index theory with applications to mathematics and physics...

Index Theory

with Applications to Mathematics and Physics

David Bleecker

Bernhelm Booß–Bavnbek

To David .

Contents

Synopsis xi

Preface xvi

Part I. Operators with Index and Homotopy Theory 1

Chapter 1. Fredholm Operators 21. Hierarchy of Mathematical Objects 22. The Concept of Fredholm Operator 33. Algebraic Properties. Operators of Finite Rank. The Snake Lemma 54. Operators of Finite Rank and the Fredholm Integral Equation 95. The Spectra of Bounded Linear Operators: Basic Concepts 10

Chapter 2. Analytic Methods. Compact Operators 121. Analytic Methods. The Adjoint Operator 122. Compact Operators 183. The Classical Integral Operators 254. The Fredholm Alternative and the Riesz Lemma 265. Sturm-Liouville Boundary Value Problems 286. Unbounded Operators 347. Trace Class and Hilbert-Schmidt Operators 53

Chapter 3. Fredholm Operator Topology 631. The Calkin Algebra 632. Perturbation Theory 653. Homotopy Invariance of the Index 684. Homotopies of Operator-Valued Functions 725. The Theorem of Kuiper 776. The Topology of F 817. The Construction of Index Bundles 828. The Theorem of Atiyah-Jänich 889. Determinant Line Bundles 9110. Essential Unitary Equivalence and Spectral Invariants 108

Chapter 4. Wiener-Hopf Operators 1201. The Reservoir of Examples of Fredholm Operators 1202. Origin and Fundamental Significance of Wiener-Hopf Operators 1213. The Characteristic Curve of a Wiener-Hopf Operator 1224. Wiener-Hopf Operators and Harmonic Analysis 1235. The Discrete Index Formula. The Case of Systems 1256. The Continuous Analogue 129

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Part II. Analysis on Manifolds 133

Chapter 5. Partial Differential Equations in Euclidean Space 1341. Linear Partial Differential Equations 1342. Elliptic Differential Equations 1373. Where Do Elliptic Differential Operators Arise? 1394. Boundary-Value Conditions 1415. Main Problems of Analysis and the Index Problem 1436. Numerical Aspects 1437. Elementary Examples 144

Chapter 6. Differential Operators over Manifolds 1561. Differentiable Manifolds — Foundations 1572. Geometry of C∞ Mappings 1603. Integration on Manifolds 1664. Exterior Differential Forms and Exterior Differentiation 1715. Covariant Differentiation, Connections and Parallelity 1766. Differential Operators on Manifolds and Symbols 1817. Manifolds with Boundary 190

Chapter 7. Sobolev Spaces (Crash Course) 1931. Motivation 1932. Definition 1943. The Main Theorems on Sobolev Spaces 2004. Case Studies 203

Chapter 8. Pseudo-Differential Operators 2061. Motivation 2062. Canonical Pseudo-Differential Operators 2103. Principally Classical Pseudo-Differential Operators 2144. Algebraic Properties and Symbolic Calculus 2285. Normal (Global) Amplitudes 231

Chapter 9. Elliptic Operators over Closed Manifolds 2371. Mapping Properties of Pseudo-Differential Operators 2372. Elliptic Operators — Regularity and Fredholm Property 2393. Topological Closure and Product Manifolds 2424. The Topological Meaning of the Principal Symbol — A Simple Case

Involving Local Boundary Conditions 244

Part III. The Atiyah-Singer Index Formula 251

Chapter 10. Introduction to Topological K-Theory 2521. Winding Numbers 2522. The Topology of the General Linear Group 2573. Elementary K-Theory 2624. K-Theory with Compact Support 2665. Proof of the Periodicity Theorem of R. Bott 269

Chapter 11. The Index Formula in the Euclidean Case 2751. Index Formula and Bott Periodicity 275

CONTENTS ix

2. The Difference Bundle of an Elliptic Operator 2763. The Index Theorem for Ellc(Rn) 281

Chapter 12. The Index Theorem for Closed Manifolds 2841. Pilot Study: The Index Formula for Trivial Embeddings 2852. Proof of the Index Theorem for Nontrivial Normal Bundle 2873. Comparison of the Proofs 301

