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Lifting Theorems for Tuples of 3-Isometric and 3-Symmetric Operators with Applications

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Title:
Lifting Theorems for Tuples of 3-Isometric and 3-Symmetric Operators with Applications
Creator:
Russo, Benjamin Peter
Place of Publication:
[Gainesville, Fla.]
Florida
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University of Florida
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english
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1 online resource (109 p.)

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Mathematics
Committee Chair:
MCCULLOUGH,SCOTT A
Committee Co-Chair:
BROOKS,JAMES K
Committee Members:
ROBINSON,PAUL L
JURY,MICHAEL THOMAS
FRY,JAMES N
Graduation Date:
4/30/2016

Subjects

Subjects / Keywords:
Algebra ( jstor )
Commuting ( jstor )
Eigenvalues ( jstor )
Hilbert spaces ( jstor )
Linear transformations ( jstor )
Mathematical theorems ( jstor )
Mathematics ( jstor )
Matrices ( jstor )
Pencils ( jstor )
Polynomials ( jstor )
3-isometry -- 3-symmetric -- dilation -- disconjugacy -- extension -- lifting -- multi-variable -- operators -- tuple
Mathematics -- Dissertations, Academic -- UF
Genre:
bibliography ( marcgt )
theses ( marcgt )
government publication (state, provincial, terriorial, dependent) ( marcgt )
born-digital ( sobekcm )
Electronic Thesis or Dissertation
Mathematics thesis, Ph.D.

Notes

Abstract:
An operator T is called a 3-isometry if there exists a B_1(T^*,T) and B_2(T^*,T) such that Q_T(n)=T^{*n}T^n=1+nB_1(T^*,T)+n^2 B_2(T^*,T) for all natural numbers n. A related class of operators, called 3-symmetric operators, have a similar definition. These operators have a connections with Sturm-Liouville theory and are natural generalizations of isometries and self-adjoint operators. We call an operator J a Jordan operator of order 2 if J=A+N, where A is either unitary or self-adjoint, N is nilpotent order 2, and A and N commute. As shown in the work of Agler, Ball and Helton, and joint work with McCullough, 3-symmetric and 3-isometric operators can be modeled as Sub-Jordan operators. We develop the extension of these theorems to the multi-variable case in relation to a conjecture of Ball and Helton. More specifically, we cover connections between the lifting theorems via spectral theory and the necessity of an extra condition unique to the multi-variable case. We also develop applications of both the one variable and multi-variable lifting theorem to disconjugacy for Sturm-Liouville operators and Schrodinger operators. ( en )
General Note:
In the series University of Florida Digital Collections.
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Includes vita.
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Includes bibliographical references.
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Description based on online resource; title from PDF title page.
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This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis:
Thesis (Ph.D.)--University of Florida, 2016.
Local:
Adviser: MCCULLOUGH,SCOTT A.
Local:
Co-adviser: BROOKS,JAMES K.
Statement of Responsibility:
by Benjamin Peter Russo.

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UFRGP
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Copyright Russo, Benjamin Peter. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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LIFTINGTHEOREMSFORTUPLESOF3-ISOMETRICAND3-SYMMETRICOPERATORS WITHAPPLICATIONS By BENJAMINPETERRUSSO ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2016 1

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c 2016BenjaminPeterRusso 2

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Tomylovedonesandsupporters 3

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ACKNOWLEDGMENTS Foremost,Iwanttothankmyadvisor,Dr.ScottMcCullough.Iamcontinuallyamazedby hisinsightasamathematicianandbyhisadeptnessasamentor.Iwouldalsoliketothankmy committeemembers:Dr.JamesBrooks,Dr.JamesFry,Dr.MichaelJury,andDr.Robinson. Moreover,IamthankfultothesupportprovidedtomebytheMathematicsDepartmentat UF;boththeprofessorsandthestudentshaveshapedmeasamathematician.Thankyou tomyparents,RichardandAnhRusso,andtomyfamily.IhopeImakeyouproud.Finally, Iwanttothankmysignicantother,ElizabethWiggins.Iwouldnotbeherewithoutyour encouragement. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS...................................................................................4 ABSTRACT..................................................................................................7 CHAPTER 1INTRODUCTION....................................................................................8 1.1SpectralTheory................................................................................9 1.1.1SpectralTheoryforaSingleOperator............................................9 1.1.2SpectralTheoryforTuplesofOperators.........................................10 1.2CompletelyPositiveMapsandDilationTheorems......................................13 1.3Lifting3-Isometricand3-SymmetricOperators..........................................21 2MULTI-VARIABLELIFTINGTHEOREMS......................................................27 2.1ConjectureofBallandHelton..............................................................27 2.2ExtensionsofTheorems......................................................................30 3SPECTRALCONSIDERATIONSAND3-SYMMETRICLIFTING..........................45 3.1SpectralConsiderations.......................................................................45 3.23-SymmetricOperatorsTuples..............................................................53 4ACOUNTEREXAMPLETOFACTORIZATION..............................................60 4.1ConstructingThreeIsometries..............................................................60 4.2APositivebutnotCompletelyPositiveMap.............................................66 4.3StrengtheningtheCounter-Example.......................................................72 5APPLICATIONSTODISCONJUGACYINONEVARIABLE................................76 5.1SobolevSpaces.................................................................................77 5.23-Isometriesand3-SymmetricOperators................................................78 5.2.1A3-SymmetricOperatoranditsAssociated3-IsometryonaFormDomain78 5.2.2A3-IsometryonaFormDomain..................................................80 5.3FormulasfortheLifts.........................................................................81 5.4ConnectionswithSturm-LiouvilleTheory.................................................84 5.5Disconjugacy..................................................................................87 6LIFTINGFOR3-ISOMETRIESONSOBOLEVSPACES.....................................92 6.1DierentialPencils............................................................................92 6.2Liftingfor3-Isometricand3-SymmetrictupleswithDierentialPencils...........93 5

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7APPLICATIONSTODISCONJUGACYINSEVERALVARIABLES........................101 7.1DirichletBoundaryConditions..............................................................101 7.2PolynomialApproximations..................................................................104 7.3UsingtheLifts.................................................................................105 REFERENCES...............................................................................................107 BIOGRAPHICALSKETCH................................................................................109 6

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy LIFTINGTHEOREMSFORTUPLESOF3-ISOMETRICAND3-SYMMETRICOPERATORS WITHAPPLICATIONS By BenjaminPeterRusso May2016 Chair:ScottMcCullough Major:Mathematics Anoperator T iscalleda 3-isometry ifthereexistsa B 1 T , T and B 2 T , T such that Q T n = T n T n =1+ nB 1 T , T + n 2 B 2 T , T forallnaturalnumbers n .Arelatedclassofoperators,called 3-symmetric operators,have asimilardenition.TheseoperatorshaveaconnectionswithSturm-Liouvilletheoryand arenaturalgeneralizationsofisometriesandself-adjointoperators.Wecallanoperator J a Jordan operatoroforder 2 if J = A + N ,where A iseitherunitaryorself-adjoint, N is nilpotentorder 2 ,and A and N commute.AsshownintheworkofAgler,BallandHelton, andjointworkwithMcCullough,3-symmetricand3-isometricoperatorscanbemodeledas Sub-Jordanoperators.Wedeveloptheextensionofthesetheoremstothemulti-variablecase inrelationtoaconjectureofBallandHelton.Morespecically,wecoverconnectionsbetween theliftingtheoremsviaspectraltheoryandthenecessityofanextraconditionuniquetothe multi-variablecase.Wealsodevelopapplicationsofboththeonevariableandmulti-variable liftingtheoremtodisconjugacyforSturm-LiouvilleoperatorsandSchrodingeroperators. 7

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CHAPTER1 INTRODUCTION OperatortheoryisthestudyoflinearmapsfromoneBanachspacetoanother.In particular,itincludesthestudyofmatrices,whicharelinearmapsbetweennitedimensional vectorspaces.However,theemphasisinoperatortheoryisthetopologicalpropertiesofthe linearmap.Innitedimensions,alllinearmapsarecontinuous.Awellknownexamplewhich separatesthestudyofinniteandnitedimensionaloperatorsistheshiftoperator S . Example1. Let L 2 [ )]TJ/F24 11.9552 Tf 9.299 0 Td [( , ] denotethesetofFourierseries f P 1 n = f n e int withsquare summablecoecientswiththenorm k f k 2 = P 1 n = j f n j 2 .Theoperator U ofmultiplication by e it on L 2 whichsends e int to e i n +1 t iscontinuousandinfactunitary U U = I = UU . Thesubspace H 2 of L 2 [ )]TJ/F24 11.9552 Tf 9.298 0 Td [( , ] consistingofthose f forwhich f n =0 for n < 0 isinvariant for U . Let S : H 2 ! H 2 denotetherestrictionof U to H 2 .Since U iscontinuousandan isometry,sois S .Ontheotherhand, S isanexampleoftheinnitedimensionalphenomenaof anisometrywhichisnotunitary,i.e. S S = I ,but SS 6 = I ,incontrastswiththefactthatin nitedimensions,allisometriesareunitary. Oneviewisthatoperatortheorybeginswithdierentialandintegralequations.Liebniz wasthersttoviewdierentiationasalinearoperator.Everylineardierentialequation canbecastasalinearoperatorfromandtoappropriatespaces.Sturm-Liouvilleequations providearichclassofexampleswithphysicalapplicationsappearinginstringoscillations, orbitstability,andperhapsmostnotablyinquantummechanicsasSchrodingeroperators. SchrodingeroperatorsandmoregenerallySturm-Liouvilleoperatorshaveeigenfunctionsand eigenvaluessimilartowhatisseeninmatrixtheory.Theseeigenfunctionsaresolutionstothe Schrodingerequationscorrespondingtocertaindiscreteenergystatesgivenbytheeigenvalues. TheworkofvonNeumannonthemathematicalunderpinningsofquantummechanics continuestohaveaprofoundinuenceonoperatortheory.Aconsequenceofthespectral theorem,generalizedbyvonNeumann,isthemodelforunitaryandself-adjointoperatorsas multiplicationby e it and t respectivelyon L 2 spaces.Inanothervein,aby-productofthe 8

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vonNeumann-Wolddecompositionforisometriesisthatallisometriesaretherestrictionofa unitaryoperatortoaninvariantsubspace.Anoperator S : H ! H onaHilbertspace H isan isometry S S = I ifandonlyifthereisaHilbertspace K ,anisometry V : H ! K anda unitaryoperator U : K ! K U U = I = UU suchthat VS = UV .Woldwasinterested inisometriesfromtheviewpointofstationarystochasticprocesses.Operatortheoryremains avibrantsubjectbecauseofitsconnectionstootherareasofmathematicssuchasHarmonic Analysisandotherdisciplinesincludingphysicsandsystemsengineering. Throughouttheremainderofthechapterwewillilluminatesomeofthebackground materialneededforthetheoremswhichlieahead.Atthesametime,wewishtoplacesome contextonthecurrentworkwithrespecttosomeofthemathematicsdiscussedabove.Here weworkexclusivelyincomplexHilbertspace.Thus,byHilbertspacewemeancomplexHilbert spaceandbyoperatorwemeanboundedlinearoperatoronHilbertspace. 1.1SpectralTheory Inthissectionwewillcoversomeofthenecessarybackgroundinformationrelatedtothe spectraltheoryforbothasingleoperatorandtuplesofoperators. 1.1.1SpectralTheoryforaSingleOperator RecallsomebasicdenitionsandfactsasseeninArveson[7]. Denition1. Let A denoteaunitalBanachalgebra.Foreveryelement x 2A ,thespectrum of x isdenedastheset x = f 2 C : x )]TJ/F24 11.9552 Tf 11.955 0 Td [( I isnotinvertible g . Wenotethatthespectrumasasetiscompactandnon-empty.Moreover,forself-adjoint elementsintheBanachalgebra A thespectrumisasubsetoftherealline,andforunitary elementsthespectrumisasubsetoftheunitcircle.Asseenfromthedenition,thespectrum dependsontheunitalBanachalgebra A .Thefollowingresultisknownas spectralpermanence forC -algebras[7]. 9

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Theorem1. Let A beaunitalC -algebraandlet BA beaC -subalgebraof A that containstheunitof A , thenforevery x 2B wehave B x = A x Herewelistoneversionofthespectraltheoremfornormaloperatorsasfoundin[22]. Theorem2. If A isanormaloperatoronaseparablespace,thenthereexistsanitemeasure space A , and 2L 1 A , suchthat A isunitarilyequivalenttotheoperator M on L 2 A , . 1.1.2SpectralTheoryforTuplesofOperators Thenotionofjointspectrumforatupleofoperatorsismorecomplicatedthanthenotion ofspectraforasingleoperator.Here,wewillgothroughsomeofthebasicsofjointspectrum, butforthemostpartwewillrefertoapaperofCurto's[12].Wewillreviewtwonotionsof jointspectrum,thealgebraicjointspectrumandtheTaylorjointspectrum. Denition2. Let A beacommutativeunitalBanachalgebraandlet a = a 1 ,... a n 2A n . Wesaythat a isinvertiblewithrespectto A ifthereexists b 1 ,..., b n suchthat a b := P n i =1 a i b i =1 .The algebraicspectrum of a in A is A := f 2 C n : a )]TJ/F24 11.9552 Tf 11.956 0 Td [( isnotinvertiblein Ag . Whilethealgebraicspectrumseemstobethemostnaturalextensionofspectrumtoa commutingtuple,itisclearthisnotionofspectrumdependsonthechoiceofBanachalgebra A anditmaybediculttocompute.Thespatialnotionofjointspectrum,alsoknownas theTaylorspectrumofatuple,usesadierentapproach.Ingeneral,thealgebraicandTaylor spectrumwillbedierentforatupleofoperators.Moreover,if A isanycommutativeBanach algebracontaining a 1 ,... a n ,thenwehavethatfor a = a 1 ,..., a n Tay a A a 10

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where Tay denotestheTaylorspectrum.However,therearesomeinstancesinwhichthetwo notionsofspectrawillcoincide.Forinstance,if N isacommutingtupleofnormaloperators, then C N N = Tay N , where C N istheC -algebrageneratedby N .Ingeneral,whilethesetwospectrasharesome properties,theTaylorspectrumhastheaddedadvantageofbeingindependentofaBanach algebra.Weusetheconstructionfoundin[12]todenetheTaylorspectrum. Denition3. Let =[ e ]= n [ e ] betheexterioralgebraon n generators e 1 ,..., e n ,with identity e 0 =1 .Wehavethat isthealgebraofforms e 1 ,..., e n withcomplexcoecients, subjecttotheproperty e i e j + e j e i =0 for 1 i , j n . Notethat n formsa 2 n -dimensionalHilbert-spacewithorthonormalbasis f e i 1 ... e i k :1 i 1 < i 2 < ... < i k n g[f e 0 g Let E i : ! bethecreationoperatorsdenedby E i = e i for i =1,..., n .Clearly, E i E j + E j E i =0 for 1 i , j n . Let X beanyvectorspace.Foranycommuting n -tuple T = T 1 ,... T n ofboundedoperators on X dene D T : X n ! X n by D T = n X i =1 T i E i . TheTaylorspectrumof T is Tay T = f 2 C n : ran D T )]TJ/F25 7.9701 Tf 6.586 0 Td [( 6 = ker D T )]TJ/F25 7.9701 Tf 6.587 0 Td [( g . 11

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Considerthefollowingcalculation D 2 T v = n X i , j =1 T i T j v E i E j = n X i < j T i T j v E i E j + E j E i =0. Thusran D T ker D T . Example2. Take X = H aHilbertspaceand n =1. Theoperator D T takestheform D T = 0 B @ 00 T 0 1 C A withrespecttothedirectsumdecomposition X 1 = X [ e 0 ] X [ e 1 ]. Notethat elementsintherangeof D T )]TJ/F25 7.9701 Tf 6.587 0 Td [( taketheform , T )]TJ/F24 11.9552 Tf 10.685 0 Td [( x > forsome x 2 X .Hereby a , b > wemeanthe a b .Ifran D T )]TJ/F25 7.9701 Tf 6.587 0 Td [( = ker D T )]TJ/F25 7.9701 Tf 6.586 0 Td [( ,thenforall x , y > 2 ker D T )]TJ/F25 7.9701 Tf 6.586 0 Td [( wehave 0 B @ x y 1 C A = 0 B @ 0 T )]TJ/F24 11.9552 Tf 11.955 0 Td [( x 0 1 C A forsome x 0 2 X .Thus T )]TJ/F24 11.9552 Tf 12.447 0 Td [( isontoandhastrivialkernel.Henceforasingleoperatoron HilbertspacetheTaylorspectrumcoincideswiththeusualnotionofspectrum. Example3. For n =2 ,theoperator D T takesthefollowingform. D T = 0 B B B B B B B @ 0000 T 1 000 T 2 000 0 )]TJ/F47 11.9552 Tf 9.299 0 Td [(T 2 T 1 0 1 C C C C C C C A e 0 e 1 e 2 e 1 e 2 . TheTaylorspectrumhasthefollowingproperties.Let T beatupleofoperatorsona Hilbertspace H . iTheTaylorspectrumiscompactandnon-empty 12

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iiIf p : C n ! C m isapolynomial,then Tay p T = p Tay T iiiThereisaholomorphicfunctionalcalculusforfunctionanalyticinaneighborhoodof Tay T . ivTheTaylorspectrumhastheprojectionproperty,i.e.if m < n , then Tay T 1 ,... T m is theprojectionontotherst m coordinatesof Tay T 1 ,... T n . Weconcludewiththespectraltheoremfortuplesofnormaloperatorsfoundin[3]. Theorem3. Let N = N 1 ,..., N m beacommutingtupleofnormaloperatorsonaHilbert space H ,andassume C N iscyclic.Thenthereisacompactsubset Y of C m ,aBorel measure on Y ,andaunitaryoperator U : L 2 ! H sothat U N i U = M z i 1 i m 1.2CompletelyPositiveMapsandDilationTheorems Twoofthemaintheoremsofthisthesisareliftingtheoremfortuplesof3-isometricand 3-symmetricoperators.Arvesonprofoundlyreinterpretedliftinganddilationtheoremsinterms ofcompletelypositivemaps,wediscussthisinterpretationasfoundin[3]and[21]. Denition4. Let B beaunitalC algebraandlet A beasub-algebrawith 1 2A and suppose H and K areHilbertspaces.Aunitalhomomorphism : A!B H dilatestoa unitalhomomorphism : A! B K ifthereexistsanisometry V : H ! K suchthat a = V a V forall a 2A . Notethatonecanview H asasubspaceof K inwhichcase a = P H a j H .Moreover, choosing A = C [ x ] givesthenotionofapowerdilation. 13

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Denition5. Let T beaboundedoperatoronaHilbertspace H ,where H isaHilbert subspaceofaHilbertspace K .Aboundedoperator S on K isa powerdilation of T if P H S n j H = T n forall n 2 N where P H istheorthogonalprojectiononto H . Asimpler,yetmorepowerfulnotionisthatofextension. Denition6. Let T beaboundedoperatoronaHilbertspace H ,where H isaHilbert subspaceofaHilbertspace K .Aboundedoperator S on K isanextensionof T if S j H = T forall n 2 N where H isinvariantunder S . Itfollowsthat p T = p S j H forallpolynomials p .Further,if S isinvertibleand H is invariantfor S )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 ,then T isinvertibleand T n = S n j H forall n 2 Z .Inthiscasethenatural algebraisthealgebraofpolynomialsin x and x )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 ,theLaurentpolynomials.Wealsohavean analogousdenitionto4forextensionsofrepresentationsofunitalsubalgebrasof B H . Denition7. Let B beaunitalC algebraandlet A beasub-algebrawith 1 2A and suppose H and K areHilbertspaces.Aunitalhomomorphism : A!B H extendstoa unitalhomomorphism : A! B K ifthereexistsanisometry V : H ! K suchthat V a = a V forall a 2A . Notice,if extendsto asinDenition7,thentherangeof V isinvariantfor meaninginvariantfor a foreach a 2A . Similartothewayextensionscanbeformulatedintermsofrestrictionstoaninvariant subspace,wecanalsoreformulatedilationsintermsofcompressionstosemi-invariant subspaces. 14

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Denition8. Asubspace H issemi-invariantfor S if H = N M where M and N are invariantsubspacesfor S . ThefollowingtheoremisattributedtoSarason,theproofofwhichcanbefoundin[3]. Theorem4 Sarason . Anoperator S isadilationoftheoperator T ifandonlyif T isthe compressionof S toasemi-invariantsubspace H of S .Moregenerally, : A! B K dilates : A! B K as, a = V a V ifandonlyiftherangeof V issemi-invariantfor A . Weconsidertwowellknowndilationtheoremsfoundin[21]. Theorem5 Theunitarydilationofanisometry . Everyisometryisarestrictionofaunitary operatortoaninvariantsubspace. Proof. Let V beanisometryonaHilbertspace H andlet P = I H )]TJ/F47 11.9552 Tf 12.659 0 Td [(VV .Thus P isthe projectionontotheorthocomplementofran V .Dene U on H H tobe U = 0 B @ VP 0 V 1 C A . Wesee U isaunitaryoperatorandifweidentify H with H 0 ,then V = P H U j H . Anoperator T isa contraction if k T k 1 .Thisconditionhasaconvenientformulation intermsofpositivity.Anoperator A onaHilbertspace H is positivesemidenite ,denoted A 0 ifeitheroftheequivalentconditions, h Ah , h i 0 forall h 2 H or A = B B forsome operator B on H ,hold.Itisroutinetoverify T isacontractionifandonlyif I )]TJ/F47 11.9552 Tf 12.142 0 Td [(T T 0 if andonlyif I )]TJ/F47 11.9552 Tf 11.955 0 Td [(TT 0 . Theorem6 Sz.-Nagy'sdilationtheorem . Let T beacontractionoperatoronaHilbertspace H .ThenthereisaHilbertspace K containing H asasubspaceandaunitaryoperator U on K 15

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suchthat T n = P H U n j H for n 2 N ;i.e.,acontractionoperatorpowerdilatestoaunitaryoperator. Proof. Weprovethisresultbyshowingthateverycontractiondilatestoanisometryand applyingTheorem5.Givenacontraction T ,let D T denotetheuniquepositivesquarerootof I )]TJ/F47 11.9552 Tf 11.955 0 Td [(T T .Thus D T = )]TJ/F47 11.9552 Tf 11.955 0 Td [(T T 1 2 . Theoperator D T isknownasthedefectoperator.Weconstructtheisometricdilationof T as follows, V = 0 B B B B B B B B B B @ T 00 D T 00 0 I 0 . . . 00 I . . . . . . . . . . . . . . . 1 C C C C C C C C C C A . ItwasArveson'sprofoundinsightthatdilationscanbereformulatedintermsofcompletely positivemaps.ThiswillleadtoanotherproofoftheSz.Nagydilationtheorem,Theorem6. WecontinuetoreferencePaulsen[21]. Denition9. Let S beasubsetofaC -algebra A andset S = f a : a 2Sg . Wecall S selfadjoint when S = S .Inthecase A hasaunitand S isaself-adjointsubsetof A containingtheunitwecall S an operatorsystem. Recall,anelement a 2A ispositiveifandonlyif a = b b forsome b 2A .Let M n denotethe n n matriceswithcomplexentries.Forasubset S ofaC -algebra A ,note that M n S canbeidentiedwith M n S ,the n n matriceswithentriesfrom S .Bythe 16

