then called a Greensymbol of typed N, ifg(y, ) =d

j=0gj(y, )jtforgj(y, ) 1j,0G( Rn1),j = 0, . . . , d.Similarly as (3.23) we also dene 2 2 block matricesa(y, ) :=_op+(p)(y, ) +g11(y, ) g12(y, )g21(y, ) g22(y, )_:Hs(R+)CjHs(R+)Cj+(4.23)for arbitrary p(y, t, , ) Str(RRn,) (independent of t for [t[ > const), andg11(y, ) 1,dG( Rq),s > d 12, while (say, for the casej = j+ = 1)g12(y, ) Scl( Rn1; C, o(R+))Boundary Value Problems with the Transmission Property 33(with C being endowed with the trivial group action),g21(y, )u(t) =d

l=0g21,l(y, )ltu(t)forg21,l(y, ) Slcl(Rn1; C, o(R+)) with the (y, )-wise adjoint in the fol-lowing sense:(g21,l(y, )v, c)C = (v, g21,l(y, )c)L2(R+),for arbitrary v L2(R+),c C, andg22(y, ) Scl( Rn1).The denition for arbitrary j is analogous. We call g21(y, ) a trace symbolof orderd N andg12(y, ) a potential symbol.From the denition it follows altogether thata(y, ) S( Rn1; Hs(R+) Cj, Hs(R+) Cj+) (4.24)for alls > d 12. Forg(y, ) := (gij(y, ))i,j=1,2we haveg(y, ) Scl( Rn1; Hs(R+) Cj, o(R+) Cj+) (4.25)for s >d 12. Let 1,dG(Rn1; j, j+) denotethe set of all suchg(y, ).Moreover, let 1,d(Rn1; j, j+) denote the set of all symbolsa(y, ) of theform (4.23).Now letXbe aCmanifold with boundaryY . Dene B,d(X; j, j+) tobe the space of smoothing operators of type d. For simplicity let again j = j+ = 1(the general case is analogous).Based on the Riemannian metrics onXandY= Xwe identify the spacesC(XX), C(XY ),etc.,withcorresponding integral operators withsuchkernels, for instance, u_X c(x, xt)u(xt)dxtandv_Yk(x, yt)v(yt)dytforc(x, xt) C(XX) and k(x, yt) C(XY ), respectively. Let B,0(X; 1, 1)denote the space of all operators( = (Cij)i,j=1,2 :C0(X)C0(Y )C(X)C(Y )such that C11 has a kernel in C(XX), C12 a kernel in C(XY ), C21 a kernelinC(Y X) andC22a kernel inC(YY ). Moreover, by B,d(X; 1, 1) ford Nwe denotethespace ofall22block matrixoperators (whereC12andC22are as before butC11 =d

l=0C11,lDl, C21 =d

l=0C21,lDlforC11,landC21,lasinthecased = 0andarst-order dierentialoperatorDonXthat is close toYequal tot, the dierentiation in normal direction. In ananalogous manner we dene B,d(X; j, j+) for arbitrary j N.Let us x a collar neighbourhood Vof YinX, let (U)Ibe a locally niteopencovering of V ,and let:U Rn+becharts, I.Those inducecharts34 B.-W. Schulzet : UY Rn1on Y . For every a(y, ) 1,d(Rn1Rn1; j, j+) we havean operator Opy(a), and we form the pull-back diag(1, t1)Opy(a) which isa 22 block matrix operator over U. Let us x a system of functions C0(U)such that

I 1 near Y , set := [Y , moreover, choose C0(U) thatare equal to 1 on supp, sett = [Y , and form/ := diag(, t)diag(1, t1)Opy(a)diag(, t). (4.26)Moreover, choose functions, , C0(V ), that are equal to 1 close toY , suchthat = 1 on supp , and 1 on supp .Denition4.15. Let B,d(X; j, j+) for Z, d N; denote by / thespace ofall operators/ = (Aij)i,j=1,2 :C0(X)C(Y, Cj)C(X)C(Y, Cj+)of the form/ = diag(, 1)

I/diag( , 1) + diag((1 )A(1 ), 0) +( (4.27)for arbitrary operators / as in (4.26),A Lcl(intX), and ( B,d(X; j, j+).The denition applies in particular toX = Rn+with the variablesx = (y, t).In this case the shape of the operators is easier, since the sum on the right-handside of(4.27) can be replaced bydiag(, 1)Op(a)diag( , 1) (4.28)for ana(y, ) 1,d(Rn1Rn1; j, j+).Let us dene the principal symbolic structure(/) = ((/), (/))consisting of the interior and the boundary symbol (/) and (/), respectively.Theupperleftcorner A11of anoperator / B,d(X; j, j+)belongstoLcl(intX), and we simply dene(/) for (x, ) TX 0 as thehomogeneousprincipal