Chapter 13. Classical Applications (Survey) 3101. Cohomological Formulation of the Index Formula 3112. The Case of Systems (Trivial Bundles) 3163. Examples of Vanishing Index 3174. Euler Characteristic and Signature 3195. Vector Fields on Manifolds 3256. Abelian Integrals and Riemann Surfaces 3297. The Theorem of Hirzebruch-Riemann-Roch 3338. The Index of Elliptic Boundary-Value Problems 3379. Real Operators 35610. The Lefschetz Fixed-Point Formula 35711. Analysis on Symmetric Spaces: The G-equivariant Index Theorem 36012. Further Applications 362

Part IV. Index Theory in Physics and the Local Index Theorem 363

Chapter 14. Physical Motivation and Overview 3641. Classical Field Theory 3652. Quantum Theory 373

Chapter 15. Geometric Preliminaries 3941. Principal G-Bundles 3942. Connections and Curvature 3963. Equivariant Forms and Associated Bundles 4004. Gauge Transformations 4095. Curvature in Riemannian Geometry 4146. Bochner-Weitzenböck Formulas 4347. Characteristic Classes and Curvature Forms 4438. Holonomy 455

Chapter 16. Gauge Theoretic Instantons 4601. The Yang-Mills Functional 4602. Instantons on Euclidean 4-Space 4663. Linearization of the Moduli Space of Self-dual Connections 4894. Manifold Structure for Moduli of Self-dual Connections 496

Chapter 17. The Local Index Theorem for Twisted Dirac Operators 5131. Clifford Algebras and Spinors 5132. Spin Structures and Twisted Dirac Operators 5253. The Spinorial Heat Kernel 5384. The Asymptotic Formula for the Heat Kernel 5495. The Local Index Formula 576

x CONTENTS

6. The Index Theorem for Standard Geometric Operators 594

Chapter 18. Seiberg-Witten Theory 6431. Background and Survey 6432. Spinc Structures and the Seiberg-Witten Equations 6553. Generic Regularity of the Moduli Spaces 6634. Compactness of Moduli Spaces and the Definition of S-W Invariants 685

Appendix A. Fourier Series and Integrals - Fundamental Principles 7051. Fourier Series 7052. The Fourier Integral 707

Appendix B. Vector Bundles 7121. Basic Definitions and First Examples 7122. Homotopy Equivalence and Isomorphy 7163. Clutching Construction and Suspension 718

Bibliography 723

Index of Notation 741

Index of Names/Authors 749

Subject Index 757

Synopsis

Preface. Target Audience and Prerequisites. Outline of History. FurtherReading. Questions of Style. Acknowledgments and Dedication.

Chapter 1. Fredholm Operators. Hierarchy of Mathematical Objects. TheConcept of a Bounded Fredholm Operator in Hilbert Space. Algebraic Properties.Operators of Finite Rank. The Snake Lemma of Homological Algebra. ProductFormula. Operators of Finite Rank and the Fredholm Integral Equation. TheSpectra of Bounded Linear Operators (Terminology).

Chapter 2. Analytic Methods. Compact Operators. Adjoint and Self-Adjoint Operators - Recalling Fischer-Riesz. Dual Characterization of FredholmOperators. Compact Operators: Spectral Decomposition, Why Compact Oper-ators also are Called Completely Continuous, K as Two-Sided Ideal, Closure ofFinite-Rank Operators, and Invariant under ∗. Classical Integral Operators. Fred-holm Alternative and Riesz Lemma. Sturm-Liouville Boundary Value Problems.Unbounded Operators: Comprehensive Study of Linear First Order DifferentialOperators Over S1: Sobolev Space, Dirac Distribution, Normalized IntegrationOperator as Parametrix, The Index Theorem on the Circle for Systems. Closed Op-erators, Closed Extensions, Closed (not necessarily bounded) Fredholm Operators,Composition Rule, Symmetric and Self–Adjoint Operators, Formally Self-Adjointand Essentially Self-Adjoint. Spectral Theory. Metrics on the Space of ClosedOperators. Trace Class and Hilbert-Schmidt Operators.

Chapter 3. Fredholm Operator Topology. Calkin Algebra and Atkinson’sTheorem. Perturbation Theory: Homotopy Invariance of the Index, Homotopiesof Operator-Valued Functions, The Theorem of Kuiper. The Topology of F : TheHomotopy Type, Index Bundles, The Theorem of Atiyah-Jänich. DeterminantLine Bundles: The Quillen Determinant Line Bundle, Fredholm Determinants, TheSegal-Furutani Construction. Spectral Invariants: Essentially Unitary Equivalence,What Is a Spectral Invariant? Eta Function, Zeta Function, Zeta RegularizedDeterminant.