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Gelfand-Niamark-SegalGNSconstruction,thereisaHilbertspace H suchthat A canbe identiedwithasubalgebraof B H .Hence, M n A isnaturallyasubalgebraof M n B H whichinturnisnaturallyidentiedwith B n 1 H ,theboundedlinearoperatorson n copiesof H .Hence,inthisway M n S isasubspaceofaC -algebra. Denition10. Let SA beanoperatorsystem, B aC -algebra,and : S!B alinear map.Wesayamap is positive ifitsendspositiveelementstopositiveelements.Wesaya map is completelypositive ifitsampliation 1 : M n S ! M n B ispositiveforall n 2 N . Example4. Let f V i g n i =1 beanitesetofmatricesin M n C .Themap : M n C ! M n C ; A = n X i =1 V i AV i iscompletelypositive. Infact,Example4typiescompletelypositivemapsonmatrixalgebras. Theorem7 Choi'stheorem . Alinearmap : M n ! M k iscompletelypositiveifandonlyif thereexistfewerthan nk linearmaps, V i : C k ! C n ,suchthat A = X i V i AV i forall A in M n . Completelypositivemapshaveapplicationsinsystemstheoryandquantuminformation theory,wherethe V i arecalledKrausoperators.Forinstance,quantumchannelsaretrace preservingcompletelypositivemaps. Theorem7canbegeneralizedtocompletelypositivemapsonC -algebras,aresultknown asStinespring'sTheorem. Theorem8 Stinespring . Let A beaunital C -algebraand H aHilbertspace.Alinearmap : A!B H iscompletelypositiveifandonlyifthereexistsaHilbertspace K ,aunital 17

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-homomorphism : A! B K ,andaboundedoperator V : H !K with k k = k V k 2 suchthat a = V a V . Stinespring'stheoremcanbeviewedasageneralizationoftheGNSrepresentationof states.InfactamodicationoftheproofofStinespring'sTheoremfoundin[21]yieldsthe standardGNSconstruction.Moreover,if isaunitalmap,then V isanisometry.Henceits notsurprising,wecanformulateanalternativeproofofSz.Nagy'sdilationtheorem,Theorem 6.ItreliesonthefactthatapositivemaponacommutativeC -algebrasiscompletely positive. ProofofTheorem6usingcompletepositivity. Let T beacontractionandlet S bethe operatorsystemdenedby S C T ; S = f p e i + q e i : p , q arepolynomials g . Themap p + q = p T + q T where p and q arepolynomialsispositiveandextends toapositivemapof C T into B H .Thismapisalsocompletelypositivesincethedomain is C T iscommutative[21]Thm3.11.Hence,bytheSteinspringtheorem,thereexistsa Hilbertspace K ,a -homomorphism : C T ! B K ,andanisometry V suchthat = V V . Thus, p T = V p U V forallpolynomials p .Since z 2 C T isunitaryand isarepresentationwehavethat z = U isaunitaryoperator.Hence U isaunitarypowerdilationof T . Arveson'sExtensionTheorem[21]isoffundamentalimportancetodilation/extension theory.Aswewillsoonsee,whencombinedwithSteinspring'stheoremitallowsustoreduce 18

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thequestionofdilation/extensiontothatoftheexistenceofacompletelypositivityofamap onacanonicaloperatorsystem. Theorem9. [Arveson'sExtensionTheorem]Let A beaC -algebra, SA anoperator system,and : S!B H acompletelypositivemapintoboundedoperatorsonaHilbert space H .Thereexistsacompletelypositivemap : A!B H extending . Withthistheoreminplace,wecannowdiscusstheroleofhereditarypolynomials[3]. Denition11. A hereditarypolynomial isapolynomialwithcomplexcoecientsintwo non-commutingvariables z and w oftheform h z , w = N X i , j =0 c i , j w j z i Denition12. Dene h x = h x , x foranyhereditarypolynomial h .Let A beaunitalC algebraand x 2A .Thehereditarymanifoldof x written H x is H x := f h x : h isahereditarypolynomial g Thefollowingwellknowntheoremconnectstheexistenceofanextensionwiththatofa completelypositivemapon H x . Theorem10. [3]Let A beaunitalC algebraandlet x 2A .Thefollowingconditionsonan operator T 2B H areequivalent: iThereexistsaHilbertspace K containing H ,andarepresentation : A!B K with = I ,suchthat H isinvariantfor x and T isequalto x j H . iiThereexistsacompletelypositivelinearmap : H x !B H with = I , x = T , and x x = T T . Wealsohaveasimilartheoremfordilations. Theorem11. [3]Let A beaunitalC algebraandlet x 2A .Thefollowingconditionsonan operator T 2B H areequivalent: 19

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iThereexistsaHilbertspace K containing H ,andarepresentation : A! B K with = I ,suchthat H issemi-invariantfor x and T isequalto P H x j H . iiThemap : p x + q x 7! p T + q T iscompletelypositive. WerecastTheorem10intheformnecessaryforthemainextensiontheoremsforthis thesis.Amulti-variableversionisproveninChapter2. Theorem12. [20]Suppose T and J areinvertibleoperatorsonHilbertspaces H and K respectively.ThereisaHilbertspace K , arepresentation : B K ! B K andan isometry V : H !K suchthat VT j = J j V forall j 2 Z ifandonlyifthemapping : H J ! H T determinedby J J = T T iscompletelypositive. Proof. Since : H J ! H T determinedby J J = T T iswelldenedand completelypositive,bythestandardapplicationoftheArvesonExtensionTheoremtogether withStinespringDilationTheoremsee[21],thereisaHilbertspace K , arepresentation : B K ! B K andanisometry V : H !K suchthat T T = J J = V J J V . Foreach , 2 Z , V J J V = T T = V J VV J V . Thus,as I )]TJ/F47 11.9552 Tf 11.956 0 Td [(VV isaprojection, V J I )]TJ/F47 11.9552 Tf 11.955 0 Td [(VV 2 J V =0 20

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andtherefore I )]TJ/F47 11.9552 Tf 12.19 0 Td [(VV J V =0 .Consequently, VV J V = J V . Itfollowsthat, foreach , VT = VV J V = J V . Theproofoftheconverseisroutine. 1.3Lifting3-Isometricand3-SymmetricOperators Wenowbegintodiscusstheoperatorswhicharethemainstudyofthisthesis. Denition13. Let H denoteacomplexseparableHilbertspace.Let T 2B H and m bea positiveinteger.Wedene yx )]TJ/F21 11.9552 Tf 11.955 0 Td [(1 m T 2B H by yx )]TJ/F21 11.9552 Tf 11.955 0 Td [(1 m T = m X k =0 )]TJ/F21 11.9552 Tf 9.299 0 Td [(1 m )]TJ/F48 7.9701 Tf 6.587 0 Td [(k m k T k T k . If yx )]TJ/F21 11.9552 Tf 11.955 0 Td [(1 m T =0 ,thenwesaythat T isan m -isometry or m -isometricoperator. Inasimilarfashionwehavethefollowing. Denition14. Dene y )]TJ/F47 11.9552 Tf 11.955 0 Td [(x m T by y )]TJ/F47 11.9552 Tf 11.955 0 Td [(x m T = m X k =0 )]TJ/F21 11.9552 Tf 9.299 0 Td [(1 m )]TJ/F48 7.9701 Tf 6.587 0 Td [(k m k T k T m )]TJ/F48 7.9701 Tf 6.587 0 Td [(k . Wesay T isa m -symmetricoperator if y )]TJ/F47 11.9552 Tf 11.955 0 Td [(x m T =0 . Thereareafewcommonexamples.Foremost,anyisometryisautomaticallya m -isometry andanyself-adjointoperatorisa m -symmetricoperatorforany m 1 .Anothercommon exampleistheDirichletshift. Example5. Let D = f f 2 Hol D : k f k 2 D = P n n +1 j ^ f n j 2 < 1g betheDirichletspace ofholomorphicfunctions.Theoperator M z denedby M z : D ! D ; M z f z = zf z isanexampleofa2-isometry. Ingeneral, m -isometrieshavebeenstudiedin-depthinaseriesofpapersbyAglerand Stankus[4,5,6].Sinceoneofthemainpointsofthisthesisistheapplicationofthetheoryof 21

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m -isometricand m -symmetricoperatorstothetheoryofdierentialequations,weshalldraw specialattentiontothecase m =3 whichconnectstosecondorderdierentialequations.Here wegivealternativecharacterizationsof 3 -isometricand 3 -symmetricoperatorsrespectively. Theorem13. [20]Let H denoteacomplexseparableHilbertspace.Anoperator T 2B H is a3-isometryifthereexistoperators B 1 T , T , B 2 T , T 2B H suchthat, T n T n = I + nB 1 T , T + n 2 B 2 T , T {1 forpositiveintegers n . Similarly, Theorem14. [15] T2B H isa3-symmetricoperatorifandonlyif exp )]TJ/F47 11.9552 Tf 9.298 0 Td [(is T exp is T = I + sB 1 T , T + s 2 B 2 T , T {2 forsome B 1 T , T , B 2 T , T 2B H andallrealnumbers s . Inparticular,if T isa3-symmetricoperator,then T =exp i T isa3-isometricoperator. Letusconsiderafewcanonicalexamplesoftheseoperators. Example6. Let H 1 [ )]TJ/F24 11.9552 Tf 9.299 0 Td [( , ] bethestandardHilbert-Sobolevspaceoffunctionstobe denedexplicitlylater.Thespace H 1 [ )]TJ/F24 11.9552 Tf 9.298 0 Td [( , ] comesequippedwiththefollowingnorm, k f k H 1 = Z )]TJ/F25 7.9701 Tf 6.586 0 Td [( j f 0 j 2 + j f j dx . Considerthefollowingcalculationfor t 2 R , e int f , e int f H 1 = n 2 Z )]TJ/F25 7.9701 Tf 6.586 0 Td [( j f j 2 dx + n Z )]TJ/F25 7.9701 Tf 6.587 0 Td [( )]TJ/F47 11.9552 Tf 5.48 -9.684 Td [(f f 0 )]TJ/F47 11.9552 Tf 11.955 0 Td [(f 0 f dx + Z )]TJ/F25 7.9701 Tf 6.586 0 Td [( j f 0 j 2 + j f j dx . Anapplicationofpolarizationshows M e it : H 1 ! H 1 ; M e it f t = e it f t M t : H 1 ! H 1 ; M t f t = tf t are3-isometricand3-symmetricoperatorsrespectively, 22

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Example7. Considerthefollowingmatriceson C 2 , A = 0 B @ 2 02 1 C A , U = 0 B @ e i 1 0 e i 1 C A where , 2 R .Easycalculationsshow A isa 3 -symmetricoperatorand U isa3-isometric operator. ThematricesinExample7arespecicexamplesofamoreimportantphenomena.We statetherelevantdenitionshere. Denition15. Anoperator J is s -Jordanoforder 2 if J = S + N ,where S and N commute, N isnilpotentordertwo,and S isself-adjoint. Denition16. Anoperator J is u -Jordanoforder 2 if J = U + N ,where U and N commmute, U isunitary,and N isnilpotentofordertwo. Onecancheckthat u -Jordanoperatorsare3-isometricand s -Jordanoperatorsare 3-symmetric.Moreover,if J is s -Jordan,then exp iJ is u -Jordan.Fortheremainderof thisthesiswewillreferto u -Jordanand s -JordanoperatorsassimplyJordanwhenitisclear fromcontextwhichtypeisbeingdiscussed.Theseoperatorsareprototypicalexamplesof 3-symmetricand3-isometricoperatorsinthesenseofthefollowingtheorems. Theorem15. T2B H isa3-symmetricoperatorifandonlyif T hasanextensiontoan operatoroftheform J = 0 B @ A 1 0 A 1 C A where A isself-adjointand 2 C . Given c > 0 ,let F c denotetheclassof3-isometricoperators T suchthat ^ Q T , s := I + sB 1 T , T + s 2 B 2 T , T )]TJ/F21 11.9552 Tf 15.798 8.088 Td [(1 c 2 B 2 T , T 0 forall s 2 R . Theorem16. [20][3-isometricliftingtheorem]Anoperator T onaHilbertspace H isin theclass F c ifandonlyifthereisaunitaryoperator U onaHilbertspace K andanisometry 23

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V : H ! K K suchthat VT = JV ,where J = 0 B @ UcU 0 U 1 C A . Moreover,if T isinvertible,then, VT )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 = J )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 V , thespectrumof T isasubsetoftheunit circle,and U canbechosensothat T = U = J . ApreliminaryversionofTheorem15wasinitiallyprovenbyHeltonin[14,15]using classicalconjugatepointtheoryandlaterimprovedfurtherbyBallandHeltonin[8].Agler establishedTheorem15inthegeneralcasein[1].Theorem16wasproveninjointworkwith McCulloughin[20].ByuseofafunctionalcalculusargumentTheorem15canberecovered fromTheorem16.Itshouldbenotedthattheapproachusedin[20]containsnouseof dierentialequations,insteadrelyingonthetheoryofcompletelypositivemaps.Insomecases theliftcanbeconstructedrathereasily. Example8. BorrowingfromanexampleofAglerandStankus,considertheSturm-Liouville operatorwithDirichletboundaryconditions, L = )]TJ/F47 11.9552 Tf 12.646 8.088 Td [(d 2 dt 2 + W {3 where W isrealandcontinuouson [0,1] .Itfollowsviaintegrationbyparts,onthesubset C 2 0,0 = f f 2 C 2 [0,1] j f = f =0 g , that h Lf , g i L 2 = Z 1 0 )]TJ/F47 11.9552 Tf 9.298 0 Td [(f 00 g + Wf gdt = Z 1 0 f 0 g 0 + Wf gdt )]TJ/F47 11.9552 Tf 11.956 0 Td [(f 0 g 1 0 = Z 1 0 f 0 g 0 + Wf gdt . Thepairing h f , g i = Z L f g . {4 24

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denesabilinearformon C 2 0,0 whichweassumeispositivedeniteie.,fornonzero f 2 C 2 0,0 , h f , f i > 0. {5 Iffurther,thereisa c > 0 suchthat h tf , tf i c 2 h f , f i whenever f 2 C 2 0,0 , {6 then,letting :[0,1] ! R denotethefunction t = t ,theoperator M , M f = t f t = tf t denselydenedon C 2 0,0 extendstoaboundedoperatorofnormatmost c ,stilldenoted M ,on thecompletionof C 2 0,0 intheinnerproductabove.Itisstraightforwardtoverifythat M isa 3-symmetricoperatorandhencehasaliftasinTheorem15. If W w > 0 forsome w 2 R and W iscontinuouson [0,1] aliftcanbeconstructed easily.Theinnerproduct h f , g i = Z 1 0 Lf gdt = h Lf , g i L 2 ispositivedenite.Let A 0 = M on L 2 [0,1]= K 0 , A 1 = M on L 2 Wdt = K 1 . Theinnerproducton K 1 is h f , g i K 1 = Z 1 0 f gVdt . Dene V : C 2 0,0 ! K 0 K 1 by V f = 0 B @ f 0 f 1 C A .Observe, h Vf , Vg i = * 0 B @ f 0 f 1 C A , 0 B @ g 0 g 1 C A + = h f 0 , g 0 i K 0 + h f , g i K 1 25

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= Z 1 0 f 0 g 0 + Wf gdt = h f , g i . So V isanisometryand 0 B @ tf 0 f 1 C A = V T f = 0 B @ A 0 I 0 A 1 1 C A 0 B @ f 0 f 1 C A = 0 B @ tf 0 + f tf 1 C A . Theremainderofthisthesiswillcomeinessentiallytwoparts.First,wewillextendthe singlevariableversionsoftheabovetheoremstothemulti-variablecase.Secondly,wewill exploreapplicationsofthesetheoremsbothsingleandmultivariabletodisconjugacyfor2nd orderequations. 26

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CHAPTER2 MULTI-VARIABLELIFTINGTHEOREMS ItseemsreasonabletotryandextendTheorems2316tothemulti-variablecase.This wasinitiallyattemptedfor3-symmetricoperatortuplesbyBallandHeltonin[8]anditisstill andopenproblem.Here,wegeneralizeTheorem16tothemulti-variablecase.However,aswe willsee,thereisafeatureofthemulti-variablecasewhichisnotpresentinthesinglevariable case.Later,inChapter3,bytheuseofafunctionalcalculusargument,weuncoveranecessary conditionwhichmustbemetinordertoachieveanextension. 2.1ConjectureofBallandHelton FollowingBallandHelton[8],let f J n = S n + N n g beanitecollectionofcommutingJordanoperatorssuchthatthenilpotentpartshavethe followingrelation, N i N j =0 forall i and j andthe S n areself-adjoint.Wewillcallthisa commutingJordanfamily .Let f T n g beanitecollectionofcommuting3-symmetricoperatorsthatsatisfythefollowing, Q s = e )]TJ/F48 7.9701 Tf 6.587 0 Td [(is k T k ... e )]TJ/F48 7.9701 Tf 6.586 0 Td [(is 1 T 1 e is 1 T 1 ... e is k T k = X j 1 ,... j k j 1 + + j k 2 B j 1 ,..., j k s j 1 1 ... s j k k . Wewillcallthisa commutingfamilyof3-symmetricoperators . Conjecture1. [8]Acollectionofoperators f T n g canbeextendedtoacommutingJordan family f J n g ifandonlyif f T n g isacommutingfamilyof3-symmetricoperators. BallandHeltonestablishedaspecialcaseofthisresultusingdisconjugacytheoryfor multivariableSturm-Liouvilleoperatorsfortuples T of3-isometricoperatorswithacyclic vectorandsatisfyingacertainsmoothnesshypothesis.Thesetwohypothesestogetherroughly saythat T ismultiplicationby t ontheformdomainofasucientlysmoothpositivedenite 27

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secondorderellepticoperator.Inthispaperweshowthatananalogofthisconjecturefor tuplesof3-isometricoperatorsisfalseandgiveacounter-example. Denition17. Acommuting2-tupleofoperators T = T 1 , T 2 ofoperatorsonthe Hilbertspace H isa2-tupleof3-isometriesifthereexistsboundedoperators B i , j on H for 0 i + j 2 and i , j 0 suchthat Q T n , m = T m 2 T n 1 T n 1 T m 2 = X 0 i + j 2 m i n j B i , j forall n , m 2 N .Wewillcall Q T theassociated quadraticpencil . Denition18. Fixpositiverealnumbers c , d .A2-tupleofcommuting3-isometries T = T 1 , T 2 isintheclass F c , d if ^ Q T , = Q T , )]TJ/F21 11.9552 Tf 15.798 8.088 Td [(1 c 2 B 2,0 )]TJ/F21 11.9552 Tf 16.141 8.088 Td [(1 d 2 B 0,2 0 forall , 2 R 2 . Thefollowingdenitionidentiesacanonicalclassofmodeloperatorsfortheclass F c , d . Denition19. Given c , d > 0 a2-tuple J = J 1 , J 2 isintheclass J c , d ontheHilbertspace K ifthereexistsaHilbertspace H suchthat K = H H H and,withrespecttothis decomposition, J 1 = 0 B B B B @ U 1 cU 1 0 0 U 1 0 00 U 1 1 C C C C A , J 2 = 0 B B B B @ U 2 0 dU 2 0 U 2 0 00 U 2 1 C C C C A . {1 forsomeunitaryoperators U 1 , U 2 thatcommute. Given J 2 J c , d ,compute,fornon-negativeintegers m , n , J n 1 = 0 B B B B @ U n 1 ncU n 1 0 0 U n 1 0 00 U n 1 1 C C C C A , J m 2 = 0 B B B B @ U m 2 0 mdU m 2 0 U m 2 0 00 U m 2 1 C C C C A 28

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and J m 2 J n 1 J n 1 J m 2 = 0 B B B B @ 1 ncmd ncn 2 c 2 +1 ncmd mdncmdm 2 d 2 +1 1 C C C C A . {2 Itfollowsthat J c , d F c , d . Theorem17. A3-isometric2-tuple T = T 1 , T 2 intheclass F c , d liftstoa2-tuple J = J 1 , J 2 intheclass J c , d ifandonlyifthethequadraticpencil ^ Q T , factorsinthe form, ^ Q T , = V 0 + V 1 + V 2 V 0 + V 1 + V 2 forsomeoperators V 0 , V 1 and V 2 in B H . Theorem17isprovedinSection2.2. Theproofoftherstpartofthefollowingremarkfor 3 -symmetricoperatorsappears in[8].Theproofoftheresultfor 3 -isometriesissimilar.Theproofofthesecondpartofthe remarkcanbefoundinChapter4. Remark1. If H isnitedimensionaland T 2 F c , d ,then T isapairofcommuting u -Jordan operatorsandthesucientconditionofTheorem17iseasilyveried.Otherwise H isinnite dimensionaland ^ Q T factorsintheformabovewith V j : H !H ,where H isanauxiliary Hilbertspace,ifandonlyifitfactorswith V j 2B H . Chapter4exhibits,byconstruction,a3-isometric2-tuple T intheclass F c , d forwhich ^ Q T doesnotfactorintheformgiveninTheorem17.Weshowthatthis T doesnotliftto a J 2 J cd andfurtherthat T doesnotlifttoanyJordanoperatorinanyclass J ~ c , ~ d forany ~ c and ~ d .Inthissensethe3-isometricanalogoftheconjectureofBall-Heltonisfalse.InChapter 3weshow,byafunctionalcalculusargument,thata2-tupleof3-symmetricoperatorslifttoa tupleofs-Jordanoperatorsifandonlyifitsassociatedoperatorpolynomialfactors. 29