symbol ofA of order in the standard sense (here we take into accountthat the symbols are smooth up to the boundary). What concerns the boundarysymbol we rst look at the situation of the half-space, cf. (4.28). In this case wedene(/)(y, ) := (a)(y, )for (y, ) TRn1 0 by(a)(y, ) := diag(op+(p()[t=0)(y, ), 0) +(g)(y, ) (4.29)wherep()(y, t, , ) is the homogeneous principal symbol ofp(y, t, , ), and(g)(y, ) is the homogeneous principal symbol of(4.25) as a classical operator-valued symbol. Together withop+(p()[t=0)(y, ) = op+(p()[t=0)(y, )1Boundary Value Problems with the Transmission Property 35for R+we obtain(/)(y, ) = diag(, 1)(/)(y, )diag(1, 1)for all R+.Theconstructionof theoperatorspaces B,d(X; j, j+) intermsof localrepresentations and subsequent pull-backs to themanifoldis possiblebecause ofnaturalinvariance propertiesundercoordinatechanges. Thesameistrueoftheprincipalsymbols,andthenweobtain,inparticular,also aninvariantly denedprincipal boundary symbol on a manifold with boundary, using the local descrip-tions (4.29). In other words, we have(/) C(TX0), (/) C(TY 0, L(Hs(R+)Cj, Hs(R+)Cj+)).(4.30)Inmanycontexts it is adequate toadmit operators betweensections ofsmooth complex vector bundlesE, FonXandJ, J+onY , respectively,/ :C0(X, E)C0(Y, J)C(X, F)C(Y, J). (4.31)The generalisation of the scalar case in the upper left corner to systems and thento the case of bundles, andE, Fof the other entries from trivial to general vectorbundlesJ, J+, is straightforward and left to the reader.If Mis a C manifold, by Vect(M) we denote the set of all smooth complexvectorbundlesoverM.If MisCwithboundary,thenweassumethateveryE Vect(M)istherestrictionof some E Vect(2M)toM. Thenthereisastandard denition of Sobolev spaces of distributional sections in E Vect(M) incomp/loc-version denoted byHscomp/loc(M, E), s R,whenMis an open manifold. IfMis compact, then we simply writeHs(M, E).Moreover, ifMisCwith boundary, we deneHs[comp/[loc)(intM, E) := Hscomp/loc(2M, E)intM.For the vector bundles E, F Vect(X), J, J+ Vect(Y ), in (4.31) we writev := (E, F; J, J+) and denote by B,d(X; v) the set of all operators (4.31).Fromthevector-valuedanalogueof (4.24)togetherwithTheorem4.7andcorresponding invariance properties we obtain that every / B,d(X; , v) inducescontinuous operators/ :Hs[comp)(intX, E)Hscomp(Y, J)Hs[loc)(intX, F)Hsloc(Y, J+)36 B.-W. Schulzefor all reals > d 12. In particular, ifXis compact, we have/ :Hs(int(X, E)Hs(Y, J)Hs(intX, F)Hs(Y, J+). (4.32)The pair of principal symbols = (, ) in this case means(/)(x, ) : XE XFwith the pull-back X of bundles under the canonical projection X : TX0 X,and(/)(y, ) : Y__Hs(R+) EtJ__ Y__Hs(R+) FtJ+__forEt:=E[Y , Ft:=F[YandthecanonicalprojectionY: TY 0 Y , fors > d 12. Alternatively we may consider(/)(y, ) : Y__o(R+) EtJ__ Y__o(R+) FtJ+__.Inthefollowingweoftendiscussoperatorsintheset-upof B,d(X; v), thoughthe reader who is mainly interested in the analytical details may consider the caseB,d(X; j, j+) which corresponds to the trivial bundlesE = X C,F= XCandJ = Y Cj, respectively.Remark4.16. Let X be compact. Then (/) = 0 implies that (4.32) is a compactoperator for everys > d 12.5. EllipticityofBoundaryValueProblemsWe now turn to the ellipticity of BVPs, more precisely, to the ShapiroLepatinskijellipticity. For elliptic operators there is also another kind of ellipticity of bound-aryconditions, knowninspecial cases, as conditions of AtiyahPatodiSingertype(APS-conditions), andingeneral asglobal projectionconditions. Whilenot every elliptic operator on aCmanifoldXwith