Chapter 4. Wiener-Hopf Operators. The Reservoir of Examples of Fred-holm Operators. Origin and Fundamental Significance of Wiener-Hopf Operators.The Characteristic Curve of a Wiener-Hopf Operator. Wiener-Hopf Operatorsand Harmonic Analysis. The Discrete Index Formula. Noether’s Theorem forthe Hilbert Transform. The Case of Systems. The Continuous Analogue.

Chapter 5. Partial Differential Equations in Euclidean Space, Re-visited. Review of Classical Linear Partial Differential Equations: Constant and

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Variable Coefficients, Wave Equation, Heat Equation, Laplace Equation, Charac-teristic Polynomial. Elliptic Differential Equations: Where Do Elliptic DifferentialOperators Arise? Boundary-Value Conditions. Main Problems of Analysis andthe Index Problem. Calculations. Elementary Examples. The Noether(-Hellwig-Vekua) Problem with Nonvanishing Index.

Chapter 6. Differential Operators over Manifolds. Motivation. Differ-entiable Manifolds — Foundations: Tangent Space. Cotangent Space. Geometryof C∞ Mappings: Embeddings, Immersions, Submersions, Embedding Theorems.Integration on Manifolds: Hypersurfaces, Riemannian Manifolds, Geodesics, Ori-entation. Exterior Differential Forms and Exterior Differentiation. Covariant Dif-ferentiation, Connections and Parallelity: Connections on Vector Bundles, ParallelTransport, Connections on the Tangent Bundle, Clifford Modules and Operatorsof Dirac Type. Differential Operators on Manifolds and Symbols: Our Data, Sym-bolic Calculus, Formal Adjoints. Elliptic Differential Operators. Definition andStandard Examples. Manifolds with Boundary.

Chapter 7. Sobolev Spaces (Crash Course). Motivation. Equivalence ofDifferent Local Definitions. Various Isometries. Global, Coordinate-Free Definition.Embedding Theorems: Dense Subspaces; Truncation and Mollification; DifferentialEmbedding; Rellich Compact Embedding. Sobolev Spaces Over Half Spaces. TraceTheorem. Case Studies: Euclidean Space and Torus; Counterexamples.

Chapter 8. Pseudo-Differential Operators. Motivation: Fourier Inver-sion; Symbolic Calculus; Quantization. Canonical and Principally Classical Pseudo-Differential Operators. Pseudo-Locality; Singular Support. Standard Examples:Differential Operators; Singular Integral Operators. Oscillatory Integrals. Kuran-ishi Theorem. Change of Coordinates. Pseudo-Differential Operators on Mani-folds. Graded ∗-Algebra. Invariant Principal Symbol; Exact Sequence; Noncanon-ical Op-Construction as Right Inverse. Coordinate-Free (Truly Global) Approach:Bokobza-Haggiag-Fourier Transformation; Bokobza-Haggiag Amplitudes; Bokobza-Haggiag Invertible Op-Construction; Approximation of Differential Operators.

Chapter 9. Elliptic Operators over Closed Manifolds. Continuity ofPseudo-Differential Operators between Sobolev Spaces. Parametrices for EllipticOperators: Regularity and Fredholm Property. Topological Closures. Outer TensorProduct on Product Manifolds. The Topological Meaning of the Principal Symbol(Simple Case Involving Local Boundary Conditions).

Chapter 10. Introduction to Topological K-Theory. Winding Numbers.One-Dimensional Index Theorem. Counter-Intuitive Dimension Two: Bending aPlane. The Topology of the General Linear Group. The Grothendieck Ring ofVector Bundles. K-Theory with Compact Support. Proof of the Bott PeriodicityTheorem.

Chapter 11. The Index Formula in the Euclidean Case. Index Formulaand Bott Periodicity: Three Integer Invariants. The Difference Bundle of an EllipticOperator: Operators Equal to the Identity at Infinity; Complexes of Vector Bundleswith Compact Support; Symbol Class in K-Theory with Compact Support. TheIndex Theorem for Ellc(Rn).