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2.2ExtensionsofTheorems Webeginbyextendingtheresultsfoundin[20]to2-tuplesofinvertiblecommuting 3-isometriesin F c , d .Whiletheproofsonlydealwith2-tuples,theextensiontogeneral n -tuples isapparent. Againwenoteasubspace A of B H is unital ifitcontainstheidentityandis self-adjoint if T 2 A implies T 2 A .Foragiven N 2 N ,let M N C bethespaceof N N matriceswith complexentries,denoted M N whenthecontextisclear.Moreover,wedenotewith M N A the spaceof N N matriceswithentriesfrom A .Note M N A canbeidentiedwithasubspace oftheboundedoperatorson H N = H H N -copiesaswellaswith M N A . Denition20. Let n , N and M begivenpositiveintegers.An hereditarypolynomial p x , y intwovariablesofsize n andbi-degreeatmost M , N ininvertiblevariables x 1 , y 1 , x 2 , and y 2 suchthat y 1 and y 2 commuteand x 1 and x 2 commute,isapolynomialofthe form p x 1 , y 1 , x 2 , y 2 = M , N X , = )]TJ/F48 7.9701 Tf 6.587 0 Td [(M , = )]TJ/F48 7.9701 Tf 6.586 0 Td [(N p , , , y 2 y 1 x 1 x 2 . {3 Herethesumisniteand p , , , are n n matricesover C .Let P n bethecollectionof 2-variablehereditarypolynomialsofsize n andlet P = P n n denotethecollectionofall hereditarypolynomials. Givenapairofcommutinginvertibleoperators T 1 and T 2 ontheHilbertspace H ,let H T 1 , T 2 = span f T 2 T 1 T 1 T 2 : , , , 2 Z g . {4 Notethat H T 1 , T 2 isaunitalself-adjointsubspaceof B H .RecallthattheGNS constructionrealizesanabstract C -algebraasasubalgebraunitalandself-adjointof some B H . WenowpresentaversionoftheArvesonExtensionTheoremfor2-tuplesofoperators. Theorem18 ArvesonExtensionTheorem . Supposethat T 1 and T 2 arecommuting invertibleoperatorsonaHilbertspace H and S 1 and S 2 arecommutinginvertibleoperators 30

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onaHilbertspace K .ThereisaHilbertspace K ,arepresentation : B K !B K ,and anisometry V : H ! K suchthat VT 1 T 2 = J 1 J 2 V forall , 2 Z ifandonlyif themapping : H J 1 , J 2 !H T 1 , T 2 determinedby J 2 J 1 J 1 J 2 = T 2 T 1 T 1 T 2 is completelypositive. Proof. Suppose : H J 1 , J 2 !H T 1 , T 2 determinedby J 2 J 1 J 1 J 2 = T 2 T 1 T 1 T 2 iswelldenedandcompletelypositive.Inthiscase,byTheorem9andTheorem8,thereisa Hilbertspace K ,arepresentation : B H !B K andanisometry V : H !K suchthat V J 2 J 1 J 1 J 2 V = J 2 J 1 J 1 J 2 = T 2 T 1 T 1 T 2 . Since isanalgebraichomomorphismwhichpreservesinvolultions, V J 2 J 1 J 1 J 2 V = T 2 T 1 T 1 T 2 . {5 Foreach , 2 Z , V J 2 J 1 J 1 J 2 V = T 2 T 1 T 1 T 2 = V J 2 J 1 VV J 1 J 2 V byEquation2{5.Hence V J 2 J 1 J 1 J 2 V )]TJ/F47 11.9552 Tf 11.955 0 Td [(V J 2 J 1 VV J 1 J 2 V =0. Since I )]TJ/F47 11.9552 Tf 11.955 0 Td [(VV isaprojectionandhenceidempotent, V J 2 J 1 I )]TJ/F47 11.9552 Tf 11.955 0 Td [(VV 2 J 1 J 2 V =0. Therefore I )]TJ/F47 11.9552 Tf 11.955 0 Td [(VV J 1 J 2 V =0. Consequently J 1 J 2 V = VV J 1 J 2 V . 31

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AgainbyEquation2{5, VT 1 T 2 = J 1 J 2 V . Sincetheconverseisnotneededforanyofourtheorems,weomitthestraightforward proof. In[20]astrongvariantofTheorem18wasprovenusingAgler'ssymmetrizationtechnique. Denition21. Givenatwo-variablehereditarypolynomial p x 1 , x 2 , y 1 , y 2 asinEquation2{3, deneits symmetrization p s by p s = X p , , , y 2 y 1 x 1 x 2 . {6 Similarly,let H s T 1 , T 2 = span f T 2 T 1 T 1 T 2 : , 2 Z g . {7 InordertoproveastrongvariantofTheorem18wewillneedseverallemmas.Theyare presentedbelow. Denition22 PairwiseRotationallySymmetric . Apairofoperators S 1 and S 2 is pairwise rotationallysymmetric ifforall t 2 R 2 , t = t 1 , t 2 ,thereexistsaunitaryoperator U t such that e it 1 S 1 = U t S 1 U t and e it 2 S 2 = U t S 2 U t . Example9. Deneon L 2 T 2 theoperators Z 1 : L 2 T 2 !L 2 T 2 Z 1 f z 1 , z 2 = z 1 f z 1 , z 2 {8 and Z 2 : L 2 T 2 !L 2 T 2 Z 2 f z 1 , z 2 = z 2 f z 1 , z 2 . {9 Given t ,dene U t on L 2 T 2 by U t f 1 , 2 = f exp it 1 1 ,exp it 2 2 .Acalculationshows U t Z j =exp it j Z j U t .Hencethepair Z 1 , Z 2 ispairwiserotationallysymmetric. 32

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Lemma1. If S 1 and S 2 arepairwiserotationallysymmetricoperatorsand T 1 and T 2 are operatorsonacommonHilbertspace,then e T 1 = T 1 S 1 and e T 2 = T 2 S 2 arepairwise rotationallysymmetric. Proof. Since S 1 and S 2 arepairwiserotationallysymmetric,foreach t = t 1 , t 2 2 R 2 there existsaunitaryoperator U t suchthat e it 1 S 1 = U t S 1 U t and e it 2 S 2 = U t S 2 U t . Since e it 1 f T 1 = T 1 e it 1 S 1 and e it 2 f T 2 = T 2 e it 2 S 2 ,toseethat f T 1 and f T 2 arepairwise rotationallysymmetric,considertheunitaryoperators f U t = I U t . Givena 2 -tuple T ,let ^ T j = T Z j ,with Z j asinExample9andlet W : H ! H L T 2 denotetheisometry Wh = h 1 . Lemma2. If J 1 and J 2 arecommutingandpairwiserotationallysymmetric, q 2P and q J 1 , J 2 0, then q s J 1 , J 2 0 . If T 1 and T 2 isacommutingpairofinvertibleoperatorsontheHilbertspace H and P 2P n ,then P s T 2 , T 1 , T 1 , T 2 = I n W P ^ T 2 , ^ T 1 , ^ T 1 , ^ T 2 I n W . Wewilloccasionallyusethenotation p T , T for p T 2 , T 1 , T 1 , T 2 . Proof. Let n denotethesizeof q i.e. q 2P n .Foreach t = t 1 , t 2 2 R 2 thereisaunitary operator U t suchthat e it 1 J 1 = U t J 1 U t and e it 2 J 2 = U t J 2 U t byacombinationofLemma1andExample9.Hence U t J 2 J 1 U t = U t J 2 U t U t J 1 U t . 33

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Itfollowsthat q e )]TJ/F48 7.9701 Tf 6.586 0 Td [(it 2 J 2 , e )]TJ/F48 7.9701 Tf 6.586 0 Td [(it 1 J 1 , e it 1 J 1 , e it 2 J 2 = I n U t q J 2 , J 1 , J 1 , J 2 U t 0. Hence, q s J 2 , J 1 , J 1 , J 2 = 1 4 2 Z 2 0 Z 2 0 q e )]TJ/F48 7.9701 Tf 6.586 0 Td [(it 2 J 2 , e )]TJ/F48 7.9701 Tf 6.586 0 Td [(it 1 J 1 , e it 1 J 1 , e it 2 J 2 dt 0. Toprovethesecondassertion,let p 2P 1 andcompute D p ^ T 2 , ^ T 1 , ^ T 1 , ^ T 2 Wh , Wf E = D p ^ T 2 , ^ T 1 , ^ T 1 , ^ T 2 h 1, f 1 E = * X , , , p , , , T 1 T 2 h e it 2 e it 1 , T 1 T 2 f e it 2 e it 1 + = D X p , , , T 2 T 1 T 1 T 2 h , f E = h p s T 2 , T 1 , T 1 , T 2 h , f i . Applyingthisresultentry-wise,wegettheresultfor P . Lemma3. Suppose T 1 , T 2 arecommutinginvertibleoperatorsonaHilbertspace H and J 1 and J 2 arecommutinginvertibleoperatorsonaHilbertspace K .If J 1 and J 2 arepairwise rotationallysymmetricandthemapping : H s J 1 , J 2 !H s T 1 , T 2 determinedby J 2 J 1 J 1 J 2 = T 2 T 1 T 1 T 2 iswelldenedandcompletelypositive,thenthemapping ^ : H J 1 , J 2 !H ^ T 1 , ^ T 2 determinedby ^ J 2 J 1 J 1 J 2 = ^ T 2 ^ T 1 ^ T 1 ^ T 2 isalsowelldenedandcompletelypositive. Proof. Fixapositiveinteger n anda p 2P n andsuppose p J , J 0 .Wearetoshow p ^ T , ^ T 0 .Givenapairofintegers M , N let P denotethe M +1 M +1 matrix 34

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whoseentriesarethe N +1 N +1 matriceswhoseentriesare n n matrices, P = I n y j 2 2 I n y j 1 1 p x , y )]TJ/F47 11.9552 Tf 5.48 -9.684 Td [(I n x k 1 1 )]TJ/F47 11.9552 Tf 12.952 -9.684 Td [(I n x k 2 2 N j 1 , k 1 = )]TJ/F48 7.9701 Tf 6.587 0 Td [(N M j 2 , k 2 = )]TJ/F48 7.9701 Tf 6.587 0 Td [(M {10 Thus P T , T isanoperatoron C n H C 2 N +1 C 2 M +1 andtheentriesof P T , T areoperatorsof C n H C 2 N +1 givenby I n T j 2 2 I n T j 1 1 p T , T )]TJ/F47 11.9552 Tf 5.479 -9.684 Td [(I n T k 1 1 )]TJ/F47 11.9552 Tf 12.951 -9.684 Td [(I n T k 2 2 N j 1 , k 1 = )]TJ/F48 7.9701 Tf 6.587 0 Td [(N . {11 Notethat P J , J 0 andthus,byLemma2, P s J , J 0 .Thus,bythehypothesesofthis lemma, P s T , T 0 .Let f e 1 ,..., e n g denotethestandardbasisfor C n .Reusingnotation, let f f )]TJ/F48 7.9701 Tf 6.587 0 Td [(N ,..., f 0 ,..., f N g and f f )]TJ/F48 7.9701 Tf 6.587 0 Td [(M ,..., f 0 ,..., f M g denotethestandardbasesfor C 2 N +1 and C 2 M +1 respectively.Agenericvectorin C n H C 2 N +1 C 2 M +1 ,thespacethat P T , T actsupon,hastherepresentation h = X h j , a , e j f a f . Let p j , k ^ T , ^ T denotethe j , k -thentryof p ^ T , ^ T .Compute,usingLemma2, 0 h P s T 2 , T 1 , T 1 , T 2 h , h i = D P ^ T 2 , ^ T 1 , ^ T 1 , ^ T 2 h 1, h 1 E = X a , b , , X j , k D ^ T b 1 ^ T 2 p j , k ^ T , ^ T ^ T a 1 ^ T 2 h j , a , 1, h k , b , 1 E . = X a , b , , X j , k h p j , k ^ T , ^ T T 2 T a 1 h j , a , b z a 1 z 2 , T 2 T b 1 h k , , z b 1 z 2 i = X j , k h p j , k ^ T , ^ T [ X a , T 2 T a 1 h j , a , b z a 1 z 2 ],[ X b , T 2 T b 1 h k , b , z b 1 z 2 ] i = h p ^ T , ^ T g , g i , {12 where g = n X j =1 N X a = )]TJ/F48 7.9701 Tf 6.587 0 Td [(N M X = )]TJ/F48 7.9701 Tf 6.587 0 Td [(M T 2 T a 1 h j , a , b z a 1 z 2 . 35

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Since T 1 and T 2 areinvertible,givenvectors g j , a , b 2 H ,thereexistsvectors h j , a , b suchthat g = n X j =1 N X a = )]TJ/F48 7.9701 Tf 6.586 0 Td [(N M X = )]TJ/F48 7.9701 Tf 6.587 0 Td [(M g j , a , b z a 1 z 2 . Finally,sincevectorsoftheform g aredensein H L 2 T 2 ,itfollowsthat p ^ T , ^ T 0 ;i.e., thatmap ^ iscompletelypositive. Lemma4. Suppose T 1 and T 2 arecommutinginvertibleoperatorsin B H . If p 2P and p ^ T 2 , ^ T 1 , ^ T 1 , ^ T 2 0 ,then p T 2 , T 1 , T 1 , T 2 0 .Inparticularthemapping : p ^ T 2 , ^ T 1 , ^ T 1 , ^ T 2 7! p T 2 , T 1 , T 1 , T 2 iswelldened. Proof. Let D NM = 1 p 2 N +1 1 p 2 M +1 N X j = )]TJ/F48 7.9701 Tf 6.586 0 Td [(N M X k = )]TJ/F48 7.9701 Tf 6.587 0 Td [(M e ijt 1 e ikt 2 2 L 2 T 2 . If f , h 2 H , thenfor , , , 2 Z , D ^ T 2 ^ T 1 ^ T 1 ^ T 2 h D N , M , f D N , M E = D ^ T 1 ^ T 2 h D N , M , ^ T 1 ^ T 2 f D N , M E = D T 1 T 2 h , T 1 T 2 f ED z 1 z 2 D N , M , z 1 z 2 D N , M E = D T 1 T 2 h , T 1 T 2 f E 1 M +1 N +1 * N + j )]TJ/F25 7.9701 Tf 6.587 0 Td [( j X j = )]TJ/F48 7.9701 Tf 6.587 0 Td [(N + j )]TJ/F25 7.9701 Tf 6.586 0 Td [( j M + j )]TJ/F25 7.9701 Tf 6.586 0 Td [( j X k = )]TJ/F48 7.9701 Tf 6.586 0 Td [(M + j )]TJ/F25 7.9701 Tf 6.587 0 Td [( j e ijt 1 e ikt 2 , N X j = )]TJ/F48 7.9701 Tf 6.587 0 Td [(N M X k = )]TJ/F48 7.9701 Tf 6.586 0 Td [(M e ijt 1 e ikt 2 + = D T 1 T 2 h , T 1 T 2 f E 2 N +1 )-222(j )]TJ/F24 11.9552 Tf 11.955 0 Td [( j 2 N +1 2 M +1 )-222(j )]TJ/F24 11.9552 Tf 11.955 0 Td [( j 2 M +1 . Thusif p 2P 1 , lim N !1 lim M !1 D p ^ T 2 , ^ T 1 , ^ T 1 , ^ T 2 h D N , M , f D N , M E = h p T 2 , T 1 , T 1 , T 2 h , f i . 36

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Henceif p ^ T 2 , ^ T 1 , ^ T 1 , ^ T 2 0, then p T 2 , T 1 , T 1 , T 2 0 aswell.Thecaseforsquare matricesiseasilyestablished. Proposition1. Suppose T 1 and T 2 arecommutinginvertibleoperatorsonaHilbertspace H ,and J 1 and J 2 arecommutinginvertibleoperatorsonaHilbertspace K .If J 1 and J 2 are pairwiserotationallysymmetricandthemapping : H s J 1 , J 2 !H s T 1 , T 2 determinedby J 2 J 1 J 1 J 2 = T 2 T 1 T 1 T 2 iswelldenedandcompletelypositive,thenthereisaHilbertspace K ,arepresentation : B K ! B K ,andaisometry V suchthat VT m 2 T n 1 = J 1 n J 2 m V for m , n 2 Z . Proof. Themapping : H ^ T 1 , ^ T 2 !H T 1 , T 2 asdescribedinLemma4,iswelldenedand completelypositive.Themapping ^ : H J 1 , J 2 !H ^ T 2 , ^ T 2 asdescribedin3isalsowell denedandcompletelypositive.Theircomposition = ^ iswelldenedandcompletelypositive.ThepropositionnowfollowsfromTheorem18. Fix c , d > 0 anddene,for 0 i + j 2 here i , j arenon-negativeintegers,the 3 3 matrices B i , j by I + X 0 < i + j 2 B i , j i j = 0 B B B B @ 1 c d c 1+ 2 c 2 cd d cd 1+ 2 d 2 1 C C C C A , {13 and B 0,0 = I )]TJ/F22 7.9701 Tf 15.167 4.707 Td [(1 c 2 B 2,0 )]TJ/F22 7.9701 Tf 15.411 4.707 Td [(1 d 2 B 0,2 .Dene, J 1 = 0 B B B B @ U 1 cU 1 0 0 U 1 0 00 U 1 1 C C C C A J 2 = 0 B B B B @ U 2 0 dU 2 0 U 2 0 00 U 2 1 C C C C A , {14 where U 1 = Z 1 and U 2 = Z 2 ,thepairwiserotationallysymmetricoperatorsinExample9. Wenote J 1 and J 2 arepairwiserotationallysymmetricviaLemma1.Itisclearthatfromthe 37

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calculationdoneinEquation2{2that J = J 1 , J 2 2 J c , d and Q J , = )]TJ/F47 11.9552 Tf 5.479 -9.684 Td [(I + X 0 < i + j 2 B i , j i j I . {15 Inparticular B i , j J = B i , j I andwedene B 0,0 J = B 0,0 I . Lemma5. If T = T 1 , T 2 isintheclass F c , d ,and ^ Q T , = Q T , )]TJ/F21 11.9552 Tf 15.798 8.088 Td [(1 c 2 B 2,0 T )]TJ/F21 11.9552 Tf 16.141 8.088 Td [(1 d 2 B 0,2 T 0 factorsintheform, ^ Q T , = V 0 + V 1 + V 2 V 0 + V 1 + V 2 , {16 thenthemap J 2 J 1 J 1 J 2 = T 2 T 1 T 1 T 2 iswelldenedandcompletelypositive. Proof. Supposethe2-tuple T = T 1 , T 2 isintheclass F c , d andfornotationalconvenience let B 0,0 T = I )]TJ/F21 11.9552 Tf 15.798 8.088 Td [(1 c 2 B 2,0 T )]TJ/F21 11.9552 Tf 16.142 8.088 Td [(1 d 2 B 0,2 T = I )]TJ/F21 11.9552 Tf 15.798 8.088 Td [(1 c 2 B 2 T 1 , T 1 )]TJ/F21 11.9552 Tf 16.141 8.088 Td [(1 d 2 B 2 T 2 , T 2 . Notethat B 0,0 T 0 since Q T , 0 for = =0. Thespaces H s J 1 , J 2 and H s T 1 , T 2 arespannedby f B 0,0 J , B 1,0 J , B 0,1 J , B 1,1 J , B 2,0 J , B 0,2 J g and f B 0,0 T , B 1,0 T , B 0,1 T , B 1,1 T , B 2,0 T , B 0,2 T g respectively.Forpositiveintegers n ,let M n denotethe n n matrices.Theelements X 2 M n H s J 1 , J 2 havetheform X = X 0 i + j 2 X i , j B i , j J . 38

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Equivalently, X = 0 B B B B @ X 0,0 cX 1,0 dX 0,1 cX 1,0 c 2 X 2,0 cdX 1,1 dX 0,1 cdX 1,1 d 2 X 0,2 1 C C C C A I . If X 0 ,theneach X i , j isself-adjoint.Further X 0 ifandonlyif Y = 0 B B B B @ X 0,0 X 1,0 X 0,1 X 1,0 X 2,0 X 1,1 X 0,1 X 1,1 X 0,2 1 C C C C A isaswell.Inthiscase,thereexists 3 n n matrices Y 0 , Y 1 , Y 2 suchthat 0 B B B B @ X 0,0 X 1,0 X 0,1 X 1,0 X 2,0 X 1,1 X 0,1 X 1,1 X 0,2 1 C C C C A = 0 B B B B @ Y 0 Y 1 Y 2 1 C C C C A Y 0 Y 1 Y 2 . Usingthefactorization2{16, 1 m X = X X i , j B i , j T = X 0,0 V 0 V 0 + X 1,0 V 0 V 1 + V 1 V 0 + X 0,1 V 0 V 2 + V 2 V 0 + X 1,1 V 1 V 2 + V 2 V 1 + X 2,0 V 1 V 1 + X 0,2 V 2 V 2 = Y 0 V 0 + Y 1 V 1 + Y 2 V 2 Y 0 V 0 + Y 1 V 1 + Y 2 V 2 . {17 Sincetherighthandsideisevidentlypositive,themap iscompletelypositive. ByProposition1andLemma5,since J 1 and J 2 arepairwiserotationallysymmetric,we haveshownafactorization2{16impliesthereisarepresentation suchthatthe2-tuple T liftstothe2-tuple J .Itremainstoshowthatanyrepresentationappliedto J = J 1 , J 2 producesa2-tupleofthesameformasinequation2{14. Lemma6. Let E betheHilbertspacethat J 1 and J 2 actupon.If e E isalsoaHilbertspace and : B E ! B e E isaunital -representation,then J 1 = J 1 and J 2 = J 2 have,up 39

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tounitaryequivalence,thesameformas J 1 and J 2 givenbyEquation 2{1 andinparticular areintheclass J c , d . Proof. Theproofproceedsmuchinthesamewayasitdoesin[20]butwithsomeminor dierences.Thefollowingrelationsareevident. i J = W i + N i where W i isunitary, N 2 i =0 for i =1,2 . ii W i N i = N i W i for i =1,2 . iii N 1 N 1 = N 2 N 2 . iv N 1 N 1 + N 1 N 1 + N 2 N 2 =1 . v N i N j =0 for i , j =1,2 . vi N i N j =0 for i , j =1,2 . Fromtheserelations, N 1 N 1 , N 2 N 2 , N 1 N 1 = N 2 N 2 arepairwiseorthogonalprojections.Let J i = J i , N i = N i ,and W i = W i for i =1,2 . Thesemustsatisfythesamealgebraicrelations,i.e. i J = W i + N i where W i isunitary, N 2 i =0 for i =1,2 . ii W i N i = N i W i for i =1,2 . iii N 1 N 1 = N 2 N 2 . iv N 1 N 1 + N 1 N 1 + N 2 N 2 =1 . v N i N j =0 for i , j =1,2 . vi N i N j =0 for i , j =1,2 . Fromtheserelations, N 1 N 1 , N 2 N 2 , 40