boundary admits ShapiroLopatinskijellipticboundaryconditions,thereare always global projection con-ditions (whenXis compact), see [24]where bothconcepts are unied to an op-erator algebra, containing also Boutet de Monvels calculus. LetLtr(X; E, F) forE, F Vect(X)denotethesetofalloperatorsA = r+Ae+, A Ltr(2X; E, F),with Ltr(2X; E, F) being the space of classical pseudo-dierential operators on thedouble 2X, referring to E, F Vect(2X) withE = E[X,F= F[X, and with thetransmission property atY= X.For convenience we assume thatYis compact. The nature of elliptic bound-aryconditionsforanellipticoperator A + G B,d(X; E, F)(i.e., forellipticBoundary Value Problems with the Transmission Property 37A Ltr(X; E, F), Z,andaGreenoperatorGonXoforderandtyped)depends on the principal boundary symbol ofA,(A)(y, ) : YHs(R+) Et Y Hs(R+) Ft(5.1)for any xeds > max, d 12, but not so much on(G)(y, ) : YHs(R+) Et Y Hs(R+) Ft. (5.2)(5.2)isafamilyofcompactoperators thatcannotaectthepossibilitytoposeShapiroLopatinskij elliptic conditions for the operatorA.Denition 5.1. An operator / B,d(X; v) for v = (E, F; J, J+) is called ellipticif both the principal interior symbol(/) : XE XF, (5.3)X : TX 0 X, and the principal boundary symbol(/) : Y__Hs(R+) EtJ__ Y__Hs(R+) FtJ+__, (5.4)Y: TY 0 Y , dene isomorphisms.The second condition is just what we call ShapiroLopatinskij ellipticity. Thesmoothness s > max, d12 is xed, but the choice is unessential. The bijectivityof (/) holds if and only if its restriction to Schwartz functions in the upper leftcorner induces an isomorphism(/) : Y__o(R+) EtJ__ Y__o(R+) FtJ+__.If / B,d(X; v), v=(E, F; J, J+), isellipticinthesenseofDenition5.1,thenforthepairof inverses1(/)and1(/)wendanoperator /(1)B,(d)+(X; v1), v1= (F, E; J+, J),(d )+= maxd , 0,such that(/(1)) =1(/), (/(1)) =1(/). This is aconsequenceof amoregeneral operatorconventiontondoperatorsforaprescribedpairof principalsymbols (those can be described independently of the operator level, similarly asin the case of classical pseudo-dierential operators on an open manifold).In that case we have compact remainders( := 1 /(1)/ B1,max,d](X; (E, E; J, J)), (5.5)T := 1 //(1) B1,(d)+(X; (F, F; J+, J+)) (5.6)intherespective Sobolev spaces, since(() = 0, (T) = 0, (withreferring toorder 0). This shows the rst part of the following theorem.38 B.-W. SchulzeTheorem 5.2. Let / B,d(X; v) be elliptic;then/ :Hs(intX, E)Hs(Y, J)Hs(intX, F)Hs(Y, J+)(5.7)is a Fredholm operator for everys > max, d 12. Conversely,if (5.7) is Fred-holm for some s = s0> max, d12, then / is elliptic which entails the Fredholmproperty for all s > max, d 12.The second part of thelatter theorem requires arguments thatare omittedhere; details may be found in [15].Remark5.3. 1. If / B,d(X, v) is elliptic, then there is a parametrix /(1)B,(d)+(X; v1) whichmeans that theabove-mentionedremainders (and T belong to B,max,d]and B,(d)+, respectively.2. Let / B,d(X; v)beanoperatorsuchthat(5.7)isanisomorphismforsomes=s0>max, d 12. Then(5.7)isanisomorphismforall s>max, d 12, and for theinverse (which is a special parametrix of /)wehave /1 B,(d)+(X; v1).In fact, 1. can be obtained by improving /(1)of(5.5), (5.6) by applying aformal Neumann series argument. The property 2. is a consequence of the secondassertion of Theorem 5.2 (more details may be found in [17]).Example. LetA=, theLaplacianonX(withrespecttoaRiemannianmet-ric),moreover, letT0u :=u[Y .Then,forevery order-reducing isomorphismR L3/2cl(Y ) on the boundary we have_T_:=_1 00 R__T0_ B2,0(X; 1, 1; 0, 1)where 1 on