SYNOPSIS xiii

Chapter 12. The Index Theorem for Closed Manifolds. K-TheoreticProof of the Index Theorem by Embedding: Pilot Study — The Index Theoremfor Embeddings with Trivial Normal Bundle. Proof of the Index Theorem for Non-Trivial Normal Bundle: The Difference Element Construction, Revisited; SymbolClass; Thom Isomorphism of K-Theory; Definition of the Topological Index; Defi-nition of the Analytic Index; Foundations of Equivariant K-Theory. MultiplicativeProperty: Formulation; How it Fits into the Embedding Proof; Proof of the Mul-tiplicative Property. Short Comparison of the Cobordism, the Embedding and theHeat Equation Proof; Outlook to Spectral Theory, Asymmetry, and Inverse Prob-lems.

Chapter 13. Classical Applications (Survey). General Appraisal. Coho-mological Formulation of the Index Formula: Comparison of K-Theoretic and Co-homological Thom Isomorphisms; Orientation Class; Chern Classes; Chern Charac-ter; Todd Class; Chern Character Defect. The Case of Systems (Trivial Bundles).Examples of Vanishing Index. Euler Number and Signature. Vector Fields onManifolds. Abelian Integrals and Riemann Surfaces. The Theorem of Riemann-Roch-Hirzebruch. The Index of Elliptic Boundary-Value Problems. Real Opera-tors. The Lefschetz Fixed-Point Formula. Analysis on Symmetric Spaces. FurtherApplications.

Chapter 14. Physical Motivation and Overview. Mode of Reasoning inPhysics. String Theory and Quantum Gravity. The Experimental Side. ClassicalField Theory: Newton-Maxwell-Lorentz, Faraday 2-Form, Abstract Flat MinkowskiSpace-Time, Relativistic Mass, Relativistic Kinetic Energy, Inertial System, LorentzTransformations and Poincaré Group, Relativistic Deviation from Flatness, TwinParadox, Variational Principles. Kaluza-Klein Theory: Simultaneous Geometriza-tion of Electro-Magnetism and Gravity, Other Grand Unified Theories, String The-ory. Quantum Theory: Photo-Electric Effect, Atomic Spectra, Quantizing Energy,State Spaces of Systems of Particles, Basic Interpretive Assumptions. HeisenbergUncertainty Principle. Evolution with Time — The Schrödinger Picture. Nonrela-tivistic Schrödinger Equation and Atomic Phenomena. Minimal Replacement andCovariant Differentiation. Anti-Particles and Negative-Energy States. Unreason-able Success of the Standard Model. Dirac Operator vs. Klein-Gordon Equation.Feynman Diagrams.

Chapter 15. Geometric Preliminaries. Principal G-Bundles; Hopf Bun-dle. Connections and Curvature: Connection 1-Form; Maurer-Cartan Form; Hori-zontal Lift. Equivariant Forms and Associated Bundles: Associated Vector Bundles,Equivariance, and Basic Forms; Horizontal Equivariant Forms; Covariant Differen-tiation and the General Bianchi Identity; Inner Products, Hodge Star Operator,and Formal Adjoints. Gauge Transformations: Distinguishing Gauge Transforma-tions from Automorphisms; The Group of Gauge Transformations; The Actionof Gauge Transformations on Connections; Lie Algebra Analogy and InfinitesimalAction. Curvature in Riemannian Geometry: The Bundle of Linear Frames; Con-nections and Forms; Kozul Connection; The Orthonormal Frame Bundle; MetricConnections; The Fundamental Lemma of Riemannian Geometry and the Levi-Civita Connection; Local Coordinates and Christoffel Symbols; The Curvature ofthe Levi-Civita Connection; First and Second Bianchi Identities; Ricci and Scalar

xiv SYNOPSIS

Curvature as Contractions, and Einstein’s Equation; All Possible Curvature Ten-sors on Rn and the Kulkarni-Nomizu Product; Curvature Parts on 4-Manifolds andSelf-Duality. Bochner-Weitzenböck Formulas: Fibered Products; Contractions andComponents; The Connection Laplacian, the Hodge Laplacian, and the Bochner-Weitzenböck Formula; Special Cases. Characteristic Classes and Curvature Forms:Chern Classes as Curvature Forms; The Pfaffian; Pontryagin Classes; Other Char-acteristic Classes Related to Index Theory; Multiplicative Classes; Todd Classand L-Polynomials; Recalculating Characteristic Classes; Unifications on Almost-Complex Manifolds. Holonomy.