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N 1 N 1 = N 2 N 2 arepairwiseorthogonalprojectionson e E .Forinstance, N 1 N 1 = N 1 N 1 N 1 + N 2 N 2 + N 1 N 1 N 1 = N 1 N 1 2 . Nowdecomposethespace H as H =ran N 1 N 1 ran N 1 N 1 ran N 2 N 2 .Themappings N j areunitarymaps Q j fromtherangeof N j totherangeof N j .Hence,withrespecttothe orthogonaldecompositionof H as H =ran N 1 N 1 ran N 1 N 1 ran N 2 N 2 , N 1 = 0 B B B B @ 0 Q 1 0 000 000 1 C C C C A andlikewise, N 2 = 0 B B B B @ 00 Q 2 000 000 1 C C C C A . Thus,uptounitaryequivalence,itmaybeassumedthat Q j = I andeachofthesummandsin thedirectsumdecompositionisthesameHilbertspace.Write W 1 = 0 B B B B @ A 1 B 1 C 1 D 1 E 1 F 1 G 1 H 1 J 1 1 C C C C A forsome A 1 , B 1 , C 1 , D 1 , E 1 , F 1 , G 1 , H 1 ,and J 1 operators.Since W 1 N 1 = N 1 W 1 , W 1 N 1 = 0 B B B B @ A 1 B 1 C 1 D 1 E 1 F 1 G 1 H 1 J 1 1 C C C C A 0 B B B B @ 0 I 0 000 000 1 C C C C A = 0 B B B B @ 0 A 1 0 0 D 1 0 0 G 1 0 1 C C C C A , 41

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and N 1 W 1 = 0 B B B B @ 0 I 0 000 000 1 C C C C A 0 B B B B @ A 1 B 1 C 1 D 1 E 1 F 1 G 1 H 1 J 1 1 C C C C A = 0 B B B B @ D 1 E 1 F 1 000 000 1 C C C C A , weconclude A 1 = E 1 and D 1 = F 1 = G 1 =0. Similarly,since W 1 N 2 = N 2 W 1 , A 1 = J 1 and H 1 =0. Hence W 1 = 0 B B B B @ A 1 B 1 C 1 0 A 1 0 00 A 1 1 C C C C A . Since W 1 isaunitaryoperator, W 1 W 1 = 0 B B B B @ A 1 B 1 C 1 0 A 1 0 00 A 1 1 C C C C A 0 B B B B @ A 1 00 B 1 A 1 0 C 1 0 A 1 1 C C C C A = 0 B B B B @ I 00 0 I 0 00 I 1 C C C C A , where I istheidentityoperator.Hence, A 1 A 1 + B 1 B 1 + C 1 C 1 = I , A 1 A 1 = I , A 1 B 1 =0, A 1 C 1 =0. 42

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Notethatthe rsttwo relationsaboveshowthat B 1 = C 1 =0 and A 1 isanisometry.Hence W isdiagonalwith A 1 downthediagonal.Since W isunitary, A 1 isunitary.Itfollowsthat W 1 = 0 B B B B @ U 1 00 0 U 1 0 00 U 1 1 C C C C A , where U 1 isaunitaryoperator.Asimilarargumentshowsthat W 2 = 0 B B B B @ U 2 00 0 U 2 0 00 U 2 1 C C C C A , where U 2 isaunitaryoperator.Since [ W 1 , W 2 ]=0, itfollowsthat [ U 1 , U 2 ]=0 .Hence,upto unitaryequivalence,the J i havetheformclaimed. TheforwarddirectionofTheorem17hasbeenestablished.Wenowneedonlytoprove theconverse.Namely,thatliftingimpliesfactorizationoftheassociatedoperatorpencil. However,thisisreadilyestablished.If T = T 1 , T 2 liftsto J = J 1 J 2 ,then V )]TJ/F47 11.9552 Tf 5.48 -9.684 Td [(Q J 1 , J 2 , )]TJ/F21 11.9552 Tf 15.798 8.088 Td [(1 c 2 B 2,0 J )]TJ/F21 11.9552 Tf 16.142 8.088 Td [(1 d 2 B 0,2 J V = Q T 1 , T 2 , )]TJ/F21 11.9552 Tf 15.798 8.088 Td [(1 c 2 B 2,0 T )]TJ/F21 11.9552 Tf 16.142 8.088 Td [(1 d 2 B 0,2 T . Henceanyfactorizationof ^ Q J , = K 0 + K 1 + K 2 K 0 + K 1 + K 2 givesthefactorizationof ^ Q T as Q T , = V K 0 + K 1 + K 2 K 0 + K 1 + K 2 V . Since ^ Q J factorsas ^ Q J , = 0 B B B B @ 0 B B B B @ 1 c d 1 C C C C A 1 c d 1 C C C C A I , 43

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theconclusionfollows. 44

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CHAPTER3 SPECTRALCONSIDERATIONSAND3-SYMMETRICLIFTING Givena2-tupleof3-isometries T inaclass F c , d thatliftstoa2-tuple J ofoperators intheclasswewillrstshowthatitispossibletochoose J suchthat Tay T = Tay J , where Tay denotestheTaylorspectrum.Secondly,usingthisresultonthespectrum,we willestablish,byaholomorphicfunctionalcalculusargument,aliftingtheoremanalogousto Theorem17for3-symmetric2-tuples. 3.1SpectralConsiderations Let U = U 1 , U 2 bethecommutingtupleofunitaryoperatorsappearingin J = J 1 , J 2 . Bytheformof J itiseasytosee, U i = J i . Proposition2. ForJordan2-tupleoftheform 2{1 Tay U = Tay J where U = U 1 , U 2 isthe2-tupleofunitaryoperatorsappearingin J = J 1 , J 2 . Suppose T isa2-tupleofinvertibleoperatorsand c , d > 0 .If T liftstoa2-tuple J 2 J c , d ,then Tay T Tay J .Moreover,inthiscasethereexistsa2-tuple J 2 J c , d such that T liftsto J and Tay T = Tay J . TheproofofProposition2occupiestheremainderofthissectionandisbrokendowninto aseriesofsubresults.ThesectionconcludeswiththeproofofProposition2 Foracompactset K ,let co K denotetheconvexhullof K .If K C n iscompact, then,byCaratheodory'sTheorem, co K isalsocompactandhenceclosed.Foraclosed convexset K , let Ext K denotethesetofextremepointsofthe K . Lemma7. Thesetofextremepointsof co T 2 is T 2 . Proof. Theconvexhullofacartesianproductisthecartesianproductoftheconvexhulls.The setofextremepointsofacartesianproductisthecartesianproductoftheextremepoints. Sincetheextremepointsof co T = T theresultfollows. 45

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Lemma8. If K isacompactsubsetof T 2 C 2 , then Extco K = K . Proof. Since K T 2 ,if z 2 K ,then z isanextremepointof co T 2 byLemma7and thereforeof co K .Hence K Extco K .Ontheotherhand, Extco K K forany compactsubset K of C n . Denition23. Thejointapproximatepointspectrumfora2-tuple T isdenedtobetheset ofpoints 2 C 2 suchthatthereexistunitvectors f x k g suchthat k T i )]TJ/F24 11.9552 Tf 11.955 0 Td [( i x k k! 0 for i =1,2. Wedenotejointapproximatepointspectrumas ap T . Thefollowingtwolemmasarewellknown.Amongthemanyreferences,see[10,12].The theoremfollowingtheselemmascanbefoundinapaperofWrobel[23]. Lemma9. Theapproximatepointspectrumofacommutingtuple T ofoperatorsonHilbert spaceliesintheTaylorspectrumof T . Lemma10. TheTaylorspectrumofacommutingtuple T ofoperatorsonHilbertspaceis nonemptyandcompact. Theorem19. If T isacommutingtupleofoperatorsonHilbertspace,then Extco Tay T =Extco ap T . Lemma11. Suppose T isacommuting2-tupleofinvertibleoperatorsonaHilbertspace H and c , d > 0 and T liftstoa2-tuple J 2 J c , d actingontheHilbertspace K ,i.e.thereisan isometry V : H ! K suchthat VT = J V foreverymulti-index .If 2 ap T , then 2 ap J ; i.e., ap T ap J . 46

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Proof. For i =1,2, V T i )]TJ/F24 11.9552 Tf 11.955 0 Td [( i = J i )]TJ/F24 11.9552 Tf 11.956 0 Td [( i V . If k T i )]TJ/F24 11.9552 Tf 11.973 0 Td [( i x k k! 0 as k !1 ,then k V T i )]TJ/F24 11.9552 Tf 11.973 0 Td [( i x k k! 0 as k !1 since V isanisometry. Hencefortheunitvectors y k = Vx k , k J i )]TJ/F24 11.9552 Tf 11.955 0 Td [( i Vx k k! 0 as k !1 . Wearenowinpositiontoshow Tay T J .Since T i and J i areinvertiblefor i =1,2 , both Tay T and Tay J aresubsetsof T 2 ,sinceforinstance Tay T T 1 T 2 T 2 .Inparticular,byTheorem19andLemma9, Tay A =Extco Tay A =Extco ap A = ap A , where A iseither T or J .AnapplicationofLemma11nowgives Tay T Tay J , completingtheproofofthesecondpartofProposition2. Toprovetherstpartoftheproposition,notethatsince U liftsto J ,bywhathasalready beenproved, Tay U Tay J .Ontheotherhand,asseenin[12],foroperators A , B and C onHilbertspace, Tay 0 B @ 0 B @ AC 0 B 1 C A 1 C A Tay A [ Tay B . Inourcasethisshowsthat Tay J Tay U andtheproofoftherstpartofthelemmais complete. WewillnowcompletetheproofofProposition2byshowingthatwecanalterthe 2-tuple J sothat Tay J Tay T .Wewillstatethisasapropositionwhoseproofwill requireseverallemmasandoccupytheremainderofthissection.Suppose T = T 1 , T 2 isa commutingtupleofinvertibleoperatorswhichlifttoacommutingtupleofinvertibleoperators 47

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J = J 1 , J 2 2 J c , d oftheform2{1i.e.thereexistsanisometry V suchthat VT 1 T 2 = J 1 J 2 V . Let U = U 1 , U 2 bethetupleofunitaryoperatorsappearingin J .Asin[20]wewillshowthat each U i canbereplacedwith W i = I )]TJ/F47 11.9552 Tf 12.086 0 Td [(P U i I )]TJ/F47 11.9552 Tf 12.087 0 Td [(P ,where P isthejointspectralprojection forthecomplementof Tay T . Proposition3. Ifacommutingtupleofinvertibleoperators T liftstoacommutingtupleof operators J 2 J c , d ,thenthereexistsatupleofcommutinginvertibleoperators J = J 1 , J 2 2 J c , d suchthat T liftsto J and Tay T = Tay J . Sincetheinclusion Tay T Tay J hasalreadybeenestablished,itremainstoprove that J canbechoseninsuchawaythatthereverseinclusionholds. Assuming T 1 and T 2 arebothinvertible,byTheorem17thereisacommuting2-tupleof unitaryoperators U 1 and U 2 actingonaHilbertspace F andanisometry V : H ! F F F suchthat VT n 1 T m 2 = J n 1 J m 2 V forall m , n 2 N wherethe J i have U i asentriesfor i =1,2 .If Tay T = T 2 , thenthereis notmuchtoprovesince Tay J Tay U T 2 andtheproofiscomplete.Sofromthis pointonwardweassumeotherwise. Asshownin[20]givenanarc A inthecomplementofthespectrumofa 3 -isometryT T T ,thereisaholomorphicfunction f suchthat j f j 1 onthearc A and j f j < 1 on andinside )]TJ/F17 11.9552 Tf 6.774 0 Td [(,where )]TJ/F17 11.9552 Tf 10.677 0 Td [(isacurvecontainingthespectrum. Let D denotetheclosedunitdisk, f z 2 C : j z j 1 g ,inthecomplexplane C . Lemma12. Let p = e i 1 , e i 2 beapointof T 2 inthecomplementoftheTaylorspectrumof T .If i , for i =1,2 ,areopensetscontaining D and 2 e i i = 2 i , thenthereexistsanopenset O p T 2 openinthetopologyof T 2 suchthat O p Tay T = ; andaholomorphicfunction f p : 1 2 ! C suchthat j f p j 1 on O p and j f p j < 1 on Tay T .Moreoverthereexist 48

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holomorphicfunctions f p i : i ! C suchthat f p z 1 , z 2 = f p 1 z 1 f p 2 z 2 . Proof. Given p = e i 1 , e i 2 2 T 2 considerthefunctions h i : i ! C , h i z = 1 )]TJ/F47 11.9552 Tf 11.955 0 Td [(e )]TJ/F48 7.9701 Tf 6.587 0 Td [(i i z for i =1,2 anddene h : 1 2 ! C by h z 1 , z 2 = h 1 z 1 h 2 z 2 = 1 )]TJ/F47 11.9552 Tf 11.955 0 Td [(e )]TJ/F48 7.9701 Tf 6.586 0 Td [(i 1 z 1 )]TJ/F47 11.9552 Tf 11.955 0 Td [(e )]TJ/F48 7.9701 Tf 6.586 0 Td [(i 2 z 2 . Wenotethat h p =1 and j h z j < 1 whenever z 6 = p and z inthebidisk.Let K bea compactsubsetof T 2 notcontaining p andnote j h n j! 0 uniformlyon K as n !1 . Hence, j h N z j < 1 2 forsome N largeenoughandall z 2 Tay T .Let C beapositivenumbersuch that 1 < C < 2 andlet O p beanopensetdisjointfromtheTaylorspectrumcontaining p such that C j h N j 1 on O p .Suchanopensetexistsbycontinuity.Nowdene f p z = Ch N z andnote f p and O p satisfytheconditionsofthelemma.Itisclearthereexistsa f p i for i =1,2 suchthat f p z 1 , z 2 = f p 1 z 1 f p 2 z 2 . Wenowchoose 1 = 2 = 3 2 D .Sinceeach U i isunitarywecandene f i U i throughthe holomorphicfunctionalcalculusorbythepowerseriesfunctionalcalculus.Ofcoursebothwill givethesameoperatorvaluefor f i U i .Atthesametimewemaydeneeach f p i J i viathe powerseriescalculus.Itisstraightforwardtoverify f p 1 J 1 = 0 B B B B @ f p 1 U 1 cf 0 p 1 U 1 0 0 f p 1 U 1 0 00 f p 1 U 1 1 C C C C A , f p 2 J 2 = 0 B B B B @ f p 2 U 2 0 df 0 p 2 U 2 0 f p 2 U 2 0 00 f p 2 U 2 1 C C C C A . Dene f p J by f p J = f p 1 J 1 f p 2 J 2 . 49

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Similarlywemaydene f p i T i andhence f T bythepowerseriesfunctionalcalculusaswell. Wenotethatanyotherfunctionalcalculususedtodene f J and f T mustagreewiththe valuesgivenbythepowerseriescalculus. Nowwritewithrespecttothedecomposition F F F V = 0 B B B B @ V 2 V 1 V 0 1 C C C C A . Lemma13. Let p 2 T 2 beinthecomplementof Tay T with f p and O p T 2 asdescribed inLemma12,then E O p V ` =0 for ` =0,1,2 . Proof. Wewillsurpressthe p inthenotationforthefunctions f p , f p 1 , and f p 2 ,writing f , f 1 , f 2 instead.Bytheholomorphicfunctionalcalculusweknow f n i T i convergestozerointhe operatornormsinceeach f n i convergesto0uniformlyontheTaylorspectrumfor T .Since Vf n i T i = f n i J i V for i =1,2, f n i J i V alsotendsto0inoperatornorm.Hence f n J V alsotendsto0intheoperatornorm. Let E betheuniquejointspectralmeasureforthe2-tuple U suchthat E A B = E 1 A E 2 B where E i isthespectralmeasurefor U i , i =1,2 .Let P bethespectralprojectionfor U correspondingto O p , P = Z O p dE = E O p . Consider,withrespecttothedecomposition K = F F F 0 0 P = 0 B B B B @ 000 000 00 P 1 C C C C A , 50

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0 P 0= 0 B B B B @ 000 0 P 0 000 1 C C C C A , and P 0 0= 0 B B B B @ P 00 000 000 1 C C C C A . Since f n J V convergestozerosodo V f n J 0 P 0 P f n J V , V f n J P 0 P 0 f n J V , and V f n J P 0 0 P 0 0 f n J V . Bycalculation f n J 0 P 0 P f n J = 0 B B B B @ f n U 00 f n U 0 0 f n U 1 C C C C A 0 B B B B @ 000 000 00 P 1 C C C C A 0 B B B B @ f n U 0 f n U 0 00 f n U 1 C C C C A = 0 B B B B @ 000 000 00 f n U Pf n U 1 C C C C A . Itfollowsthat Pf n U V 0 tendsto 0 inoperatornorm.However, Pf n U f n U P = f n U Pf n U , since P isthespectralprojectionassociatedwith U .Consequently, V 0 P j f n j 2 PV 0 = V 0 f n U Pf n U V 0 kk )167(! 0. 51

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But P j f n j 2 P P since j f n j 1 onthesupport O p of P .Thus PV 0 =0 .Similarly, V f n J P 0 P 0 f n J V kk )167(! 0 and V 1 P j f n j 2 PV 1 = V 1 f n U Pf n U V 1 kk )167(! 0. Hencebysimilarargument PV 1 =0 .Lastlysince V f n J P 0 0 P 0 0 f n J V kk )167(! 0, byusingthefactthat PV 1 = PV 0 =0 andarguingsimilarlytothepreviouscaseswehavethat PV 2 =0 . Lemma14. If A isacompactsubsetof T suchthat A Tay T = ; , then E A V ` =0 for ` =0,1,2 . Proof. Since A iscoveredbynitelymany O p i ,indexedbyaniteset F wehave E A V ` E [ p i 2 F O p i ! V ` X p i 2 F E O p i V ` hence E A V ` =0 for ` =1,2 . Sincetheproofofthefollowinglemmacarriesoverfrom[20]withonlysupercial modications,wesimplystatetheresulthere. Lemma15. Suppose A 1 A 2 ... isanincreasingsequenceofBorelsubsetsof T 2 andlet A = [ j A j .If E A j V ` =0 forall j and ` =0,1,2 ,then E A V ` =0 . Thecomplementof Tay T canbewrittenasanincreasingsequenceofclosedcompact sets.ByanapplicationofLemmas15and14 E Tay T c V ` =0, for ` =0,1,2. 52

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Let P = E Tay T c . Each W i = I )]TJ/F47 11.9552 Tf 11.955 0 Td [(P U i I )]TJ/F47 11.9552 Tf 11.955 0 Td [(P isunitaryand J 1 = 0 B B B B @ W 1 cW 1 0 0 W 1 0 00 W 1 1 C C C C A , J 2 = 0 B B B B @ W 2 0 dW 2 0 W 2 0 00 W 2 1 C C C C A havetheappropriateform.Finally,bytherstpartoftheproposition, Tay J = Tay W Tay T . 3.23-SymmetricOperatorsTuples Wewillnowgomoreindepthintousingtheholomorphicfunctionalcalculusfor T and J .For i =1,2 let i beasimplyconnectedopensubsetoftheplane.Givena2-tupleof commutingoperators T = T 1 , T 2 witheach T i i andfunctions g i analyticon i , for i =1,2 ,beanalyticfunctions,byuseoftheholomorphicfunctionalcalculuswecan denetheoperators g i T i .ByRunge'sTheoremthereisasequenceofpolynomials s i , n whichconvergeuniformlyoncompactsubsetsof i to g i forboth i =1,2 .Thesequences ofoperators s i , n T i convergeinnormto g i T i for i =1,2, bythestandardpropertiesof theholomorphicfunctionalcalculus.Considera2-tupleofoperators J = J 1 , J 2 oftheforms 2{1with U i i .Straightforwardcalculationshows,forapolynomial q z , q J 1 = 0 B B B B @ q U 1 cU 1 q 0 U 1 0 0 q U 1 0 00 q U 1 1 C C C C A . Thus,as s 0 i , n convergesuniformlyto g 0 i oncompactsubsetsof i , g 1 J 1 =lim s 1, n J 1 = 0 B B B B @ g 1 U 1 cU 1 g 0 1 U 1 0 0 g 1 U 1 0 00 g 1 U 1 1 C C C C A , 53

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g 2 J 2 =lim s 2, n J 2 = 0 B B B B @ g 2 U 2 0 dU 2 g 0 2 U 2 0 g 2 U 2 0 00 g 2 U 2 1 C C C C A . Foranormaloperator T theoperator g i T isnormalaswell.Moreover,thespectrum of g i T isgivenbythespectralmappingtheoremas g i T .Hence,giventhetuple J = J 1 , J 2 andholomorphicfunctions g 1 and g 2 wehaveaformulafor g 1 J 1 and g 2 J 2 as wellastheirrespectivespectra. Togetsomeinformationabouttheindividualspectra,wewillusetheprojectionproperty fortheTaylorjointspectrumasdiscussedinChapter1.Moreover,wegivemoredetailas seenin[12].Let A and B bea n -tupleand k -tuplerespectivelyi.e. A = A 1 ,..., A n and B = B 1 ,..., B k .Let A , B denotethetuple C 1 ,..., C n + k where C i = A i for i =1,... n and C i = B i )]TJ/F48 7.9701 Tf 6.587 0 Td [(n for i = n +1,..., n + k . TheprojectionpropertyfortheTaylorjointspectrumisasfollows, 1,..., n Tay A , B = Tay A and n +1,..., n + k Tay A , B = Tay B wherewedene 1,..., n : C n C k ! C n , z 1 ,..., z n , z 1+ n ,..., z n + k 7! z 1 ,... z n andsimilarly for n +1,..., n + k .Forusthisprojectionpropertyimplies i Tay T 1 , T 2 = Tay T i = T i 54

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for i =1,2 .InthecontextofProposition3,if T = T 1 , T 2 liftstoatuple J 2 J c , d , then thereexistsaJordantuple J2 J c , d suchthat Tay J = Tay T . Since Tay J 1 , J 2 = Tay T 1 , T 2 , bytheprojectionproperty, J i = i Tay J 1 , J 2 = i Tay T 1 , T 2 = T i , for j =1,2. Let U = U 1 , U 2 betheunitarycommutingtupleappearingin J = J 1 , J 2 . Sinceitwillbeofrelevanceintheexpositiontofollowwerecallforthereadertheequality U i = J i fromProposition2. Denition24. Atupleofoperators T = T 1 , T 2 onaHilbertspace H willbecalleda commuting3-symmetrictupleifthereexistboundedoperators B j , k on H suchthat, exp is 2 T 2 exp is 1 T 1 exp is 1 T 1 exp is 2 T 2 = I + X 0 < j + k 2 s j 1 s k 2 B j , k forall s 1 , s 2 2 R 2 . Itisclearthatif T = T 1 , T 2 isacommuting3-symmetrictuple,then T = e i T 1 , e i T 2 isa commuting3-isometrictuple. Considercommuting2-tuplesof3-symmetricoperators T 1 , T 2 whosespectraliein [ a 1 , b 1 ] and [ a 2 , b 2 ] respectively.WenotethattheTaylorjointspectrumfor T 1 , T 2 mustbe containedin [ a 1 , b 1 ] [ a 2 , b 2 ] .Let G z =exp iz andlet S i = G [ a i , b i ] .Supposethe lengthofeach [ a i , b i ] isstrictlylessthan 2 . Inthiscase S i isapropersubsetoftheunitcircle T .Foreach i thereexists i [ a i , b i ] and i S i ,opensimplyconnectedsubsetsof C such that G 1 = G j 1 : 1 ! 1 G 2 = G j 2 : 2 ! 2 55