the right-hand side stands for trivial bundles of bre dimension 1 overXandY , respectively, and we have_T_1 B2,0(X; 1, 1; 1, 0).Let us now discuss the nature of ShapiroLopatinskij ellipticity in more de-tail. A closer look at (5.1) reveals some interesting structures that are useful alsoto understand the dierence to ellipticity with global projection conditions, men-tioned at the beginning of this section.Consider anoperatorA B,0(X; E, F) (i.e., Aisofthetypeofanupperleftcornerinthe22blockmatrixset-up)satisfyingtheellipticitycondition(5.3). Then (5.1) is a family of Fredholm operators, where dimker (A)(y, ) anddimcoker (A)(y, ) are independent ofs > max, d 12. The same is true of(/)(y, )1,1 = (A)(y, ) +(G)(y, ). (5.8)Boundary Value Problems with the Transmission Property 39If (5.4)isafamilyofisomorphismsthentheroleoftheadditional entries((/)(y, ))i,j for i+j> 2 is to ll up (5.8) to a family of isomorphisms. However,many operatorsA B,0(X; E, F) thatare ellipticwithrespectto()do notadmit such families of block matrix isomorphisms.As notedbefore, an example is theCauchyRiemann operator ina smoothbounded domain in C which is elliptic of order 1. Other examples are Dirac oper-ators in even dimensions.In order to illustrate the phenomenon in general we recall a few notions fromK-theorywhichareconnectedwiththeindexoffamiliesofFredholmoperatorsparametrised by a compact topological space. In the present case we consider (5.8)for (y, ) SY , the unit cosphere bundle induced, byTY 0. Observe that byvirtue of the homogeneity((/)(y, ))1,1 = ((/)(y, )1,1)1,thevaluesof (/)(y, )1,1forall(y, ) TY 0aredeterminedbythosefor(y, ) SY . The compact topological spaces that we have in mind here areSYand Y , respectively (we discuss the case that X is a smooth manifold with compactboundaryY ).First, on a compact topological space M(connected, to simplify matters) wehave the set Vect(M) of (locally trivial) continuous complex vector bundles on M.InthecaseofaCmanifoldMwemay(andwill)takesmoothcomplexvec-tor bundles. Roughly speaking, continuous vector bundles overMare topologicalspaces which are disjoint unions E =

xM Ex of bres Ex that are vector spacesisomorphic to Ckfor some k N, and every point x0 Mhas a neighbourhood Usuch that E[U=

xU Ex is homeomorphic to UCkwhere this homorphism is -brewise an isomorphism and commutes with the canonical projections p : E M,ex x forex Ex, andq :UCkU, (x, v) x forv Ck. An example isE=MCkwhichisaso-called trivialvector bundle.Thusapartofthegen-eral denition requiresE[Uto be isomorphic to a trivial bundle which is just themeaning of locally trivial. We do not repeat here everything on vector bundlessuchaswhatisavectorbundleisomorphism =, butthenotiondirectlycomesfrom vector space isomorphisms, now parametrised byx M. More generally, wehave vector bundle morphisms which are brewise vector space homomorphisms.Moreover, we have a natural notion ofa direct sumE FforE, F Vect(M),brewise dened byEx Fx,x M.Similarlywecanformtensor products E FbytakingbrewisetensorproductsEx Fx,x M.TheK-groupK(M)overMisdenedasthesetofequivalenceclassesofpairs (E, F) Vect(M) Vect(M) where(E, F) (E, F)meansthatthereisaG Vect(M)suchthat E F G =F E G. Theequivalence class represented by (E, F) is denoted by [E] [F]. The structure of40 B.