Chapter 16. Gauge Theoretic Instantons. The Yang-Mills Functional.Instantons on Euclidean 4-Space. Linearization of the Manifold of Moduli of Self-dual Connections. Manifold Structure for Moduli of Self-dual Connections.

Chapter 17. The Local Index Theorem for Twisted Dirac Operators.Clifford Algebras and Spinors: Clifford Algebra Basics; Spin Groups and DoubleCover; Spinor Representations; Supertrace. Spin Structures and Twisted Dirac Op-erators: Čech Cohomology; Admittance of Spin Structures; Standard and TwistedDirac Operators; Chirality. The Spinorial Heat Kernel: Index, Spectral Asymme-try and the Existence of the Heat Kernel; Solving the Spinorial Heat Equations;Calculating Index and Supertrace; General Heat Kernels. The Asymptotic Formulafor the Heat Kernel: Why Asymptotic Expansion? The Radial Gauge; About theGeometry of the Ball; Further Approximations. The Local Index Formula: Con-tent and Meaning of the Local Index Formula; How the Curvature Terms Arise inthe Heat Asymptotics; The case m = 1 (surfaces); The case m = 2 (4-manifolds);Proof of the Local Index Formula for Arbitrary Even Dimensions; Index Theorem

for Twisted Dirac Operators; Â Genus; Rokhlin’s Theorem. The Index Theoremfor Standard Geometric Operators: Index Theorem for Generalized Dirac Oper-ators; Twisted Generalized Dirac Operators; The Hirzebruch Signature Formula;The Gauss-Bonnet-Chern Formula; The Generalized Yang-Mills Index Theorem;The Hirzebruch-Riemann-Roch Formula for Kähler Manifolds.

Chapter 18. Seiberg-Witten Theory. Background and Survey: Inter-section Form and Homotopy Type of Compact Oriented Simply-Connected Four-Manifolds; Unimodular Forms and Freedman’s Theorem; Existence of Differen-tiable Structures; Donaldson’s Polynomial Invariants; Results Concerning Symplec-tic Manifolds; Purely Geometric Applications. Spinc Structures and the Seiberg-Witten Equations: Admittance of Spinc Structures, Spinc Dirac Operators; Moti-vating and Defining the Unperturbed and the Perturbed Seiberg-Witten Equations.Generic Regularity of the Moduli Spaces: Gauge Transformations; Moduli Space ofSolutions of the Perturbed S-W Equations; The Formal Dimension of the ModuliSpace; Seiberg-Witten Function; Quotient Manifolds; Manifold Structure for theParametrized Moduli Space; Generic Regularity; A Priori Bounds; Sobolev Esti-mates; Compactness of Moduli Spaces; Definition of the S-W Invariant; MetricDependence of Connections and Dirac Operators; Oriented Cobordism; FredholmTransversality; Full Invariance of the S-W Invariant.

Appendix A. Fourier Series and Integrals — Fundamental Principles.Fourier Series: The Fundamental Function Spaces on S1; Density; OrthonormalBasis; Fourier Coefficients; Plancherel’s Identity; Product and Convolution. The

SYNOPSIS xv

Fourier Integral: Different Integral Conventions; Duality Between Local and Global— Point and Neighborhood — Multiplication and Differentiation — Bounded andContinuous; Fourier Inversion Formula; Plancherel and Poisson Summation Formu-lae; Parseval’s Equality; Higher Dimensional Fourier Integrals.

Appendix B. Vector Bundles. Basic Definitions and First Examples. Ho-motopy Equivalence and Isomorphy. Clutching Construction and Suspension.

Bibliography. Key References. Classical and Recent Textbooks. Referencesto Technical Details; History; Perspectives.

Table 0.1. Suggested packages (selections, curricula) for upper-undergraduate and graduate classes/seminars and teach-yourself

Aims Logical Order by Boxed Chapter Numbers

Index Theorem andTopolog. K-Theory

App. B → 1.1-1.3, 2.1,2.2, 3.1-3.8, Thm. 5.11 → 6-7→ 8.5,9 → 10-12.2 → 13.1-13.5, 13.10,13.11 → 18.1

Index Theorem viaHeat Equation

App. A → 1.2,3.3 → 5.2, 6-7, 8.3 → 8.4,9.2 → 12.3→ 15 → 17

Gauge-TheoreticPhysics

14 → 5-6 → 9.2 → 12.3 → 13.8,13.11 → 15 → 16→ 18

Spectral Geometry 1-4 → 6,15 → 12.3 → 13.4-13.11 → 17.4-17.6 → 18

Global and Micro-Local Analysis

App. A,B → 1, 2, 3.1-3.5 → 3.10,4 → 5-6 → 14-15→ 7-9 → 10.1, 10.2 → 12.3,13 → 16,17.6,18