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arebi-analytic.Inparticular, j doesnotcontainalloftheunitcircle.Fortheoperator 2-tupleofcommuting3-symmetricoperators T = T 1 , T 2 with T i [ a i , b i ] theoperators G i T i aredenedbytheholomorphicfunctionalcalculusand G i T i S i T .Let T i = G i T i andsupposethecommuting3-isometric2-tuple T = T 1 , T 2 lifts,i.e.there existsanisometry V andaJordantuple J suchthat VT n 1 T m 2 = J n 1 J m 2 V . ByProposition3andtheprojectionpropertythereexistunitaryoperators W 1 and W 2 andan isometry V suchthat VT 1 = 0 B B B B @ W 1 cW 1 0 0 W 1 0 00 W 1 1 C C C C A V = J 1 V VT 2 = 0 B B B B @ W 2 0 dW 2 0 W 2 0 00 W 2 1 C C C C A V = J 2 V where W i = T i .Againeach G i isbi-analyticintheneighborhoodofthespectrumof each J i hence V T 1 = VG )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 1 T 1 = G )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 1 J 1 V V T 2 = VG )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 2 T 2 = G )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 2 J 2 V . {1 Let A i = G )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 i W i andnote G )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 i 0 W i = )]TJ/F47 11.9552 Tf 9.299 0 Td [(iW i .Hence, V T 1 = 0 B B B B @ A 1 )]TJ/F47 11.9552 Tf 9.299 0 Td [(ic 0 0 A 1 0 00 A 1 1 C C C C A V 56

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V T 2 = 0 B B B B @ A 2 0 )]TJ/F47 11.9552 Tf 9.298 0 Td [(id 0 A 2 0 00 A 2 1 C C C C A V . Ifthespectrumofeach T i doesnothavelengthlessthan 2 wecandothesameanalysis ontheoperators e T i = t i T i whereeach t i ischosensothat e T i isoflengthlessthan 2 .As shownin[20]thesearealso3-symmetricoperators.TheTaylorspectrumofthe3-symmetric tuple e T = e T 1 , e T 2 iscontainedinsome [ a 1 , b 2 ] [ a 2 , b 2 ] whereeach [ a i , b i ] isoflength lessthan 2 .Again e T =exp i e T 1 ,exp i e T 2 isa3-isometrictupleandsupposetheyliftby Theorem17,i.e.thereexistsanisometry V andJordantuple e J suchthat V e T n 2 e T m 1 = e J m 1 e J n 2 V andmoreover V e T i = e J i V . Byapplyingthesameargumentasin3{1wehave V e T i = e J i V andthus V T i = 1 t i J i V . Bynotingthat T and T =exp i T sharethesameoperatorpencil,weseethatthe 3-symmetricversionofTheorem17.Moreover,itturnsoutif T isa 2 -tupleof 3 -symmetric operators,then T doesinfactlifttoacommutingtupleofJordanoperators.Theseresultsare collectedinthefollowingtheorem. Theorem20. Tuplesof3-symmetricoperators T 1 , T 2 willlifttoa2-tuple J 1 , J 2 ofthe forms J 1 = 0 B B B B @ A 1 )]TJ/F47 11.9552 Tf 9.298 0 Td [(ic 0 0 A 1 0 00 A 1 1 C C C C A J 2 = 0 B B B B @ A 2 0 )]TJ/F47 11.9552 Tf 9.298 0 Td [(id 0 A 2 0 00 A 2 1 C C C C A 57

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ifandonlyifthepolynomial ^ Q T , = I + B 1,0 + B 0,1 + B 1,1 + 2 B 2,0 + 2 B 0,2 )]TJ/F21 11.9552 Tf 15.798 8.088 Td [(1 c 2 B 2,0 )]TJ/F21 11.9552 Tf 16.142 8.088 Td [(1 d 2 B 0,2 0 factorsintheform, ^ Q T , = V 0 + V 1 + V 2 V 0 + V 1 + V 2 forsomeoperators V 0 , V 1 and V 2 in B H . Moreover,if T isa 2 -tupleof 3 -symmetricoperators,thenthereexists c , d > 0 such thatthepencil ^ Q T , ispositivesemideniteforall , 2 R .Inotherwords,thereexists c , d > 0 suchthat T =exp iT 2 F c , d . Proof. Theforwardimplicationoftherstpartofthetheoremhasalreadybeenproved.The proofthataliftingimpliesafactorizationisthesameasinthecaseof3-isometrictuplessee theproofofTheorem17. Itremainstoshow,givencommuting 3 -symmetricoperators T 1 and T 2 ,with T 1 = exp i T 1 and T 2 =exp i T 2 , that T = T 1 , T 2 2 F c , d forsome c , d > 0 .Towardthisend let Q s 1 , s 2 := I + X 0 < j + k 2 s j 1 s k 2 B j , k =exp is 2 T 2 exp is 1 T 1 exp is 1 T 1 exp is 2 T 2 for s , t 2 R .Bydenition, exp it 2 T 2 exp it 1 T 1 Q s 1 , s 2 exp it 1 T 1 exp it 2 T 2 = Q s 1 + t 1 , s 2 + t 2 . Hencebytermcomparison exp it 2 T 2 exp it 1 T 1 B 0,2 exp it 1 T 1 exp it 2 T 2 = B 0,2 and exp it 2 T 2 exp it 1 T 1 B 2,0 exp it 1 T 1 exp it 2 T 2 = B 2,0 . 58

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Thereexists c and d largeenoughsuchthat I )]TJ/F21 11.9552 Tf 15.798 8.088 Td [(1 c 2 B 0,2 )]TJ/F21 11.9552 Tf 13.676 8.088 Td [(1 d B 2,0 0. Hence, ^ Q T t 1 , t 2 =exp it 2 T 2 exp it 1 T 1 I )]TJ/F21 11.9552 Tf 15.798 8.087 Td [(1 c 2 B 0,2 )]TJ/F21 11.9552 Tf 13.676 8.087 Td [(1 d B 2,0 exp it 1 T 1 exp it 2 T 2 0 asdesired. InthecontextofHeltonandBall'sconjecture1wehaveestablishedanecessaryand sucientconditioninthecase f T n g hascardinalitytwo.Hence,anyattempttosolvethis conjecturewillbemetwithourfactoringcondition. 59

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CHAPTER4 ACOUNTEREXAMPLETOFACTORIZATION Thischapterhasthreeparts.Let Q , beanarbitrarytwovariablequadraticpencil Q , = I + X 0 < j + k 2 j k B j , k {1 withcoecients B j , k operatorsonaseparableHilbertspace H suchthat ^ Q , = Q , )]TJ/F21 11.9552 Tf 15.798 8.088 Td [(1 c 2 B 2,0 )]TJ/F21 11.9552 Tf 16.141 8.088 Td [(1 d 2 B 0,2 0 {2 forall , 2 R 2 .Intherstpartweshowbyconstructionthereexistsacommuting2-tuple of 3 -isometries T 2 F cd suchthat ^ Q T factorsifandonlyif ^ Q factors.Inthesecondpartwe showthatgivenapositiveinteger n andpositivemap : Sym 3 C ! M n , ifthecanonical quadraticpencilitdeterminesfactors,then iscompletelypositive.Hence,anexampleof Choi[10]ofapositive : Sym 3 ! M n whichisnotcompletelypositiveproducesaquadratic twovariablepencilwhichdoesnotfactorwhichinturnproducesacounter-exampletoa naturalgeneralizationofthemainliftingresultof[20].Thiscounter-exampleisstrengthenedin thelastpart. 4.1ConstructingThreeIsometries. Let F beavectorspacewithbasis f f j : j 2 Z g .Inparticular,theset f f j f k : j , k 2 Z g is abasisforthetensorproduct F F .Dene,onthealgebraictensorproduct H F F the sesquilinearform [ h f j f k , h 0 f j 0 f k 0 ] = 8 > < > : h Q j , k h , h 0 i H if j = j 0 and k = k 0 0 otherwise , andthelinearmaps T h f j f k = h f j +1 f k {3 and S h f j f k = h f j f k +1 . {4 60

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Notethatthissesquilinearformispositivesemi-denitesince Q takes,byhypothesis,positive semi-denitevalues.Let H betheHilbertspaceobtainedfrom H F F bymoddingoutby thenullvectorsandformingthecompletion.Wecontinuetodenotetheinnerproducton H by [ , ] andlet h f j f k denotetheequivalenceclassitrepresentsinthequotient.Weusefreely thefactthat D ,thelinearspanof f h f j f k : j , k 2 Z , h 2 H g , isdensein H . Proposition4. Givena2-variablepencilintheformdenedby 4{1 ,ifthereexists c , d 2 R suchthat c > 0 , d > 0 and Q , )]TJ/F21 11.9552 Tf 15.798 8.088 Td [(1 c 2 B 2,0 )]TJ/F21 11.9552 Tf 16.142 8.088 Td [(1 d 2 B 0,2 0 forall , 2 R 2 , thentheoperators S and T denedin 4{3 and 4{4 arewelldened andextendtoinvertibleboundedoperators H .Moreover S and T are3-isometriesand D ^ Q T , S , h f j f k , g f a f b E = j , k , a , b D ^ Q + j , + k h , h E H , where istheKrockerdeltafunction.Inparticular, S , T isintheclass F c , d . Proof. Let h = ^ h f j f k beanelementarytensorandcompute, 2+ c 2 [ h , h ] )]TJ/F21 11.9552 Tf 11.956 0 Td [([ Th , Th ]= D Q j , k +2 c 2 Q j , k )]TJ/F47 11.9552 Tf 11.956 0 Td [(Q j +1, k ^ h , ^ h E = D Q j , k +2 c 2 Q j , k )]TJ/F47 11.9552 Tf 11.955 0 Td [(B 0,1 )]TJ/F47 11.9552 Tf 11.955 0 Td [(kB 1,1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(2 jB 2,0 )]TJ/F47 11.9552 Tf 11.956 0 Td [(B 2,0 ^ h , ^ h E = D Q j , k +2 c 2 Q j , k )]TJ/F47 11.9552 Tf 11.955 0 Td [(B 0,1 )]TJ/F47 11.9552 Tf 11.955 0 Td [(kB 1,1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(2 jB 2,0 + B 2,0 )]TJ/F21 11.9552 Tf 11.956 0 Td [(2 B 2,0 ^ h , ^ h E = D Q j )]TJ/F21 11.9552 Tf 11.955 0 Td [(1, k +2 c 2 Q j , k )]TJ/F21 11.9552 Tf 11.955 0 Td [(2 B 2,0 ^ h , ^ h E . Since Q , )]TJ/F22 7.9701 Tf 15.85 4.708 Td [(1 c 2 B 2,0 )]TJ/F22 7.9701 Tf 16.094 4.707 Td [(1 d 2 B 0,2 0 forall , 2 R 2 , certainly Q )]TJ/F22 7.9701 Tf 15.85 4.707 Td [(1 c 2 B 2,0 0 and Q 0 forall , 2 R 2 .Hence, [2+ c 2 [ h , h ] )]TJ/F21 11.9552 Tf 12.031 0 Td [([ Th , Th ] 0. Usingorthogonalityofthe subspaces f h f j f k : h 2 H g for j , k 2 Z ,itfollowsthatforeach h 2 H F F , 2+ c 2 [ h , h ] [ Th , Th ]. 61

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Thus T isboundedonthealgebraictensorproductandthusextendstoaboundedoperator, stilldenotedby T ,on H bycontinuity.Asimilarcomputationshowsthat S isalsobounded. Itisstraightforwardtoverifythat T 3 T 3 )]TJ/F21 11.9552 Tf 11.955 0 Td [(3 T 2 T 2 +3 T T )]TJ/F47 11.9552 Tf 11.955 0 Td [(I =0, aconditionwellknowntobeequivalentto T beinga3-isometry[4,20].Likewise S isa 3-isometry.Since S and T are3-isometriesthereexist B 1 T , T , B 1 S , S , B 2 T , T and B 2 S , S suchthatforallnaturalnumbers m and n , S m S m = I + mB 1 S , S + m 2 B 2 S , S T n T n = I + nB 1 T , T + n 2 B 2 T , T . Dene, ~ B 1,0 = B 1 T , T , ~ B 0,1 = B 1 S , S , ~ B 2,0 = B 2 T , T , ~ B 0,2 = B 2 S , S , and ~ B 1,1 = B 1,1 I I . {5 Directcomputationshows [ B 1 T , T h f j f k , h f a f b ] = j , k , a , b h B 1,0 + kB 1,1 +2 jB 2,0 h , h i H , {6 [ B 1 S , S h f j f k , g f a f b ] = j , k , a , b h B 0,1 + jB 1,1 +2 kB 0,2 h , g i H , {7 [ B 2 T , T h f j f k , g f a f b ] = j , k , a , b h B 2,0 h , g i H , {8 [ B 2 S , S h f j f k , g f a f b ] = j , k , a , b h B 0,2 h , g i H . {9 Bythedenitionof B 1,1 , h ~ B 1,1 h f j f k , g f a f b i = j , k , a , b h B 1,1 h , g i . {10 Fromtheaboveequationsitfollowsthat [ S m B 1 T , T S m h f j f k , g f a f b ] = j , k , a , b h B 1,0 + k + m B 1,1 +2 jB 2,0 h , g i H . {11 62

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Likewise, [ S m B 2 T , T S m h f j f k , g f a f b ] = j , k , a , b h B 2,0 h , g i . {12 Hence,byequations4{6,4{7,4{8,4{9,4{10,4{11,and4{12, [ S m T n T n S m h f j f k , g f a f b ] = S m + nB 1 T + n 2 B 2 T S m h f j f k , g f a f b = I + mB 1 S + m 2 B 2 S + nS m B 1 T S m + n 2 B 2 T h f j f k , g f a f b = h I + m ~ B 0,1 + n ~ B 1,0 + mn ~ B 1,1 + m 2 ~ B 0,2 + n 2 ~ B 2,0 h f j f k , g f a f b i . Weconclude, Q T , S , = I + ~ B 1,0 + ~ B 0,1 + ~ B 1,1 + 2 ~ B 2,0 + 2 ~ B 0,2 Theaboveequationsgivethefollowingrelationship h Q T , S , h f j f k , g f a f b i = j , k , a , b h Q + j , + k h , g i H and D ^ Q T , S , h f j f k , g f a f b E = j , k , a , b D ^ Q + j , + k h , g E H . Proposition5. Let Q , beaquadraticpenciloftheform 4{1 satisfyingthepositivitycondition 4{2 andlet Q T , S , bethequadraticpencilforthe3-isometric2-tuple T , S 2 F cd constructedinProposition4.Themodiedpencil ^ Q , factorsifandonly ifthemodiedpencil ^ Q T , S , factors. Proof. BytheconclusionofProposition4, D ^ Q T , S , h f j f k , h f a f b E = j , k , a , b D ^ Q + j , + k h , g E H . 63

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Suppose ^ Q T , S , factorsas ^ Q T , S , = V 0 + V 1 + V 2 V 0 + V 1 + V 2 where V j areboundedoperatorsfrom H intosomeauxiliaryHilbertspace.Dene U : H !H by Uh = h f 0 f 0 . {13 Toverifythat U isanisometry,note k Uh k = k h f 0 f 0 k = k Q ,0 1 2 h k = k h k . Nowforall g , h 2 H h U V 0 + V 1 + V 2 V 0 + V 1 + V 2 Uh , g i = D U ^ Q T , S , Uh , g E = D ^ Q T , S , Uh , Ug E = D ^ Q T , S , h f 0 f 0 , g f 0 f 0 E = D ^ Q , h , g E . Thus, ^ Q factorsas ^ Q , =[ V 0 + V 1 + V 2 U ] [ V 0 + V 1 + V 2 U ]. Conversely,supposethat ^ Q , factorsas ^ Q , = V 0 + V 1 + V 2 V 0 + V 1 + V 2 wherethe V j areboundedoperatorsfrom H intoanauxiliaryHilbertspace,whichwelabel K forconvenience.Let ` 2 denotetheHilbertspace ` 2 Z withthestandardorthonormalbasis f e j : j 2 Z g andlet K denotetheHilbertspacetensorproduct K ` 2 ` 2 .Dene,onthe 64

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denseset D ,equaltothespanofelementarytensors h f j f k ,of H into K thelinearmaps, W 0 X h j , k f j f k = X V 0 + jV 1 + kV 2 h j , k e j e k W ` X h j , k f j f k = X V ` h j , k e j e k , for ` =1,2. Since, h W 0 X h j , k f j f k , W 0 X g a , b f a f b i = X j , k h Q j , k h j , k , g a , b i =[ X h j , k f j f k , X h a , b f a f b ], W 0 isanisometryon D andthusextendstoanisometry,stilldenoted W 0 , from H into K . Similarly, h W 1 X h j , k f j f k , W 1 X h a , b f a f b i = X j , k h V 1 h j , k , V 1 h j , k i = X j , k h B 2 S , S h j , k , h j , k i c 2 X j , k h Q j , k h j , k , h j , k i = c 2 [ X h j , k f j f k , X h a , b f a f b ]. Thus W 1 isboundedon D andthusextendstoaboundedlinearoperator,stilldenoted W 1 , from H to K .Ofcourseasimilarstatementholdsfor W 2 . Finally, h W 0 + W 1 + W 2 W 0 + W 1 + W 2 h j , k f j f k , g a , b f a f b i = h W 0 + W 1 + W 2 h jk f j f k , W 0 + W 1 + W 2 h a , b f a f b i = h V 0 + + j V 1 + + k V 2 h j , k , V 0 + + j V 1 + + k V 2 h a , b i = j , k , a , b D ^ Q + j , + k h j , k , h j , k E = D ^ Q T , S h j , k f j f k , h a , b f a f b E . 65

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Hence ^ Q T , S hasthefactorization W 0 + W 1 + W 2 W 0 + W 1 + W 2 . 4.2APositivebutnotCompletelyPositiveMap. InthissectionanexampleofChoiisusedtoproduceatwo-variablequadraticpencilwhich takespositivesemidenitevalueson R 2 ,butdoesnotfactor.Inturnthispencilisused,in Proposition7,togiveacounter-exampletoanaturalgeneralizationofthemainresultof[20]. Denition25. An operatorsystem S isaunitalselfadjointvectorsubspaceofthebounded operatorsonaHilbertspace.Let E i , j denotethematrixunitsfor M n .Thematrix C = E ij i , j 2 M n S isthe Choimatrix ofthelinearmap : M n ! S . Thefollowinglemmacanbefoundin[21] Lemma16. Let S beanoperatorsystem.Amap : M n ! S iscompletelypositiveifand onlyif C ispositivesemidenite. Recallthedenitionsofthe 3 3 matrices B i , j fromequation2{13.Theyformabasis for Sym 3 C . Lemma17. Suppose S isanoperatorsystemand :Sym 3 C ! S isaunitalpositivelinear map.Ifthecanonicalpencil ^ Q , = " I + X 0 < i + j 2 i j B i , j # )]TJ/F21 11.9552 Tf 15.798 8.088 Td [(1 c 2 B 0,2 )]TJ/F21 11.9552 Tf 16.142 8.088 Td [(1 d 2 B 2,0 = X 0 j + k 2 j k B ij associatedto factorsas ^ Q , = V 0 + V 1 + V 2 V 0 + V 1 + V 2 , wherethe V j areoperatorsintoanauxiliaryspace,thenthemap iscompletelypositive. Conversely,ifthemap iscompletelypositive,then ^ Q factors. Proof. Supposethatthecanonicalpencilfactorsas ^ Q , = V 0 + V 1 + V 2 V 0 + V 1 + V 2 . 66

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Anelement X 2 M n Sym C hasthefollowingform X = 0 B B B B @ X 0,0 X 1,0 X 0,1 X 1,0 X 2,0 X 1,1 X 0,1 X 1,1 X 0,2 1 C C C C A . If X 0, theneach X i , j isself-adjointand 0 B B B B @ X 0,0 X 1,0 X 0,1 X 1,0 X 2,0 X 1,1 X 0,1 X 1,1 X 0,2 1 C C C C A = 0 B B B B @ Y 0 Y 1 Y 2 1 C C C C A Y 0 Y 1 Y 2 , wherethe Y j are 3 n n matrices.Thus, m X = X X i , j B i , j = X 0,0 V 0 V 0 + X 1,0 V 0 V 1 + V 1 V 0 + X 0,1 V 0 V 2 + V 2 V 0 + X 1,1 V 1 V 2 + V 2 V 1 + X 2,0 V 1 V 1 + X 0,2 V 2 V 2 = Y 0 V 0 + Y 1 V 1 + Y 2 V 2 Y 0 V 0 + Y 1 V 1 + Y 2 V 2 0. {14 Hence iscompletelypositive. Wepauseatthispointtonotesomedierencesbetweentheniteandinnitedimensional cases.ThereisaHilbertspace E suchthat S B E andthe V j mapintoanauxiliaryHilbert space K .Infact, V j : E! 2 _ i =0 ran V i . Thus,replacing K by W 2 i =0 ran V i ,itcanbeassumedthat V j mapinto E 3 .Thus,if E isnite dimensional,say S M k inwhichcasethereisnoharminassuming S = M k ,thenitcan beassumedthat V j mapintoanauxiliaryspaceofdimensionofatmost 3 k .If E isaninnite dimensionalspace,then E 3 canbeidentiedwith E . 67

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Nowsupposethatthemap :Sym 3 C ! S iscompletelypositiveand S B E .By Lemma16,theChoimatrix C ispositivesemideniteandhencefactors, C = 0 B B B B @ E 00 E 01 E 02 E 10 E 11 E 12 E 20 E 21 E 22 1 C C C C A = 0 B B B B @ V 0 V 1 V 2 1 C C C C A V 0 V 1 V 2 where V j map E intoanauxiliaryHilbertspace.Tocompletetheproof,observethat ^ Q T , = V 0 + V 1 + V 2 V 0 + V 1 + V 2 . Wenowpresentamapon Sym 3 C thatispositivebutnotcompletelypositive.By Lemma17thismapproducesapencilthatdoesnotfactor. Theorem21 Choi . Thereexistsapositivelinearmap :Sym 3 R ! Sym 3 R thatdoes notadmitanexpressionas A = P V > i AV i with 3 3 matrices V i .Themap jk jk 7! 2 0 B B B B @ 11 + 22 00 0 22 + 33 0 00 33 + 11 1 C C C C A )]TJ/F21 11.9552 Tf 11.955 0 Td [( jk jk issuchanexample. Choi'smapisnotunital,sinceitsendsthe I to 3 I .Wecorrectthisdefectbymultiplying byapositivescalar. Wewillshowthatavariationofthismapisnotcompletelypositive. Proposition6. Theunitalpositvemap :Sym 3 C ! Sym 3 C givenby jk jk 7! 2 3 0 B B B B @ 11 + 22 00 0 22 + 33 0 00 33 + 11 1 C C C C A )]TJ/F21 11.9552 Tf 13.151 8.088 Td [(1 3 jk jk jk 2 C {15 isnotcompletelypositive. 68