-W. SchulzeK(M) of a commutative group comes from the direct sum, namely,([E1] [F1]) + ([E2] [F2]) := [E1 E2] [F1 F2].Note that the tensor product between bundles turns K(M) even to a commutativering.Moreover, recallthatwhenf: M Nisacontinuousmap, wehavethebundle pull back E fE for E Vect(N) and a resulting fE Vect(M). Thisgives rise to a homomorphismf : K(N) K(M)dened byf([E] [F]) = [fE] [fF].An example isM := SY ,N= Y , with the canonical projection,1 : SY Y, 1(y, ) = y. (5.9)(Non-trivial) vector bundles may appear in connection with elliptic boundary valueproblems, or, more generally, with families of Fredholm operators. The latter onesgive rise to an equivalent denition ofK(M). The construction is closely relatedtothetasktondentries (/)(y, )i,jfor i, j =1, 2, i + j >2, foragiven(/)(y, )1,1thatcompletethelatterFredholmfamilytoafamilyof isomor-phisms, cf. (5.4). The general construction is as follows.By T(H, H)for HilbertspacesH, Hwe denote thesetofall Fredholm op-eratorsH H. Recall that T(H, H) is open in L(H, H), thespace of all linearcontinuous operators in the operator norm topology.Lemma 5.4. Leta C(M, T(H, H)), and assume thata(x) : H His surjectivefor everyx M. Then the familyof kernelskerM a := ker a(x) : x Mhas the structure of a(continuous) vector bundle overM.Proof. Let: H ker a(x1)betheorthogonalprojectiontoker a(x1)foranyxedx1 M. Then the family of continuous operators_a(x)(x1)_: H Hker a(x1)(5.10)is an isomorphism atx =x1and hence for all x in an open neighbourhoodUofx1. Therefore, by virtueof Lemma2.3 theoperator(x1)inducesisomorphisms(x1):ker a(x) ker a(x1)forall x U.Thisgivesusacontinuousfamilyofmapsker a(x) : x U U ker a(x1)whichis just thedesiredtrivialisationwhenweidentifyker a(x1) withCkfork = dimker a(x1). Boundary Value Problems with the Transmission Property 41Lemma5.5. Foreverya C(M, T(H, H))thereexistsaj Nandalinearoperator ker : Cj Hsuch that(a(x) k) :HCj H (5.11)is surjective for everyx M.Proof. For everyx1 Mthere exists a nite-dimensional subspaceW1 Handan isomorphismk1 : Cj1 W1forj1 = dimW1such that(a(x) k1) :HCj1 H (5.12)is surjective for x = x1. Then (5.12) is surjective for all x U1 for some open neigh-bourhood U1 of x1. Those neighbourhoods, parametrised by x1 M, form an opencovering of M. Since Mis compact, there are nitely many points x1, . . . , xN Msuch thatM= Nl=1Ulfor the respectiveUl.Choosing operatorsklanalogouslyas in (5.12) for every 1 l 1, with dimensionsjlrather thanj1, we obtain theassertion fork := (k1, . . . , kN), andj :=

Nl=1. Proposition 5.6. For every a C(M, T(H, H)) there exist vector bundles J, J+ Vect(M) and a continuous family of isomorphismsa :=_a(x) k(x)t(x) q(x)_:HJ,xHJ+,x, x M. (5.13)Proof. Choosek=k(x)asinLemma 5.5 forthetrivial bundleJ=MCj.Then applying Lemma 5.4 to the Fredholm family (5.11) we obtain thatkerM(a(x) k)is a nite-dimensional subbundle oft(H Cj), isomorphic toJ+for someJ+ Vect(M). Choosing a bundle isomorphismb0 : kerM(a(x) k) J+andsettingb :=b0 (x)forthefamilyoforthogonal projections(x):t(H Cj) ker(a(x) k) we obtain our result when we sett(x) := b(x)[M, q(x) := b(x)[Cj. Denition5.7. Fora C(M, T(H, H)) and any choice of(5.13) we setindMa := [J+] [J], (5.14)called theK-theoretic index of the Fredholm familya.42 B.-W. SchulzeItcanbeproved, cf. [10], thatindMaonlydependsonabutnotonthespecic choice of the family of isomorphisms (5.13).In particular, we obtain the same indMa when we replace (5.13) by isomor-phisms of the kind_a(x) h(x)b(x) d(x)_:H J,xL,xH J+,xL+,xfor someL, L+ Vect(M). Moreover, ifc C(M, L(H, H)) is a family of com-pact operators, thenindM(a +c) = indMa.Themap indM:C(M, T(H, H)) K(M)is surjective andinduces amap onlydependingonthehomotopyclassesof Fredholmfamilies. Thisgivesrisetoanequivalent denition ofK(M), cf. Janich [11].Let