Preface

Target Audience and Prerequisites. The mathematical philosophy of indextheory and all its basic concepts, technicalities and applications are explained inParts I-III. Those are the easy parts. They are written for upper undergraduatestudents or graduate students to bridge the gap between rule-based learning andfirst steps towards independent research. They are also recommended as generalorientation to mathematics teachers and other senior mathematicians with differentbackground. All interested can pick up a single chapter as bedside reading.

In order to enjoy reading or even work through Parts I-III, we expect thereader to be familiar with the concept of a smooth function and a complex separableHilbert space. Nothing more — but a will to acquire specialized topics in functionalanalysis, algebraic topology, elliptic operator theory, global analysis, Riemanniangeometry, complex variables, and some other subjects. Catching so many differentconcepts and fields can make the first three Parts a bit sophisticated for a busyreader. Instead of ascending systematically from simple concepts to complex ones inthe classical Bourbaki style, we present a patch-work of definitions and results whenneeded. In each chapter we present a couple of fully comprehensible, important,deep mathematical stories. That, we hope, is sufficient to catch our four messages:

(1) Index theory is about regularization, more precisely, the index quantifiesthe defect of an equation, an operator, or a geometric configuration frombeing regular.

(2) Index theory is also about perturbation invariance, i.e., the index is ameaningful quantity stable under certain deformations and apt to storecertain topological or geometric information.

(3) Most important for many mathematicians, the index interlinks quite di-verse mathematical fields, each with its own very distinct research tradi-tion.

(4) Index theory trains the student to recognize all the elementary topics oflinear algebra in finite dimensions in the sophisticated topics of infinite-dimensional and nonlinear analysis and geometry.

Part IV is different. It is also self-contained. Choosing one or two chapters ofthis Part IV of the book would make a suitable text for a graduate course in se-lected topics of global analysis. All concepts will be explained fully and rigorously,but much shorter than in the first Parts. This last Part is written for graduate stu-dents, PhD students and other experienced learners, interested in low-dimensionaltopology and gauge-theoretic particle physics. We try to explain the very place ofindex theory in geometry and for revisiting quantum field theory. There are thou-sands of other calculations, observations and experiments. But there is somethingspecial about the actual and potential contributions of index theory. Index theory

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PREFACE xvii

is about chirality (asymmetry) of zero modes in the spectrum and classifies connec-tions (back ground fields) and a variety of other intrinsic properties in geometryand physics. It is not just about some more calculations, some more numbers andrelations.

Outline of History. When first considering infinite-dimensional linear spaces,there is the immediate realization that there are injective and surjective linearendomorphisms which are not isomorphisms, and more generally the dimension ofthe kernel minus that of the cokernel (i.e., the index) could be any integer. However,in the classical theory of Fredholm integral operators which goes back at least to theearly 1900s (see [147]), one is dealing with compact perturbations of the identityand the index is zero. Fritz Noether (in his study [324] of singular integraloperators and the oblique boundary problem for harmonic functions, published in1920), was the first to encounter the phenomenon of a nonzero index for operatorsnaturally arising in analysis and to give a formula for the index in terms of a windingnumber constructed from data defining the operator. Over some decades, this resultwas expanded in various directions by G. Hellwig, I.N. Vekua and others (see[425]), contrary to R. Courant’s and D. Hilbert’s expectation in [116] that“linear problems of mathematical physics which are correctly posed behave like asystem of N linear algebraic equations in N unknowns”, i.e., they should satisfy theFredholm alternative and always yield vanishing index. Meanwhile, many workingmainly in abstract functional analysis were producing results, such as the stabilityof the index of a Fredholm operator under perturbations by compact operators orbounded operators of sufficiently small operator norm (e.g., first J.A. Dieudonné[120], followed by F.V. Atkinson [49], B. Yood [449], I.Z. Gohberg and M.G.Krein [178], etc.).