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Proof. Foramatrix A ,let A denotethematrixwhoseentriesaretheconjugatesofthe entriesof A .Thenotation A and A > willdenotetheconjugatetransposeandtransposeof A respectively.Nowsupposethat iscompletelypositiveandthusextends,viaArveson's extentiontheorem[21],toacompletelypositivemapalsodenotedby from M 3 C to M 3 C .Thus, C ,theChoimatrixof , ispositivesemidenite.Considerthematrix e C = C + C > 2 .Wenotethat e C istheChoimatrixforsomemap : M 3 C ! M 3 C .Fromthis pointonwardwewilldenote e C as C .Sincetranspositionisapositivemap, C isalsoa positivematrixandhence isacompletelypositivemap.HencebyChoi'sTheorem[9],there existnitelymanymatricesoftheappropriatesizesuchthat,for A 2 M 3 C , A = X i V i AV i . {16 Tobeclear,writing C = C jk 3 j , k =1 wherethe C ij 3 3 arematrices,andusing C jk = C kj since C = C C = C + C > 2 = 1 2 0 B B B B @ C 11 C 12 C 13 C 12 C 22 C 23 C 13 C 23 C 33 1 C C C C A + 1 2 0 B B B B @ C > 11 C 12 > C 13 > C > 12 C > 22 C 23 > C > 13 C > 23 C > 33 1 C C C C A . Inparticular, C = C + C 2 . {17 Werstshowthatthemap whenrestrictedto Sym 3 R isthesamemapas restricted to Sym 3 R .Let E jk bethestandardmatrixbasiselementsandnotethefollowingbasis forthesymmetriccomplexmatrices, f E jk + E kj 2 :1 j k 3 g .For i , j =1,2,3, E jk + E kj = C jk + C jk 2 Sym 3 R bydenitionasseenfrom4{15.Hence C jk + C jk = C jk + C jk > . 69

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Thus, E jk + E kj = C jk + C jk > 2 + C jk + C > jk 2 = C jk + C jk 2 + C jk + C jk > 2 = C jk + C jk = E jk + E kj . Hence, j Sym 3 R = j Sym 3 R . By4{17 C isarealsymmetricmatrix.Since C ispositiveithasafactorizationinto tworealmatrices.Thisisequivalenttothefactthat C = P i w > i w i whereeach w i isa 1 9 matrixwithrealentries.Write w i = x i 1 , x i 2 , x i 3 whereeach x i j isa 1 3 matrix.For 1 i 3, formthe 3 3 matrices W i whose j -throwis x i j .Notethat E j , k = P i W > i E j , k W i andby linearity A = P i W > i AW i Hence,thematrices V i intherepresentationof in4{16canbereplacedbyreal matrices W i and A = X i W > i AW i . Since j Sym 3 R = j Sym 3 R , thisisacontradictionofTheorem21. Proposition7. Foreach c , d > 0 thereexistsa3-isometric2-tupleofinvertibleoperators T , S intheaclass F c , d suchthatthepencil ^ Q T , S doesnotfactor.Inparticular,the2-tuple T , S doesnotlifttoa2-tuple J 1 , J 2 intheclass J c , d . 70

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Proof. Given c , d > 0 ,considerthefollowingbasisfor Sym 3 C , B 0,1 = 0 B B B B @ 0 c 0 c 00 000 1 C C C C A ; B 1,0 = 0 B B B B @ 00 d 000 d 00 1 C C C C A ; B 1,1 = 0 B B B B @ 000 00 cd 0 cd 0 1 C C C C A ; B 0,0 = 0 B B B B @ 100 000 000 1 C C C C A ; B 0,2 = 0 B B B B @ 000 0 c 2 0 000 1 C C C C A ; B 2,0 = 0 B B B B @ 000 000 00 d 2 1 C C C C A . {18 Wenote B 0,0 = I )]TJ/F22 7.9701 Tf 15.471 4.707 Td [(1 c 2 B 0,2 )]TJ/F22 7.9701 Tf 15.716 4.707 Td [(1 d 2 B 2,0 .ByProposition21thereexistsaunitalpositivebutnot completelypositivelinearmap :Sym 3 C ! M 3 C .Thus, 0 0 B B B B @ 0 B B B B @ 1 c d c 2 c 2 cd d cd 2 d 2 1 C C C C A 1 C C C C A = X 0 i + j 2 i j B i , j ! = X 0 i + j 2 i j B i , j = ^ Q , . {19 HerewehaveusedthenotationinLemma17.ByLemma17thecanonicalpencil ^ Q , doesnotfactorsince isnotacompletelypoistivemap.Let Q = I + X 0 < i + j 2 e B i , j where e B i , j = B i , j . Note Q , )]TJ/F21 11.9552 Tf 15.798 8.087 Td [(1 c 2 e B 0,2 )]TJ/F21 11.9552 Tf 16.142 8.088 Td [(1 d 2 e B 2,0 = X 0 i + j 2 j k B i , j = ^ Q , . ByProposition4,since ^ Q , 0 wecanconstructa2-tuple T , S intheclass F c , d such that ^ Q T , S , doesnotfactor.ByTheorem17,the2-tuple T , S doesnotlift. 71

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4.3StrengtheningtheCounter-Example Whilethecounter-exampleofPropostion7answersthenaturalquestionofwhether 2-tuples T in F c , d alwayslifttoa2-tuple J intheclass J c , d ,wewillactuallyconstructa strongercounter-example.Givenaquadraticpencilwhichdoesnotfactorwewillconstructa 2-tupleofcommuting3-isometriesthatdoesnotlifttoa2-tuple J inanyoftheclasses J c , d . Let Q , = X 0 i + j 2 i j B ij 0 forall , 2 R 2 {20 beanotnecessarilymonicquadraticpencilwith B ij 2B H whichdoesnotfactor.The existenceofsuchobjectsisgivenbyProposition6.Webeginwiththefollowinglemma. Lemma18. If Q , doesnotfactorintheform Q , = V 0 + V 1 + V 2 V 0 + V 1 + V 2 andif )]TJ/F20 11.9552 Tf 10.095 0 Td [(2B H ispositivesemidenite,then Q , )]TJ/F21 11.9552 Tf 11.955 0 Td [()]TJ/F52 11.9552 Tf 10.677 0 Td [(doesnotfactorintheform Q , )]TJ/F21 11.9552 Tf 11.956 0 Td [()-278(= W 0 + W 1 + W 2 W 0 + W 1 + W 2 . Proof. Weprovethecontrapositive.Accordingly,suppose Q , )]TJ/F21 11.9552 Tf 11.956 0 Td [()-278(= W 0 + W 1 + W 2 W 0 + W 1 + W 2 , inwhichcase Q , = W 0 + W 1 + W 2 W 0 + W 1 + W 2 +. Since, )]TJ/F20 11.9552 Tf 10.096 0 Td [( 0 ,thereexists 2B H suchthat )-278(= .Hence, Q , = 0 B @ 0 B @ W 0 1 C A + 0 B @ W 1 0 1 C A + 0 B @ W 2 0 1 C A 1 C A 0 B @ 0 B @ W 0 1 C A + 0 B @ W 1 0 1 C A + 0 B @ W 2 0 1 C A 1 C A . 72

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Wenowshowthereexistsamonicpencil Q , suchthat Q , )]TJ/F22 7.9701 Tf 15.534 4.707 Td [(1 c 2 B 2,0 )]TJ/F22 7.9701 Tf 15.779 4.707 Td [(1 d 2 B 0,2 doesnotfactorforall c , d forwhich Q )]TJ/F21 11.9552 Tf 15.798 8.088 Td [(1 c 2 B 2,0 )]TJ/F21 11.9552 Tf 16.142 8.088 Td [(1 d 2 B 0,2 0 forall , 2 R 2 . Theorem22. Foreach c 0 , d 0 > 0 thereexistsamonicquadraticpencil Q , = I + X 0 < i + j 2 i j B i j suchthat i Q , )]TJ/F21 11.9552 Tf 15.798 8.087 Td [(1 c 2 0 B 0,2 )]TJ/F21 11.9552 Tf 16.142 8.087 Td [(1 d 2 0 B 2,0 0 forall , 2 R 2 iiif c , d > 0 ,thentheredoesnotexistanauxiliaryHilbertspace K andoperators V 0 , V 1 , V 2 2B H , K suchthat Q , )]TJ/F21 11.9552 Tf 15.798 8.088 Td [(1 c 2 B 0,2 )]TJ/F21 11.9552 Tf 16.142 8.088 Td [(1 d 2 B 2,0 = V 0 + V 1 + V 2 V 0 + V 1 + V 2 . Proof. Let Q , bethenon-monicmatrixvaluedpencilthatdoesnotfactor,i.e. Q , := ^ Q , = X 0 i + j 2 i j B i , j ! where isthemapfromProposition6and ^ Q , isthepencildenedbyEquation4{19 intheproofofProposition7.Therststepistoshowthatwecanassumethat Q ismonic andthatthereexistsa > 0 suchthat Q , I forall , 2 R .Foranoperator A 2B H thenotation A 0 willmeanthatforall x 2 H h Ax , x i 0. 73

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Westartbyconsideringthefollowingpencil Q " , = Q , + " I 0. Hereweneedtochoose "> 0 sothatthe Q , + " I stilldoesnotfactor.ByLemma17 Q , willfactorifandonlyifthemap iscompletelypositive.Themap iscompletely positiveifandonlyifitsChoimatrix C ispositivesemidenitebyLemma16.Since isa unitalmap,andbydenitionof Q , ,wewillhavethat Q , + " I willnotfactorif C + " I isnotpositive.Since C isnotpositiveintherstplace,wesimplyneedtopickan "> 0 smallenoughsothat C + " I isnotpositive.Wenotethat Q , = 0 B B B B @ 0 B B B B @ 1 c 0 d 0 c 0 2 c 2 0 c 0 d 0 d 0 c 0 d 0 2 d 2 0 1 C C C C A 1 C C C C A where c 0 and d 0 comefromthechoiceofbasisasin4{18.Since isaunitalmap Q " , = 0 B B B B @ 0 B B B B @ 1+ " c 0 d 0 c 0 2 c 2 0 + " c 0 d 0 d 0 c 0 d 0 2 d 2 0 + " 1 C C C C A 1 C C C C A . Let Q " , = X 0 i + j 2 i j e B i , j . Inparticular Q " ,0= ~ B 00 " 0. Let = B )]TJ/F23 5.9776 Tf 7.782 3.259 Td [(1 2 0,0 0 andnotethat e Q " , := [ Q , + " I ] 0 andismonic.Nowchoosea > 0 suchthat " I .Hence e Q " , ismonicand e Q " , I . 74

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Withourassumptionsvalidatedfromthispointonwewillassumewehavea monic matrix pencil Q , suchthat Q , I forall , 2 R 2 .Let Q , = I + X 0 < i + j 2 i j B i , j . Forall c , d 2 R 2 suchthat I 1 c 2 B 0,2 + 1 d 2 B 2,0 thepencil Q ismonic, Q , )]TJ/F21 11.9552 Tf 15.798 8.087 Td [(1 c 2 B 0,2 + 1 d 2 B 2,0 0, anddoesnotfactorbyLemma18. Wesummarizeinthefollowingproposition. Proposition8. Thereexists c 0 , d 0 > 0 anda3-isometric2-tupleofinvertibleoperators T , S intheclass F c 0 , d 0 suchthat T , S doesnotlifttoany2-tuple J inanyclass J c , d . Proof. TheprooffollowsfromanapplicationofPropositions4and5andTheorem22. 75

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CHAPTER5 APPLICATIONSTODISCONJUGACYINONEVARIABLE Inthischapter,wediscussapplicationsoftheliftingtheoremstodisconjugacyin one-variable.Thetwovariablecaseistakenupinthenextchapter.Asaninformaldenition, disconjugacy istheassertionthatthentheigenfunction n 0 ofasecondorderoperator withDirichletboundaryconditionsoveracompactinterval I =[ a , b ] R hasexactlynzeros intheinteriorof I [2].Here,theDirichletboundaryconditionsforafunction f refertothe requirementthat f vanishattheendpointsoftheinterval [ a , b ] . Let L 2 = L 2 [0,1] denotethespaceofsquareintegrablecomplex-valuedfunctionsonthe interval [0,1] .Afunction f on [0,1] is absolutelycontinuous ,denoted f 2 AC [0,1] ,if f is dierentiablealmosteverywhere,thederivativeisLebesgueintegrable,and f x = f + Z x 0 f 0 t dt forall x 2 [0,1] [19].Givencontinuousreal-valuedfunctions p and r on [0,1] with p bounded belowawayfrom 0 ,let L denotetheSturm-Liouvilleoperator Lf = )]TJ/F21 11.9552 Tf 9.298 0 Td [( pf 0 0 + rf withDirichletboundaryconditionsandappropriateregularityconditionsi.e.,onthe domain g H 0,1 = 8 > < > : f :[0,1] ! C j f , f 0 2 AC [0,1], f 00 2L 2 , f = f =0 9 > = > ; . Itiswellknownthattheunboundedoperator L isself-adjointonthisdomain,haseigenvalues 0 < 1 < 2 < ... andcorrespondingeigenfunctions f n 2 g H 0,1 , Lf n = n f n whichformacompleteorthonormalbasisfor L 2 [11]. Asstatedintheintroduction,inthepaper[20]aliftingtheoremforaclassof3-isometric operatorsisestablishedandtheliftingtheoremfor3-symmetricoperatorsofAglerandHelton 76

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isrecoveredasacorollary.Itshouldbenotedthattheargumenttheredidnotappealto disconjugacyorotherresultsrelatedtodierentialequations.Rather,inwhatfollows,the liftsforthisclassof3-symmetricand3-isometricoperatorsareusedtoestablishtheknown disconjugacyresultthattheeigenvectorcorrespondingtothesmallesteigenvalueknownasthe groundstate doesnotvanishontheopeninterval. 5.1SobolevSpaces Fortheremainderofchapterweworkwiththefollowingtwovectorsubspacesof L 2 : g H 0,1 = 8 > < > : f :[0,1] ! C f , f 0 2 AC [0,1], f 00 2L 2 , f = f =0 9 > = > ; and g H 0 0,1 = 8 > < > : f :[0,1] ! C f , f 0 2 AC [0,1], f 00 2L 2 , f = f , f 0 = f 0 9 > = > ; . Given Q 2 C [0,1] dene L ontheabovespacestobetheoperator Lf = )]TJ/F47 11.9552 Tf 9.299 0 Td [(f 00 + Qf . Itfollowsviaintegrationbyparts,for Q 2 C [0,1] andfor f , g 2 g H 0,1 or f , g 2 g H 0 0,1 , h Lf , g i L 2 = Z 1 0 )]TJ/F47 11.9552 Tf 9.299 0 Td [(f 00 g + Qf gdt = Z 1 0 f 0 g 0 + Qf gdt )]TJ/F47 11.9552 Tf 11.955 0 Td [(f 0 g 1 0 = Z 1 0 f 0 g 0 + Qf gdt . Denethefollowingbilinearforms h f , g i H 0,1 = h Lf , g i L 2 for f , g 2 g H 0,1 {1 and h f , g i H 0 0,1 = h Lf , g i L 2 for f , g 2 g H 0 0,1 . {2 77

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Wemaketheadditionalassumptionthatbilinearformsarepositivedenite.Hence g H 0,1 and g H 0 0,1 withtheaboveinnerproductarepre-Hilbertspaces.Let H 0,1 and H 0 0,1 ,respectively, denotetheircompletionsundertheseinnerproducts. 5.23-Isometriesand3-SymmetricOperators Inwhatfollowswewillintroducea3-symmetricoperatorforthespace H 0,1 ,itsassociated 3-isometry,anda3-isometryforthespace H 0 0,1 .Wewillconsiderthemsidebyside,pointing outdierencesandsimilaritiesasneeded. 5.2.1A3-SymmetricOperatoranditsAssociated3-IsometryonaFormDomain Let ' :[0,1] ! R , ' t = t .Givenafunction f 2 g H 0,1 ,notethat ' f 2 g H 0,1 .Denea multiplicationoperator T f t = ' t f t = tf t {3 onthispre-Hilbertspace.Ingeneral, ' couldbereplacedbyanycontinuousrealvalued functionon [0,1] . Considerthefollowinginformalcalculation, T 3 )]TJ/F21 11.9552 Tf 11.955 0 Td [(3 T 2 T +3 T T 2 )-222(T 3 f , g = f , T 3 g )]TJ/F21 11.9552 Tf 11.955 0 Td [(3 T f , T 2 g +3 T 2 f , T g )]TJ/F29 11.9552 Tf 11.956 9.684 Td [( T 3 f , g = Z 1 0 f 0 t 3 g 0 + Qf t 3 g dt )]TJ/F21 11.9552 Tf 11.956 0 Td [(3 Z 1 0 tf 0 t 2 g 0 + Q tf t 2 g dt +3 Z 1 0 t 2 f 0 tg 0 + Q t 2 f tg dt )]TJ/F29 11.9552 Tf 11.955 16.273 Td [(Z 1 0 t 3 f 0 g 0 + Q t 3 f gdt . Because ' t = t isareal-valuedfunction, T 3 )]TJ/F21 11.9552 Tf 11.955 0 Td [(3 T 2 T +3 T T 2 )-222(T 3 f , g = Z 1 0 t 3 f 0 g 0 +3 t 2 f 0 g + Qt 3 f gdt )]TJ/F21 11.9552 Tf 11.955 0 Td [(3 Z 1 0 t 3 f 0 g 0 +2 t 2 f 0 g + t 2 f g 0 +2 tf g + Qt 3 f gdt +3 Z 1 0 t 3 f 0 g 0 +2 t 2 f 0 g + t 2 f g 0 +2 tf g + Qt 3 f gdt 78

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)]TJ/F29 11.9552 Tf 11.955 16.272 Td [(Z 1 0 t 3 f 0 g 0 +3 t 2 f 0 g + Qt 3 f gdt =0 Ifwemaketheassumptionthat T isboundedon g H 0,1 theaboveshowsthat T denesa 3-symmetricoperatorthatisnotselfadjoint.Wewillmaketheadditionalassumptionthat T extendstoaboundedlinearoperatoron H 0,1 .Itwillbeusefultolabeltheassociated 3-isometry.Since T isa3-symmetricoperatorthenwehaveautomaticallythat T =exp i T is a3-isometrydenselydenedon H 0,1 .Infact, exp i T f t =exp it f t . Considerthefollowingcalculationon g H 0,1 . h T n f , T n f i = e int f , e int f , so e int f , e int f = Z 1 0 e int f 0 e int f 0 + Q e int f e int f dt = Z 1 0 e int f 0 + infe int e int f 0 + infe int + Q j f j 2 dt = Z 1 0 e int f 0 + infe int e )]TJ/F48 7.9701 Tf 6.587 0 Td [(int f 0 )]TJ/F47 11.9552 Tf 11.955 0 Td [(in fe )]TJ/F48 7.9701 Tf 6.587 0 Td [(int + Q j f j 2 dt = Z 1 0 j f 0 j 2 + inf f 0 )]TJ/F47 11.9552 Tf 11.955 0 Td [(in ff 0 + n 2 j f j 2 + Q j f j 2 dt = n 2 Z 1 0 j f j 2 dt + n Z 1 0 if f 0 )]TJ/F47 11.9552 Tf 11.955 0 Td [(i ff 0 dt + Z 1 0 j f 0 j 2 + Q j f j 2 dt . Thiscalculationshowsthat T isa3-isometryandthat h B 2 T , T f , f i = Z 1 0 j f j 2 dt {4 andviapolarizationwehavethat h B 2 T , T f , g i = Z 1 0 f gdt . {5 79

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Westatetherelevantliftingtheoremfor3-symmetricoperatorsfoundinrecentworkof McCullough[20]. Theorem23 -symmetricliftingtheorem . If 2 B H isa3-symmetricoperator,butnot self-adjoint,then B 2 , 6 =0 .Inthiscase,with c 2 = k B 2 , k , exp )]TJ/F47 11.9552 Tf 9.298 0 Td [(is exp is )]TJ/F21 11.9552 Tf 15.798 8.088 Td [(1 c 2 B 2 , 0 foralls.MoreoverthereexistsaHilbertspace E ,aself-adjointoperator A 2 B E ,andan isometry V : H !EE suchthat V = 0 B @ Aic 0 A 1 C A V . If T isa3-symmetricoperatorand T =exp i T istheassociated3-isometry,thelift V is thesameliftforboth T and T ,andbyconstruction B 2 T , T = B 2 T , T . {6 5.2.2A3-IsometryonaFormDomain Wenowintroduceaspecialclassof3-isometries. Denition26. Givenapositivenumber c ,let F c denotethose3-isometries suchthatthe quadratic Q , s =[ I )]TJ/F21 11.9552 Tf 15.798 8.088 Td [(1 c 2 B 2 ,]+ sB 1 ,+ s 2 B 2 , ispositivesemideniteforallrealnumberss. Let :[0,1] ! C , t =exp i 2 t .Givenafunction f 2 g H 0 0,1 ,notethat f 2 g H 0 0,1 . Thusdeneamultiplicationoperator Tf = t f t =exp i 2 t f t {7 onthispre-Hilbertspace.Itcanbeshownthat T isa3-isometryandwewillmakethe additionalassumptionsthatitisintheclass F c with c 2 = k B 2 T , T k andextendstoa 80