Xbecompact, E, FVect(X), andlet A B,d(X; (E, F; 0, 0))beellipticwithrespectto(cf. therstconditionof Denition5.1). Thentherestriction of (A)(y, ) toSY(denoted briey again by (A)(y, )) gives us afamily of Fredholm operators(A)(y, ) : Hs(R+) Et Hs(R+) Ft, (5.15)s > max, d 12, parametrised by (y, ) SY . Therefore, we obtain an indexelementindSY (A) K(SY )(whichisindependentof s). Thefollowingtheoremwasrstformulatedinthecaseofdierential operatorsinthepaper[2] byAtiyahandBott, andthenforpseudo-dierential operators with the transmission property at the boundary in [4]by Boutet de Monvel, cf. also [24]. An analogue for edge operators may be foundin [20], cf. also the authors joint papers [26], [27], with Seiler, and the referencesthere.Theorem 5.8. A-elliptic operatorA B,d(X; (E, F; 0, 0)) can be completed byadditionalentries to a (, )-elliptic2 2 block matrix operator/ B,d(X; (E, F; J, J+))for suitableJ, J+ Vect(Y ) withA in the upper left corner if and only ifindSY(A) 1K(Y ) (5.16)(cf. the notation(5.9)).Proof. The condition (5.16) is necessary, since the ShapiroLopatinskij ellipticitymeans that (5.4) is a family of isomorphisms and hence, by virtue of(5.14),indSY(A) = [1J+] [1J].Conversely, thecondition(5.16)allowsustoconstructablockmatrixfamilyofisomorphismsofthekind(5.13)with(A)intheupperleftcornerandvectorBoundary Value Problems with the Transmission Property 43bundles over SYthat are pull backs of vector bundles over Y . The construction forevery (y, ) SYis practically the same at that in the proof of Proposition 5.6.In addition we guarantee that the resulting block matrix operators locally belongto B,d(R+; k, k; j, j+) and smoothly depend on (y, ), for k = dimEy = dimFy,j=dimJ,y. Thecorrespondingoperatorfunctionsk(y, ), t(y, )andq(y, )can be extended from SYtoTY 0 by-homogeneity of order . This can bedone in terms of principal parts of symbols belonging to (4.25). Then applying anoperator convention which assigns to such principal symbols associated operatorsgives us the additional entries. Remark 5.9. The proof of Theorem5.8shows howwe cannd(inprincipleall)ShapiroLopatinskijellipticboundaryvalueproblems / B,d(X; v), v=(E, F; J, J+), foranygiven-ellipticoperatorA B,d(X; (E, F; 0, 0))pro-vided that the topological condition (5.16) is satised. It turns out that, from thepoint of view of the associated Fredholm indices, for every two such /1, /2withthe same upper left corner we can construct an elliptic operator R on the boundarysuch thatind/1 ind/2 = indR.ThelatterrelationisknownastheAgranovichDynin formula(seealso[4] and[15]). The proof is close to what we did in (2.19) modied for general 22 matricesrather than column matrices, cf. [15, Section 3.2.1.3.].It may happen that(A)(y, ) is a family of isomorphisms (5.15), i.e., thatfor the ellipticity of A with respect to and no additional entries are necessary.For instance, consider the symbolr(, ) :=_(C))) i_for some xed(t) o(R) such that(0) = 1 and suppF1 R, for instance,():=c1_0eit(t)dtforsome C0(R)wherec := _0(t)dt ,= 0.Then,if C>0isasucientlylargeconstant,wehaver(, ) Str(Rn), andr(, ) is elliptic of order . This symbol can be smoothly connected with , )far fromt = 0 by formingr(t)(, ), )(1(t))for a real-valued C0(R)such that 1 in a neighbourhood oft = 0. Then, if we interprett R+as theinner normal of a collar neighbourhood ofYinXthere is obviously a-ellipticoperator R on Xwith such amplitude functions near the boundary, and (R)has thedesired property, indeed. Asimilar construction is possible inthe vectorbundle set-up, which gives us such an operatorR,E B,d(X; (E, E; 0, 0)),R,E : Hs(intX, E) Hs(intX, E). (5.17)In addition the operator convention can be chosen in such a way that (5.17) is anisomorphism for every s > max, d 12. More details on such constructions maybefoundinthepaper[9] ofGrubb, seealsotheauthorsjointmonograph withHarutyuyan [10, Section 4.1].44 B.