Around 1960, the time was ripe for I.M. Gelfand [159] to propose that theindex of an elliptic differential operator (with suitable boundary conditions in thepresence of a boundary) should be expressible in terms of the coefficients of highestorder part (i.e., the principal symbol) of the operator, since the lower order partsprovide only compact perturbations which do not change the index. Indeed, a con-tinuous, ellipticity-preserving deformation of the symbol should not affect the index,and so Gelfand noted that the index should only depend on a suitably definedhomotopy class of the principal symbol. The hope was that the index of an ellipticoperator could be computed by means of a formula involving only the topology ofthe underlying domain (the manifold), the bundles involved, and the symbol of theoperator. In early 1962, M.F. Atiyah and I.M. Singer discovered the (elliptic)Dirac operator in the context of Riemannian geometry and were busy working at

Oxford on a proof that the Â-genus of a spin manifold is the index of this Diracoperator. At that time, S. Smale happened to pass through Oxford and turnedtheir attention to Gelfand’s general program described in [159]. Drawing on thefoundational and case work of analysts (e.g., M.S. Agranovich, A.S. Dynin, L.Nirenberg, R.T. Seeley and A.I. Volpert), particularly that involving pseudo-differential operators, Atiyah and Singer could generalize Hirzebruch’s proofof the Hirzebruch-Riemann-Roch theorem of 1954 (see [207]) and discovered andproved the desired index formula at Harvard in the Fall of 1962. Moreover, the Rie-mannian Dirac operator played a major role in establishing the general case. Thedetails of this original proof involving cobordism actually first appeared in [328].A K-theoretic embedding proof was given in [44], the first in a series of five papers.

xviii PREFACE

This proof was more direct and susceptible to generalizations (to G-equivariantelliptic operators in [42] and families of elliptic operators in [47]).

The proof of the Index Theorem in [44] was inspired by Grothendieck’sproof and thorough generalization of the Hirzebruch-Riemann-Roch Theorem, ex-plained in [86]. We shall present the approach in detail in Chapters 10-12 of thisbook. The invariance of the index under homotopy implies that the index (say,the analytic index) of an elliptic operator is stable under rather dramatic, but con-tinuous, changes of its principal symbol while maintaining ellipticity. Using thisfact, one finds (after considerable effort) that the analytical index of an ellipticoperator transforms predictably under various global operations such as embed-ding and extension. Using K-theory and Bott periodicity, a topological invariant(say, the topological index ) with the same transformation properties under theseglobal operations is constructed from the symbol of the elliptic operator. One thenverifies that a general index function having these properties is unique, subject tonormalization. To deduce the Atiyah–Singer Index Theorem (i.e., analytic index= topological index ), it then suffices to check that the two indices are the same inthe trivial case where the base manifold is just a single point. A particularly niceexposition of this approach for twisted Dirac operators over even-dimensional man-ifolds (avoiding many complications of the general case) is found in E. Guentner’sarticle [194] following an argument of P. Baum.

Not long after the K-theoretical embedding proof (and its variants), thereemerged a fundamentally different means of proving the Atiyah–Singer Index The-orem, namely the heat kernel method. This is worked out here (see Chapter 17 inthe important case of the chiral half D+ of a twisted Dirac operator D. In theindex theory of closed manifolds, one usually studies the index of a chiral half D+instead of the total Dirac operator D, since D is symmetric for compatible connec-tions and then index D = 0.) The heat kernel method had its origins in the late1960s (e.g., in [291], inspired by [302] of 1949) and was pioneered in the works[331], [166], [33]. In the final analysis, it is debatable as to whether this methodis really much shorter or better. This depends on the background and taste of thebeholder. Geometers and analysts (as opposed to topologists) are likely to find theheat kernel method appealing. The method not only applies to geometric operatorswhich are expressible in terms of twisted Dirac operators, but also largely for moregeneral elliptic pseudo-differential operators, as R.B. Melrose has done in [292].Moreover, the heat method gives the index of a “geometric” elliptic differential op-erator naturally as the integral of a characteristic form (a polynomial of curvatureforms) which is expressed solely in terms of the geometry of the operator itself (e.g.,curvatures of metric tensors and connections). One does not destroy the geometryof the operator by using ellipticity-preserving deformations. Rather, in the heatkernel approach, the invariance of the index under changes in the geometry of theoperator is a consequence of the index formula itself more than a means of proof.However, considerable analysis and effort are needed to obtain the heat kernel for

e−tD2

and to establish its asymptotic expansion as t→ 0+. Also, it can be arguedthat in some respects the K-theoretical embedding/cobordism methods are moreforceful and direct. Moreover, in [273], we are cautioned that the index theoremfor families (in its strong form) generally involves torsion elements in K-theorythat are not detectable by cohomological means, and hence are not computablein terms of local densities produced by heat asymptotics. Nevertheless, when this