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boundedoperatoron H 0 0,1 .Acomputationsimilartothatin5{4shows h B 2 T , T f , f i =4 2 Z 1 0 j f j 2 dt {8 andviapolarizationwehavethat h B 2 T , T f , g i =4 2 Z 1 0 f gdt . {9 Weagainstatetherelevantliftingtheoremsfrom[20] Theorem24 -isometricliftingtheorem . Anoperator onaHilbertspace H isintheclass F c ifandonlyifthereisanoperator J oftheform = 0 B @ UcU 0 U 1 C A actingonaHilbertspace K andanisometry V : H ! K suchthat V = V . If isinvertible,then,necessarily, V )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 = )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 V .Moreover,inthiscase,thespectrumof isasubsetoftheunitcircle,and canbechosensothat = . 5.3FormulasfortheLifts Inthissectionwederiveaformulafortheisometry V appearinginTheorem23appliedto thepolynomialswhichvanishattheboundaryin g H 0,1 andforallfunctionsin g H 0 0,1 .Wecannot deriveaformulaforageneral f 2 g H 0,1 sincethiscasehasafewtechnicaldiculties,butthe formulawedohavewillbeenoughforthemaintheorem. Weconsidertherstcase.Recall g H 0,1 = 8 > < > : f :[0,1] ! C f , f 0 2 AC [0,1], f 00 2L 2 , f = f 9 > = > ; ispre-Hilbertspacewiththeinnerproductasdenedin5{1anddenoteitscompletionas H 0,1 .Consider T on H 0,1 asdenedin5{3andnotethat T isnotself-adjoint.Letthescalar 81

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c ,theHilbertspace E ,operator J ,andisometry V beasintheconclusionofTheorem23. Thus J = 0 B @ Aic 0 A 1 C A andnote J n = 0 B @ A n icnA n )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 0 A n 1 C A . Wenotethefollowing,let q = t )]TJ/F47 11.9552 Tf 11.955 0 Td [(t , V t n q = V T n q = J n V q . Let V q = 0 B @ F 0 F 1 1 C A . Forevery n 2 N , V t n q = 0 B @ A n icnA n )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 0 A n 1 C A 0 B @ F 0 F 1 1 C A . Bylinearity,forapolynomial p , V pq = 0 B @ p A F 0 + icp 0 A F 1 p A F 1 1 C A . {10 Wecanalsoderiveaformulafortheisometry V appearinginTheorem 16 .Recall g H 0 0,1 = 8 > < > : f :[0,1] ! C f , f 0 2 AC [0,1], f 00 2L 2 , f = f , f 0 = f 0 9 > = > ; isapre-Hilbertspacewithinner-productasdenedin5{2anddenoteitscompletionas H 0 0,1 . Nowconsiderthe3-isometry T = e i 2 t onthespace H 0 0,1 asdenedin5{7.Accordingto Theorem16ittoohasaliftoftheform VT = JV = 0 B @ UcU 0 U 1 C A V . 82

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Necessarilywehaveforall n 2 Z , V e i 2 nt = VT n = J n V = 0 B @ UcU 0 U 1 C A n V V e i 2 nt = 0 B @ U n cUnU n )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 0 U n 1 C A V . Wenotethattrigonometricpolynomials,i.e.polynomialsin exp i 2 t aredenseinthespace H 0 0,1 andthatforatrigonometricpolynomial p ,wehave V p = 0 B @ p U cUp 0 U 0 p U 1 C A V . Consideraxed f 2 g H 0 0,1 .Weusethefactthatfor h continuousandperiodiconthe interval, h 0 continuousontheinterval, h 00 existing,thenatrigonometricseriesconvergingto h maybedierentiatedanditisatrigonometricseriesfor h 0 . Wealsonoteforafunctionofboundedvariationourfunction f beingabsolutely continuoushenceofBVifithasatrigonometricseriesconvergeconvergingtoitthen,the serieswillconvergeuniformly.Let betheinversefor exp i 2 t :[0,1] ! C .Thus, S M t = p z e i 2 t ! f uniformly and S 0 M t = i 2 zp 0 z e i 2 t ! f 0 uniformly , hence p z ! f uniformly and zp 0 z !)]TJ/F47 11.9552 Tf 30.498 8.088 Td [(i 2 f 0 uniformly . 83

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Bycontinuitywemusthavethefollowing: V f = 0 B @ f U )]TJ/F48 7.9701 Tf 13.947 4.707 Td [(i 2 c f 0 U 0 f U 1 C A V . If V = 0 B @ F 0 F 1 1 C A , then V f = 0 B @ f U F 0 )]TJ/F48 7.9701 Tf 16.603 4.708 Td [(i 2 c f 0 U F 1 f U F 1 1 C A . {11 Againwetakenoteofthisformulaandleaveitforthetimebeing.Wewillcomebacktothese formulasafterwemakeaconnectionwithSturm-LiouvilleTheory. 5.4ConnectionswithSturm-LiouvilleTheory WerstshowthattheeigenvaluesoftheSturm-Liouvilleoperator L hasaninverse relationshipwiththeeigenvaluesof B 2 .ConsidertheSturm-Liouvilleoperator L = )]TJ/F47 11.9552 Tf 12.646 8.088 Td [(d 2 dt 2 + Q oneitherspace g H 0,1 or g H 0 0,1 .Wenotethat L onthesespacesisself-adjoint[17].ByEquations 5{1and5{2, h f , g i H 0 0,1 = h Lf , g i L 2 for f , g 2 g H 0 0,1 and h f , g i H 0,1 = h Lf , g i L 2 for f , g 2 g H 0,1 . Let e H beeither g H 0 0,1 or g H 0,1 andlet H denoteeitheroftheirrespectivecompletions.Byvirtue ofEquations5{5and5{9, h B 2 f , g i H hastheform h B 2 f , g i H = k Z 1 0 f gdt , {12 where k issomeconstantand f , g 2 e H . 84

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Theorem25. Let L : e H ! L 2 [0,1] betheSturm-Liouvilleoperator.Then 6 =0 isan eigenvaluefor L ifandonlyif 1 isaeigenvaluefor B 2 : e H ! H .Furthermoretheorthonormal basisofeigenfunctionsof L in L 2 composedoffunctionsin e H isanorthonormalbasisof eigenfunctionsof B 2 in H . Proof. Forthesimplicityoftheproofwewillignoretheconstant k in5{12.Since L is self-adjointon e H ,thereexists f n g and f n 2 e H suchthat Lf n = n f n , 0 < 1 ... ,and f f n g is anorthonormalbasisfor L 2 [0,1] .If 6 =0 isaneigenvaluefor L withassociatedeigenvector f 2 e H then,for g 2 e H h f , g i H = h Lf , g i 2 = h f , g i 2 = Z 1 0 f gdt = h B 2 f , g i H . {13 Wenotethat5{13istrueforall g 2 e H ,sowehavethat h B 2 f , g i H = 1 h f , g i H = 1 f , g H , forall g 2 e H . Since e H isdensein H , B 2 f = 1 f . Nowsuppose isaneigenvaluefor B 2 actingon e H ,wewanttoshowthat 1 isaneigenvalue for L on L 2 [0,1] .Soconsiderthefollowing,forall g 2 e H h f , g i 2 = Z 1 0 f gdt = h B 2 f , g i H = h f , g i H = h L f , g i 2 = h Lf , g i 2 = h Lf , g i 2 . Hencewehavethat h f , g i 2 = h Lf , g i 2 forall g 2 e H . Since e H isdensein L 2 [0,1] wehavethat Lf = 1 f and 1 isaneigenvaluefor L . 85

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Let f f n g betheorthonormaleigen-basiscomposedoffunctionsfrom e H .Toprove f f n g is complete,supposethat h f n , g i H =0 forall f n .Let g 2 H ,andlet g m ! g ,whereeach g m is in e H .Fixan n 0 thenconsiderthefollowing, h f n 0 , g m i H = h L f n 0 , g m i 2 = h n 0 f n 0 , g i 2 . So n 0 h f n 0 , g m i 2 = h f n 0 , g m i H !h f n 0 , g i =0. {14 Since5{14holdsforallfunctionsinanorthonormalbasisof L 2 weseethat g m ! 0 weakly in L 2 . Ontheotherhandif h 2 Dom L then, L h 2L 2 andthus h g m , h i H = h g m , L h i 2 ! 0. But h g m , h i H !h g , h i H , so h g , h i H =0 forall h 2 Dom L .SinceDomLisadensesubset g H 0,1 or g H 0 0,1 wehavethat g =0 on H . Fromtheaboverelationsorthogonalityisnotdiculttoprove,since h f n , f m i H = h L f n , f m i 2 = n h f n , f m i = 8 > < > : n : n = m 0: n 6 = m Since f f n g iscompletein H wehavethat n 1 n f n o isanorthonormalbasisfor H . When L isself-adjoint,itseigenvaluesarereal,hencetheeigenvaluesof B 2 mustalso bereal.Sincetheeigenvaluesof L ,say f n g musttendtowardinnitywehavethat B 2 is compact.Wenotethatsince B 2 iscompactandself-adjoint,itslargesteigenvalueis k B 2 k . Let f betheeigenfunctionassociatedwiththelargesteigenvalueof B 2 whichwedenotewith 86

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c 2 = k B 2 k .Thus, c 2 )]TJ/F47 11.9552 Tf 11.955 0 Td [(B 2 f =0 andbyTheorem25wehavethat 1 c 2 )]TJ/F47 11.9552 Tf 11.955 0 Td [(L f =0. Hence f isasolutionfortheeigenvalueproblem, L f = 1 c 2 f . Wewillshowthatthis f hastosatisfyarstorderdierentialequation. WeapplytheFredholmtheoryforcompactoperatorsfoundin[7].Forthecase T on H 0 0,1 ,since B 2 T , T iscompact,theneither ker I )]TJ/F21 11.9552 Tf 15.798 8.088 Td [(1 c 2 B 2 T , T 6 =0 or I )]TJ/F21 11.9552 Tf 15.798 8.088 Td [(1 c 2 B 2 T , T isinvertible . Since c 2 ischosentobethenormof B 2 T , T ,thatmeansthenormof c )]TJ/F22 7.9701 Tf 6.587 0 Td [(2 B 2 T , T is1.Sincethespectralradiusisequaltothenormforpositiveoperators,wehavethat 1 2 c )]TJ/F22 7.9701 Tf 6.587 0 Td [(2 B 2 T , T ,hence I )]TJ/F22 7.9701 Tf 15.997 4.707 Td [(1 c 2 B 2 T , T cannotbeinvertible.Sotheremustbe somenon-zero f inthekernel.Wenotethatthis f mustbeaneigenfunctionforthesmallest eigenvalueof L on L 2 .Asimilarargumentwillholdfor T and B 2 T , T .Wesumthese argumentsupinthefollowinglemmas Lemma19. Thereexistsannon-zerofinthekernelof I )]TJ/F22 7.9701 Tf 15.167 4.708 Td [(1 c 2 B 2 T , T Lemma20. Thereexistsannon-zerofinthekernelof I )]TJ/F22 7.9701 Tf 15.167 4.707 Td [(1 c 2 B 2 T , T 5.5Disconjugacy Let'sreiteratethefactthatif T isa3-symmetricoperatorand T =exp i T isthe associated3-isometry,thenthelift V isthesameliftforboth T and T ,andbyconstruction B 2 T , T = B 2 T , T . 87

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Asimpleformulaexistsforcalculating B 2 T , T asapolynomialin T and T foundin[20] andwenotebytheliftthat B 2 T , T = V B 2 J , J V .Itfollowsfromcalculationthat B 2 T , T = V B 2 J , J V = V 0 B @ 00 0 c 2 1 C A V and I )]TJ/F21 11.9552 Tf 15.798 8.088 Td [(1 c 2 B 2 T , T = I )]TJ/F21 11.9552 Tf 15.798 8.088 Td [(1 c 2 B 2 T , T = V I )]TJ/F21 11.9552 Tf 15.798 8.088 Td [(1 c 2 B 2 J , J V . So I )]TJ/F21 11.9552 Tf 15.798 8.087 Td [(1 c 2 B 2 T , T = I )]TJ/F21 11.9552 Tf 15.798 8.087 Td [(1 c 2 B 2 T , T = V 0 B @ 10 00 1 C A V . {15 Let W betheunitaryoperatorgivenbyTheorem2,i.e. W AWf t = M f t = t f t . {16 Itisnotdiculttoseebythedensityofpolynomialsin L 2 A , thatforacontinuous function F , W F A Wf t = F t f t . {17 Itshouldalsobenotedthat M ismultiplicationby t on A . Wewillalsomakeuseofthefollowinglemma. Lemma21. Polynomialsaredensein g H 0,1 . Proof. Let f 2 g H 0,1 ,so f 0 iscontinuousandthereexistspolynomials r n ! f 0 uniformly. Denote, r n t = a m , n t m + a m )]TJ/F22 7.9701 Tf 6.586 0 Td [(1, n t m )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 +...+ a 1, n t + a 0, n andlet p n t = a m , n m +1 t m +1 + a m )]TJ/F22 7.9701 Tf 6.586 0 Td [(1, n m t m +...+ a 1, n 2 t 2 + a 0, n t . Since, p 0 n = r n ! f 0 uniformlyand p n ! f ,wehave p n ! f uniformly.Wethenlet, n = p n 88

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and q n = p n )]TJ/F21 11.9552 Tf 11.955 -0.166 Td [( n t . Wehavethat n ,convergesto 0 bytheuniformconvergenceof p n to f , i.e.since p n ! f =0 .Wehavethat, q 0 n ! f 0 uniformly q n ! f =0 so q n ! f uniformly and q n = q n =0 foralln . Wenotethatthepolynomials q n arein g H 0,1 ,and q n ! f and q 0 n ! f 0 uniformly,thus q n ! f in g H 0,1 .Hencepolynomialsaredenseinthespace g H 0,1 . Let f beanon-zerofunctionin ker )]TJ/F47 11.9552 Tf 5.48 -9.684 Td [(I )]TJ/F22 7.9701 Tf 15.166 4.707 Td [(1 c 2 B 2 T , T ,whichisguaranteedbyLemma20. Thereexists, f p n g ,asequenceofpolynomialsapproaching f 2 g H 0,1 ,suchthateach p n = g n q , where q = t t )]TJ/F21 11.9552 Tf 12.674 0 Td [(1 .Wealsohavethat V p n ! V f .Notethatbytheabovederived formula5{10havethat V p n = 0 B @ g n A F 0 + icg 0 n A F 1 g n A F 1 1 C A . By5{15wehavethat V 0 B @ 10 00 1 C A V f = V 0 B @ 10 00 1 C A 0 B @ 10 00 1 C A V f =0. Wenotethat I )]TJ/F22 7.9701 Tf 15.167 4.707 Td [(1 c 2 B 2 T , T 0 ,hence 0 B @ 10 00 1 C A V f =0. {18 89

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Since )]TJ/F47 11.9552 Tf 5.48 -9.684 Td [(I )]TJ/F22 7.9701 Tf 15.166 4.707 Td [(1 c 2 B 2 J , J isaboundedoperatorwehavethat 0 B @ 10 00 1 C A V p n ! 0 B @ 10 00 1 C A V f =0. Futhermore, 0 B @ 10 00 1 C A V p n = g n A F 0 + icg 0 n A F 1 . BytheSpectralTheorem2andbyEquation5{17thereexistssomeunitaryoperator W such that W g n A WF 0 t + icW g 0 n A WF 1 t = g n t F 0 t + icg 0 n t F 1 t . {19 Since f p n g convergesto f in H 0,1 , 0 B @ 10 00 1 C A V p n ! 0 B @ 10 00 1 C A V f =0 {20 andhenceby5{19and5{20, g n t F 0 t + icg 0 n t F 1 t ! 0. Sothereisaquestionofwhatthe g n mustconvergeto,butifweconsideranycompact sub-intervalof [0,1] ,wecananswerthis. g n ! f q uniformlyonanycompactsub-intervalsince p n ! f and p 0 n ! f 0 uniformly.Sowemusthave thatonanycompactsubintervalof [0,1] thefollowing: f q t F 0 t + ic f q 0 t F 1 t =0. {21 So f satisesthisrst-orderdierentialequationonanycompactsubintervalof [0,1] . 90

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Thecasefor T ismucheasiersinceweknowwhattheisometrydoestoall f 2 g H 0 0,1 . AgainsimilartoEquation5{15, I )]TJ/F21 11.9552 Tf 15.798 8.088 Td [(1 c 2 B 2 T , T = V 0 B @ 10 00 1 C A V . {22 ByLemma19wehavea ~ f 6 =0 suchthat I )]TJ/F22 7.9701 Tf 16.602 4.707 Td [(1 c 2 B 2 T , T ~ f =0 ,where I )]TJ/F22 7.9701 Tf -404.913 -19.201 Td [(1 c 2 B 2 T , T 0 ,i.e. ~ f isaneigenfunctionforboth B 2 and L .Thiseigenfunctionagain livesinthedomainof L whichis g H 0 0,1 .Usingsimilarlogictotheabovewemusthavethat ~ f t F 0 t )]TJ/F47 11.9552 Tf 17.878 8.088 Td [(i 2 c ~ f 0 t F 1 t =0 {23 butthistimefor t 2 U = @ D . 91

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CHAPTER6 LIFTINGFOR3-ISOMETRIESONSOBOLEVSPACES Inthissectionalternateclassesof 2 -tuplesofcommuting 3 -isometricor 3 -symmetric operatorsareintroducedandliftingtheoremsanalogoustoTheorem17andTheorem20are established.Theseliftingtheoremsaretailoredforuseinthefollowingsectiontoestablisha 2 -variabledisconjugacyresult. 6.1DierentialPencils Aswewillseeinthenextsection,thefollowingclassof 2 -tupleshasaconnectionto secondorderellepticdierentialoperators. Denition27. Fixapositiverealnumber c .A2-tupleofcommuting3-isometries T = T 1 , T 2 isintheclass D c if Q T , = I + B 0,1 + B 1,0 + 2 + 2 B 2 and ^ Q T , = Q T , )]TJ/F21 11.9552 Tf 15.798 8.088 Td [(1 c 2 B 2 0 forall , 2 R 2 .Wewillsometimesreferto Q T asa dierential pencil,and ^ Q T asa modied dierentialpencil. Thefollowingdenitionidentiesacanonicalclassofmodeloperatorsfortheclass D c . Denition28. Given c > 0 a2-tuple J = J 1 , J 2 isintheclass DJ c if J 1 = 0 B B B B @ U 1 0 cU 1 0 U 1 0 00 U 1 1 C C C C A , J 2 = 0 B B B B @ U 2 00 0 U 2 cU 2 00 U 2 1 C C C C A . {1 forsomeunitaryoperators U 1 , U 2 thatcommute. 92

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Given J 2 DJ c ,compute,fornon-negativeintegers m , n , J n 1 = 0 B B B B @ U n 1 0 ncU n 1 0 U n 1 0 00 U n 1 1 C C C C A , J m 2 = 0 B B B B @ U m 2 00 0 U m 2 mcU m 2 00 U m 2 1 C C C C A and J m 2 J n 1 J n 1 J m 2 = 0 B B B B @ 10 nc 01 mc ncmcc 2 n 2 + m 2 +1 1 C C C C A . {2 6.2Liftingfor3-Isometricand3-SymmetrictupleswithDierentialPencils Theorem26. A3-isometric2-tuple T = T 1 , T 2 intheclass D c liftstoa2-tuple J = J 1 , J 2 intheclass DJ c ifandonlyifthethequadraticpencil ^ Q T , factorsintheform, ^ Q T , = V 0 + V 1 V 0 + V 1 + W 0 + W 1 W 0 + W 1 Inourfactorizationofthepencil ^ Q T notetheabsenceofacrossaswellas V 0 V 0 + W 0 W 0 = I )]TJ/F21 11.9552 Tf 15.798 8.087 Td [(1 c 2 B 2 and V 1 V 1 = W 1 W 1 . Theproofofthistheoremwilloccupytheremainderofthissection.Wewillrelyheavilyonthe workdoneinChapter2. Let T 2 D c andlet J 1 = 0 B B B B @ U 1 0 cU 1 0 U 1 0 00 U 1 1 C C C C A , J 2 = 0 B B B B @ U 2 00 0 U 2 cU 2 00 U 2 1 C C C C A . {3 93

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bepairwiserotationallysymmetricJordanoperators.WemodifyLemmas5and6forour dierentialpencils. Lemma22. If T = T 1 , T 2 isintheclass D c ,and ^ Q T , = Q T , )]TJ/F21 11.9552 Tf 15.798 8.088 Td [(1 c 2 B 2 T 0 factorsintheform, ^ Q T , = V 0 + V 1 V 0 + V 1 + W 0 + W 1 W 0 + W 1 {4 thenthemap J 2 J 1 J 1 J 2 = T 2 T 1 T 1 T 2 iswelldenedandcompletelypositive. Proof. Supposethe2-tuple T = T 1 , T 2 isintheclass D c and ^ Q T factorsasindicated.For notationalconveniencelet B 0,0 T = I )]TJ/F21 11.9552 Tf 15.798 8.088 Td [(1 c 2 B 2 T . Notethat B 0,0 T 0 since Q T , 0 for = =0. Thespaces H s J 1 , J 2 and H s T 1 , T 2 arespannedby f B 0,0 J , B 1,0 J , B 0,1 J , B 2 J g and f B 0,0 T , B 1,0 T , B 0,1 T , B 2 T g respectively.Forpositiveintegers n ,recallthat M n denotesthe n n matrices.Theelements X 2 M n H s J 1 , J 2 havetheform X = X 00 B 00 J + X 01 B 01 J + X 10 B 10 J + X 2 B 2 J 94

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equivalently X = 0 B B B B @ X 0,0 0 cX 0,1 0 X 0,0 cX 1,0 cX 0,1 cX 1,0 c 2 X 2 1 C C C C A I . If X 0 ,theneach X i , j isselfadjoint.Further X 0 ifandonlyif X = 0 B B B B @ X 0,0 0 X 0,1 0 X 0,0 X 1,0 X 0,1 X 1,0 X 2 1 C C C C A I . isaswell.Inthiscase,thereexists 3 n n matrices Y 0 , Y 1 , Y 2 suchthat 0 B B B B @ X 0,0 0 X 0,1 0 X 0,0 X 1,0 X 0,1 X 1,0 X 2 1 C C C C A = 0 B B B B @ Y 0 Y 1 Y 2 1 C C C C A Y 0 Y 1 Y 2 . Inparticular, Y 0 Y 1 =0= Y 1 Y 0 . Usingthefactorization6{4, 1 m X = X X i , j B i , j T = X 0,0 V 0 V 0 + X 0,1 V 0 V 1 + V 1 V 0 + X 1,0 W 0 W 1 + W 1 W 0 + X 2 V 1 V 1 + W 1 W 1 = Y 0 V 0 + Y 2 V 1 Y 0 V 0 + Y 2 V 1 + Y 1 W 0 + Y 2 W 1 Y 1 W 0 + Y 2 W 1 . {5 Sincetherighthandsideisevidentlypositive,themap iscompletelypositive. ByProposition1andLemma22since J 1 and J 2 arepair-wiserotationallysymmetric, wehaveshownafactorizationimpliesthereisarepresentation suchthatthe2-tuple T lifts tothe2-tuple J .Itremainstoshowthatanyrepresentationappliedto J = J 1 , J 2 producesa2-tupleofthesameform. 95