-W. SchulzeUsing thefactthatthere are also order-reducing operators ofany order onthe boundary (which is a compact C manifold, cf. the formulas (2.5), (2.12)) wecan compose any (, )-elliptic operator / B,d(X; (E, F; J, J+)) by diago-nal matrices of order reductions to a (, )-elliptic operator /0 B0,0(X; (E, F;J, J+)). For many purposes it is convenient to deal with operators of order andtype zero, and we will assume that for a while, in order to illustrate other inter-esting aspects of elliptic pseudo-dierential boundary value problems.Let us set := SX[Y N(5.18)withSX[Ydenoting the restriction of the unit cosphere bundle to the boundary,andN = (y, 0, 0, ) TX[Y: 1 1which refers to the splitting of variables x = (y, t) near the boundary. The intervalbundleNistrivialanditsbresNy= (y, 0, 0, ) : 1 1connectthesouthpoles(= 1)withthenorthpoles(= +1)of SX[y, y Y .Inotherwords, is a kind of cage with barsNy, called the conormal cage. Letc : Ydenote the canonical projection.Remark5.10. Let A B0,0(X; (E, F; 0, 0)) be -elliptic; then (A)X[Yextendsto an isomorphismt(A) : cEt cFt. (5.19)Infact, (A)(y, 0, , ) : EyFyis afamilyof isomorphisms for all(y, 0, , ) SX[Y. By virtue of the transmission property we have(A)(y, 0, 0 1) = (A)(y, 0, 0, +1). (5.20)Theprincipal symbol (A)(x, )isaltogether (positively)homogeneous oforder zero in ,= 0; in particular, we have(A)(y, 0, 0, ) =(A)(y, 0, 0, 1) for all< 0,(A)(y, 0, 0, ) =(A)(y, 0, 0, +1) for all> 0.Nowtherelation(5.20)showsthat(A)(y, 0, 0, )doesnotdependon ,=0,and hence it extends toNywhen we denett(A)(y) := (A)(y, 0, 0, 0) := (A)(y, 0, 0, 1).We obtain an isomorphismtt(A) : Et Ft. (5.21)Let us now return to operators on the half-axisop+((A)[t=0)(y, ) : L2(R+) Ety L2(R+) Ftyparametrised by(y, ) SY .By virtueof (5.21) wemay replaceFtbyEt.Asusualweinterpret yYL2(R+) EtyasL2(R+) EtwhichisaHilbertspaceBoundary Value Problems with the Transmission Property 45bundle over Y(by Kuipers theorem it is trivial). Set a(y, , ) := (A)(y, 0, , )which is a family of isomorphismsa(y, , ) : Ety Ety,(y, , ) SX[Y . By virtue of the homogeneity we havea(y, , ) =a(y, , )for all R+, in particular,a(y,[[,[[) = a(y, , )for all ,= 0. Thus (5.20) gives uslima(y, , ) = a(y, 0, 1) = a(y, 0, +1) = lim+a(y, , ).This ts to the picture of symbols with the transmission property indescribedin Section 3. In other words, we havea(y, , ) S0tr(R) Iso(Ey, Ey)for every xed (y, ) SY(here Iso(, ) means the space of isomorphisms betweenthe vector spaces in parenthesis). The operatorsop+(a)(y, ) : L2(R+) Ey L2(R+) EyareFredholmandtheirpointwiseindexisequal tothewindingnumberof thecurveL(a) := det a(y, , ) : R C.This is a useful information for the construction of extra trace and potentialconditions in an elliptic BVP. It would be optimal to know the dimensions of kerneland cokernel; of course, those are not necessarily constant iny.6. TheAnti-TransmissionPropertyIn this section we return to scalar symbols (for simplicity). Recall that the trans-mission property of a symbol a() Scl(R) means the condition (3.11). In general,the curveL(a) = a() C : R (6.1)is not closed. Leta(y, t, , ) S0cl(R+Rn,) be an elliptic symbol,a(0)itshomogeneous principal part, anda() := a(0