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Lemma23. Let E betheHilbertspacethat J 1 and J 2 acton.If e E isalsoaHilbertspace and : B E ! B e E isaunital -representation,then J 1 = J 1 and J 2 = J 2 have,up tounitaryequivalence,thesameformas J 1 and J 2 givenbyEquation 6{1 andinparticular areintheclass DJ c . Proof. Theproofproceedsmuchinthesamewayasitdoesin[20]butwithsomeminor dierences.Thefollowingrelationsareevident. i J = W i + N i where W i isunitary, N 2 i =0 for i =1,2 . ii W i N i = N i W i for i =1,2 . iii N 1 N 1 = N 2 N 2 . iv N 1 N 1 + N 2 N 2 + N 2 N 2 =1 v N i N j =0 for i , j =1,2 . Fromtheserelations, N 1 N 1 , N 2 N 2 , N 1 N 1 = N 2 N 2 arepairwiseorthogonalprojections.Let J i = J i , N i = N i and W i = W i for i =1,2 . Thesemustsatisfythesamealgebraicrelations,i.e. i J = W i + N i where W i isunitary, N 2 i =0 for i =1,2 . ii W i N i = N i W i for i =1,2 . iii N 1 N 1 = N 2 N 2 . iv N 1 N 1 + N 2 N 2 + N 2 N 2 =1 v N i N j =0 for i , j =1,2 . Fromtheserelations, N 1 N 1 , N 2 N 2 , 96

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N 1 N 1 = N 2 N 2 arepairwiseorthogonalprojectionson e E .Forinstance, N 1 N 1 = N 1 N 1 N 1 + N 2 N 2 + N 1 N 1 N 1 = N 1 N 1 2 Nowdecomposethespace H as H =ran N 1 N 1 ran N 2 N 2 ran N 2 N 2 .Themappings N j areunitarymaps Q j fromtherangeof N j totherangeof N j .Hence,withrespecttothe orthogonaldecompositionof H as H =ran N 1 N 1 ran N 2 N 2 ran N 2 N 2 , N 1 = 0 B B B B @ 00 Q 1 000 000 1 C C C C A andlikewise, N 2 = 0 B B B B @ 000 00 Q 2 000 1 C C C C A . Thus,uptounitaryequivalence,itmaybeassumedthat Q j = I andeachofthesummandsin thedirectsumdecompositionisthesameHilbertspace.Write W 1 = 0 B B B B @ A 1 B 1 C 1 D 1 E 1 F 1 G 1 H 1 J 1 1 C C C C A forsome A 1 , B 1 , C 1 , D 1 , E 1 , F 1 , G 1 , H 1 ,and J 1 operators.Since W 1 N 1 = N 1 W 1 , W 1 N 1 = 0 B B B B @ A 1 B 1 C 1 D 1 E 1 F 1 G 1 H 1 J 1 1 C C C C A 0 B B B B @ 00 I 000 000 1 C C C C A = 0 B B B B @ 00 A 1 00 D 1 00 G 1 1 C C C C A 97

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and N 1 W 1 = 0 B B B B @ 00 I 000 000 1 C C C C A 0 B B B B @ A 1 B 1 C 1 D 1 E 1 F 1 G 1 H 1 J 1 1 C C C C A = 0 B B B B @ G 1 H 1 J 1 000 000 1 C C C C A weconclude A 1 = J 1 and D 1 = H 1 = G 1 =0. Similarly,since W 1 N 2 = N 2 W 1 E 1 = J 1 and B 1 =0. Hence W 1 = 0 B B B B @ A 1 0 C 1 0 A 1 F 1 00 A 1 1 C C C C A . Since W 1 isaunitaryoperator, W 1 W 1 = 0 B B B B @ A 1 0 C 1 0 A 1 F 1 00 A 1 1 C C C C A 0 B B B B @ A 1 00 0 A 1 0 C 1 F 1 A 1 1 C C C C A = 0 B B B B @ I 00 0 I 0 00 I 1 C C C C A , where I istheidentityoperator.Hence, A 1 A 1 + C 1 C 1 = I , A 1 A 1 + F 1 F 1 = I , A 1 A 1 = I , 98

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Notethattherelationsaboveshowthat F 1 = C 1 =0 and A 1 isanisometry.Hence W is diagonalwith A 1 downthediagonal.Since W isunitary, A 1 isunitary.Itfollowsthat W 1 = 0 B B B B @ U 1 00 0 U 1 0 00 U 1 1 C C C C A , where U 1 isaunitaryoperator.Asimilarargumentshowsthat, W 2 = 0 B B B B @ U 2 00 0 U 2 0 00 U 2 1 C C C C A where U 2 isaunitaryoperator.Since [ W 1 , W 2 ]=0 wehavethat [ U 1 , U 2 ]=0 .Hence,upto unitaryequivalencewehavethecorrectformsforeach J i . WenowneedtoprovethebackwardsassertionofTheorem26.Howeverthisisreadily available.If T = T 1 , T 2 2 D c liftsto J = J 1 J 2 2 DJ c ,then V )]TJ/F47 11.9552 Tf 5.48 -9.684 Td [(Q J 1 , J 2 , )]TJ/F21 11.9552 Tf 15.798 8.088 Td [(1 c 2 B 2 V = Q T 1 , T 2 , )]TJ/F21 11.9552 Tf 15.798 8.088 Td [(1 c 2 B 2 . Henceanyfactorizationof ^ Q J , = K 0 + K 1 K 0 + K 1 + L 0 + L 1 L 0 + L 1 givesthefactorizationof ^ Q T as Q T , = V [ K 0 + K 1 K 0 + K 1 + L 0 + L 1 L 0 + L 1 ] V Since ^ Q J factorsas ^ Q J , = 0 B B B B @ 0 B B B B @ 1 0 c 1 C C C C A 10 c + 0 B B B B @ 0 1 c 1 C C C C A 01 c 1 C C C C A I , 99

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theconclusionfollows. Denition29. Fixapositiverealnumber c .A2-tupleofcommuting3-symmetric T = T 1 , T 2 isintheclass D s c if Q T , = I + B 0,1 + B 1,0 + 2 + 2 B 2 and ^ Q T , = Q T , )]TJ/F21 11.9552 Tf 15.798 8.088 Td [(1 c 2 B 2 0 forall , 2 R 2 . Wenotethat T2 D s c ifandonlyif exp i T 1 ,exp i T 2 2 D c . Thefollowingdenitionidentiesacanonicalclassofmodeloperatorsfortheclass D s c . Denition30. Given c > 0 a2-tuple J = J 1 , J 2 isintheclass DJ s c if J 1 = 0 B B B B @ A 1 0 ic 0 A 1 0 00 A 1 1 C C C C A , J 2 = 0 B B B B @ A 2 00 0 A 2 ic 00 A 2 1 C C C C A . {6 forsomeselfadjointoperators A 1 , A 2 thatcommute. Wenotethat J2 DJ s c ifandonlyif exp i J 1 ,exp i J 2 2 DJ c . Theproofofthefollowingtheoremiseasilyestablishedfor D s c and DJ s c byagainrelying ontheworkinChapter2.Thenecessarychangesaresupercial. Theorem27. Tuplesof3-symmetricoperators T 1 , T 2 2 D s c willlifttoa2-tuple J 1 , J 2 2 DJ s c oftheforms J 1 = 0 B B B B @ A 1 0 ic 0 A 1 0 00 A 1 1 C C C C A , J 2 = 0 B B B B @ A 2 00 0 A 2 ic 00 A 2 1 C C C C A . ifandonlyifforsomeoperators V 0 , V 1 and V 2 in B H thepolynomial ^ Q T , factorsin theform, ^ Q T , = V 0 + V 1 V 0 + V 1 + W 0 + W 1 W 0 + W 1 . 100

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CHAPTER7 APPLICATIONSTODISCONJUGACYINSEVERALVARIABLES WeTheorem27to3-symmetricoperatortuplestoobtainatwovariabledisconjugacy theorem. 7.1DirichletBoundaryConditions HerewewillbegintousethestandardSobolevspacenotation.Webeginwiththe followingdenitionsfrom[13]. Denition31. Let beanopensetin R n .Fix p with 1 p < 1 and k apositiveinteger. Dene W k , p = f u 2 L p j D u 2 L p forall j j k g where isamulti-indexand D istheweakderivative.Furthermore,if u 2 W k , p deneits normtobe k u k W k , p = 0 @ X j j k Z U j D u j p dx 1 A 1 = p . Itisawellknownthat,for p =2 , W k ,2 isaHilbertspaceandwereferto H k = W k ,2 astheHilbert-Sobolevspace.Further,toworkwithwhatareknownasDirchletboundary conditionswewillfocusonasubspaceof H k . Denition32. H k 0 = W k , p 0 = C 1 c kk W k , p , where C 1 c arethesmoothfunctionswithcompactsupport. ConsiderthefollowingsimpletwovariableversionoftheSturm-Liouvilleoperators.Let =,1 ,1 anddene L = )]TJ/F21 11.9552 Tf 9.299 0 Td [(+ Q 101

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where Q iscontinuouson and istheLaplacian.Asbefore L determinesabilinearformon H 1 0 H 1 0 .Givenfunctions u x , y and v x , y in H 1 0 dene, [ u , v ] = h Lu , v i 2 = ZZ )]TJ/F21 11.9552 Tf 9.298 0 Td [( u v + Qu vdxdy = ZZ [ @ u @ x @ v @ x + @ u @ y @ v @ y + Qu v ] dxdy . {1 Weneedthefollowingdenition. Denition33. Wesaythepartialdierentialoperator L = )]TJ/F48 7.9701 Tf 18.545 14.944 Td [(n X i , j =1 a i , j x u x i x j + n X i =1 b i u x i + cu subjecttothecondition a i , j = a j , i i , j =1,... n isuniformlyellipticifthereexistsaconstant > 0 suchthat n X i , j =1 a i , j x i j j j 2 foralmostevery x 2 andall 2 R n Theoperator L = )]TJ/F21 11.9552 Tf 9.298 0 Td [(+ Q isclearlyuniformlyelliptic.Inaddition,uniformlyelliptic operators,haveaorthonormalsetofeigen-functionswhicharesolutionsintheweaksense.We statethefollowingtheoremwhoseproofisfoundin[16]. Theorem28. Considerthesymmetric,uniformlyellipticoperatoroftheform Lu = )]TJ/F48 7.9701 Tf 18.545 14.944 Td [(n X i , j =1 @ i a i , j @ j u + cu where a i , j = a j , i and a i , j , c 2 L 1 . Itdeterminesthesymmetricbilinearform, [ , ] : H 1 0 H 1 0 ! R [ u , v ] = Z n X i , j =1 i , j @ i u @ j v + cuv . 102

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Thereisanorthonormalbasis f n : n 2 N g of L 2 consistingofeigenfunctions n 2 H 1 0 suchthatfor f 2 H 1 0 , [ n , f ] = n h n , f i 2 wheretheeigenvaluesareofnitemultiplicity 1 2 ... n ... and n !1 . Wewillassumethisbilinearformtobepositivedeniteon H 1 0 andhenceaninner product.Aswedidinonevariable,weconsidertheformdomainbytakingthecompletion of H 1 0 undertheinner-product [ f , g ] ,wewillcallthisspace H D .Wenotethatif Q q > 0 then H 1 0 undertheinner-productdenedinEquation7{1isalreadycomplete. Indeed,with Q =1 wehavethesubspaceoftheHilbertSobolevspaceinDenition32. Moreover,if Q q > 0 ,aliftingforthe3-symmetricand3-isometricoperatorstobedened belowaretrivialtoproduceasintheonevariablecase.Weformalizethedenitioninthe aboveparagraph. Denition34. Denethefollowinginner-producton H 1 0 usingtheinner-productdenedin 7{1 h f , g i H D = [ f , g ] andlet H D = H 1 0 kk H D Theoperators X : H 1 0 ! H 1 0 ; X f = xf x , y {2 and Y : H 1 0 ! H 1 0 ; Y f = yf x , y {3 determinedenselydenedmultiplicationoperatorson H D .Notethat X and Y commute. Assume X and Y areboundedcommuntingoperatorsandthusextendtoboundedcommuting operatorson H D . 103

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For f 2 H 1 0 andconstants , 2 R , e i X e i Y f , e i X e i Y f H = ZZ @ @ x e i x e i y f @ @ x e )]TJ/F48 7.9701 Tf 6.587 0 Td [(i x e )]TJ/F48 7.9701 Tf 6.586 0 Td [(i y f dxdy + ZZ @ @ y e i x e i y f @ @ y e )]TJ/F48 7.9701 Tf 6.586 0 Td [(i x e )]TJ/F48 7.9701 Tf 6.587 0 Td [(i y f dxdy + ZZ Q j f j 2 dxdy . {4 Thus, e i X e i Y f , e i X e i Y f H = 2 + 2 ZZ j f j 2 dxdy + ZZ if @ f @ x )]TJ/F47 11.9552 Tf 11.955 0 Td [(i f @ f @ x + ZZ if @ f @ y )]TJ/F47 11.9552 Tf 11.955 0 Td [(i f @ f @ y dxdy + ZZ [ @ f @ x 2 + @ f @ y 2 + Q j f j 2 ] dxdy . Thus X , Y isa3-symmetrictuple.Moreover,wewillassumewith c 2 = k B 2 k that X , Y areintheclass D s c . 7.2PolynomialApproximations ThefollowingtheoremisthetwovariableanalogofTheorem21. Theorem29. Polynomials p 2 C [ x , y ] suchthat p x , y =0 forall x , y 2 @ [0,1] 2 are densein H 1 0 ,1 2 . Proof. Note C 1 c ,1 2 isdensein H 1 0 ,1 2 bydenition.Let f 2 H 1 0 ,1 2 and "> 0 . Choosea 2 C 1 c ,1 2 suchthat k f )]TJ/F24 11.9552 Tf 12.299 0 Td [( k W 1,2 <" .Notesupp ,1 2 ,since has compactsupport.Alsonotethat is C 2 [0,1] 2 andzeroontheboundaryofthesquare.By applicationofBernsteinpolynomials,thereisasequenceofpolynomials p n x , y suchthat p n ! and @ i p n ! @ i uniformlyon [0,1] 2 .Morespecically,aresultofKingsley[18]states if B m , n istheBernsteinpolynomialassociatedwiththefunction B m , n x , y = n X p =0 m X q =0 p n , q m n , p x m , q y where n , p z = C n , p z p )]TJ/F47 11.9552 Tf 11.955 0 Td [(z n )]TJ/F48 7.9701 Tf 6.586 0 Td [(p , 104

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then,for i =0,1,2 , lim m , n !1 @ 2 B m , n @ x i @ y 2 )]TJ/F48 7.9701 Tf 6.587 0 Td [(i = @ 2 @ x i @ y 2 )]TJ/F48 7.9701 Tf 6.586 0 Td [(i andtheconvergenceisuniform.Notethesepolynomialsvanishontheboundaryof [0,1] 2 by denition. Theorem29impliespolynomialsvanishingontheboundaryofthesquare [0,1] 2 aredense intheformdomain H D ,sinceconvergenceintheHilbertSobolevnormimpliestheindividual L 2 convergenceof f n ! f and D f n ! D f .Byinspectionofthenorm kk H D convergencein theHilbertSolbolevnormimpliesconvergencein kk H D . WenotethattheproofsofTheorem25andLemmas19and20carryoverwithout modication.Wewillusethemfreely.Wealsoneedtorecallthespectraltheoremfortuplesof normaloperatorsfoundinChapter1. 7.3UsingtheLifts Inthissectionwederiveasucientconditionfordisconjugacyonthesquare.Let J 1 = 0 B B B B @ A 1 00 0 A 1 ic 00 A 1 1 C C C C A and J 2 = 0 B B B B @ A 2 0 ic 0 A 2 0 00 A 2 1 C C C C A , where A 1 and A 2 arecommutingself-adjointoperators.Thus J 1 and J 2 commuteaswell.We willassumeitispossibletolift X and Y to J 1 and J 2 . Equivalently,themodiedquadratic pencilassociatedwith X , Y denedinDenition30factorswith c 2 = k B 2 k byTheorem27. Thereexistsanisometry V : H D ! K suchthat V X Y = J 1 J 2 V . Let q x , y = xy )]TJ/F47 11.9552 Tf 11.956 0 Td [(x )]TJ/F47 11.9552 Tf 11.955 0 Td [(y andlet V q = F 0 , F 1 , F 2 > .Let f 2 ker )]TJ/F22 7.9701 Tf 15.168 4.707 Td [(1 c 2 B 2 bethe non-zerofunctionguaranteedbyapplicationsofTheorem25andLemma20viathearguments inSection5.4;i.e.let f beaneigenfunctionof L intheeigenspaceofthethersteigenvalue. ByTheorem29polynomialsoftheform pq where p isapolynomialaredenseinthespace. 105

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Let p n x , y = g n x , y q x , y beasequenceofpolynomialsapproaching f .Note,bythe proofofTheorem29, p n and @ i p n approach f and @ i f uniformly.Wehave V p n = 0 B B B B @ g n A 1 , A 2 0 ic @ g n A 1 , A 2 @ y 0 g n A 1 , A 2 ic g n A 1 , A 2 @ x 00 g n A 1 , A 2 1 C C C C A 0 B B B B @ F 0 F 1 F 2 1 C C C C A = 0 B B B B @ F 0 g n + icF 2 @ g n A 1 , A 2 @ x F 1 g n + icF 2 @ g n A 1 , A 2 @ y F 2 g n 1 C C C C A . ByTheorem25 I )]TJ/F22 7.9701 Tf 15.167 4.707 Td [(1 c 2 B 2 T f =0 andbysimilarargumentstoSection5.5, 0 B B B B @ 100 010 000 1 C C C C A V f =0 Since V p n H D )167(! V f , 0 B B B B @ 100 010 000 1 C C C C A V p n ! 0 B B B B @ 100 010 000 1 C C C C A V f =0. HencebyapplicationofTheorem3,both F 0 g n x , y + icF 2 @ g n x , y @ x ! 0 F 1 g n x , y + icF 2 @ g n x , y @ y ! 0 Sincetheconvergenceto f anditspartialswasuniform,oncompactsubsetsof [0,1] 2 F 0 f q x , y + icF 2 @ f = q @ x =0 F 1 f q x , y + icF 2 @ f = q @ y =0 Here,wehaverecoveredthefactthat f mustsatisfyarstorderdierentialequationineach variable.Thisisouranalogousconditiontothesinglevariablecase. 106

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REFERENCES [1]JimAgler. SUBJORDANOPERATORS .ProQuestLLC,AnnArbor,MI,1980.Thesis Ph.D.{IndianaUniversity. [2]JimAgler.AdisconjugacytheoremforToeplitzoperators. Amer.J.Math. ,112:1{14, 1990. [3]JimAglerandJohnE.McCarthy. PickinterpolationandHilbertfunctionspaces , volume44of GraduateStudiesinMathematics .AmericanMathematicalSociety, Providence,RI,2002. [4]JimAglerandMarkStankus. m -isometrictransformationsofHilbertspace.I. Integral EquationsOperatorTheory ,21:383{429,1995. [5]JimAglerandMarkStankus. m -isometrictransformationsofHilbertspace.II. Integral EquationsOperatorTheory ,23:1{48,1995. [6]JimAglerandMarkStankus. m -isometrictransformationsofHilbertspace.III. Integral EquationsOperatorTheory ,24:379{421,1996. [7]WilliamArveson. Ashortcourseonspectraltheory ,volume209of GraduateTextsin Mathematics .Springer-Verlag,NewYork,2002. [8]JosephA.BallandJ.WilliamHelton.Nonnormaldilations,disconjugacyandconstrained spectralfactorization. IntegralEquationsOperatorTheory ,3:216{309,1980. [9]ManDuenChoi.Completelypositivelinearmapsoncomplexmatrices. LinearAlgebraand Appl. ,10:285{290,1975. [10]Man-DuenChoi.Positivesemidenitebiquadraticforms. LinearAlgebraandits Applications ,12:95{100,1975. [11]JohnB.Conway. Acourseinfunctionalanalysis ,volume96of GraduateTextsin Mathematics .Springer-Verlag,NewYork,secondedition,1990. [12]RaulE.Curto.Applicationsofseveralcomplexvariablestomultiparameterspectraltheory. In Surveysofsomerecentresultsinoperatortheory,Vol.II ,volume192of PitmanRes. NotesMath.Ser. ,pages25{90.LongmanSci.Tech.,Harlow,1988. [13]LawrenceC.Evans. Partialdierentialequations ,volume19of GraduateStudiesin Mathematics .AmericanMathematicalSociety,Providence,RI,secondedition,2010. [14]H.Helton.Operatorswitharepresentationasmultiplicationby onaSobolevspace.In HilbertspaceoperatorsandoperatoralgebrasProc.Internat.Conf.,Tihany,1970 ,pages 279{287.looseerrataColloq.Math.Soc.JanosBolyai,5.North-Holland,Amsterdam, 1972. [15]J.WilliamHelton.JordanoperatorsininnitedimensionsandSturmLiouvilleconjugate pointtheory. Bull.Amer.Math.Soc. ,78:57{61,1971. 107

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[16]AntoineHenrot. Extremumproblemsforeigenvaluesofellipticoperators .Frontiersin Mathematics.BirkhauserVerlag,Basel,2006. [17]V.HutsonandJ.S.Pym. Applicationsoffunctionalanalysisandoperatortheory ,volume 146of MathematicsinScienceandEngineering .AcademicPress,Inc.[HarcourtBrace Jovanovich,Publishers],NewYork-London,1980. [18]EdwardH.Kingsley.Bernsteinpolynomialsforfunctionsoftwovariablesofclass C k . Proc.Amer.Math.Soc. ,2:64{71,1951. [19]GiovanniLeoni. ArstcourseinSobolevspaces ,volume105of GraduateStudiesin Mathematics .AmericanMathematicalSociety,Providence,RI,2009. [20]ScottMcCulloughandBenjaminRusso.The3-isometricliftingtheorem. Integral EquationsandOperatorTheory ,pages1{19,2015. [21]VernPaulsen. Completelyboundedmapsandoperatoralgebras ,volume78of Cambridge StudiesinAdvancedMathematics .CambridgeUniversityPress,Cambridge,2002. [22]HeydarRadjaviandPeterRosenthal. Invariantsubspaces .DoverPublicationsInc., Mineola,NY,secondedition,2003. [23]VolkerWrobel.TheboundaryofTaylor'sjointspectrumfortwocommutingBanachspace operators. StudiaMath. ,84:105{111,1986. 108

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BIOGRAPHICALSKETCH BenjaminwasborninFlorida.HegraduatedfromDeereldBeachHighSchoolIn2006. HereceivedhisBachelorofScienceinmathematicsandphysicsfromtheUniversityofFlorida andcontinuedontoreceivehismaster'sandPh.Dinmathematics. 109