PREFACE xix

difficulty does not arise, the K-theoretical expression for the topological index maybe less appealing than the integral of a characteristic form, particularly for thosewho already understand and appreciate the geometrical formulation of characteris-tic classes. More importantly, the heat kernel approach exhibits the index as justone of a whole sequence of spectral invariants appearing as coefficients of termsof the asymptotic expansion (as t → 0+) of the trace of the relevant heat kernel.(On p. 118, we guide the reader to the literature about these particular spectralinvariants and their meaning in modern physics. The required mathematics forthat will be developed in Section 17.4.) All disputes aside, the student who learnsboth approaches and formulations to the index formula will be more accomplished(and probably a good deal older).

Further Reading. What the coverage of topics in this book is concerned,we hope our table of contents needs no elaboration, except to say that space limi-tations prevented the inclusion of some important topics (e.g., the index theoremfor families; index theory for manifolds with boundary, other than the Atiyah-Patodi-Singer Theorem; L2-index theory and coarse geometry of noncompact man-ifolds; R. Nest’s and B. Tsygan’s algebraic and operator theoretic index theoryof [317, 318]; P. Kronheimer’s and T. Mrowka’s visionary work on knot ho-mology groups from instantons; lists of all calculated spectral invariants; aspectsof analytic number theory). However, we now provide some guidance for furtherstudy. A fairly complete exposition, by Atiyah himself, of the history of indextheory from 1963 to 1984 is found in Volume 3 of [26] and duplicated in Volume4. Volumes 3, 4 and 5 contain many unsurpassed articles written by Atiyah andcollaborators on index theory and its applications to gauge theory. In the infor-mative — and charming [448], S.-T. Yau collected The founders of index theory:reminiscences of and about Sir Michael Atiyah, Raoul Bott, Friedrich Hirzebruch,and I. M. Singer. N. Hitchin’s short text [216] on the 2004 Abel Prize Laureatesdescribes the index theorem, where it came from, its different manifestations and acollection of applications. It indicates how one can use the theorem as a tool in aconcrete fashion without necessarily retreating into the details of the proof. We allowe a debt of gratitude to H. Schröder for the definitive guide to the literature onindex theory (and its roots and offshoots) through 1994 in Chapter 5 of the excellentbook [167] of P.B. Gilkey. We have benefited greatly not only from this book,but also from the marvelous work [273] by H.B. Lawson and M.L. Michelsohn.In that book, there are proofs of index formulas in various contexts, and numerousbeautiful applications illustrating the power of Dirac operators, Clifford algebrasand spinors in the geometrical analysis of manifolds, immersions, vector fields, andmuch more. The classical book [389] of P. Shanahan is also a masterful, elegantexposition of not only the standard index theorem, but also the G-index theoremand its numerous applications. A fundamental source on index theory for certainopen manifolds and manifolds with boundary is the authoritative book [292] ofR.B. Melrose. In [366], Th. Schick reviews coarse index theory, in particular,for complete partitioned manifolds. It has been introduced by J. Roe and pro-vides a theory to use tools from C∗-algebras to get information about the geometryof non-compact manifolds via index theory of Dirac type operators. [204] of N.Higson and J. Roe gives a well-written presentation of the underlying ideas ofanalytic K-homology and develops some of its applications. For a concrete calcula-tion see also the concise [309, Section 7.4.2] and the Notes (forthcoming) [205]. See

xx PREFACE

also [137, 138, 139] of J. Eichhorn for heat kernel asymptotics on non-compactmanifolds and [316] of B.-W. Schulze and collaborators for index theory on sin-gular spaces. [457] of W. Zhang gives an excellent introduction to various aspectsof Atiyah-Singer index theory via the Bismut/Witten-type deformations of ellipticoperators. Very close to our own view upon index theory is the plea [157] of M.Furuta for reconsidering the index theorem, with emphasis on the localizationtheorem. In the case of boundary-value problems for Dirac operators, we put quitesome care in the writing of our [83] jointly with K.P. Wojciechowski. The recentbook [155] of D. Fursaev and D. Vassilevich contains a detailed descriptionof main spectral functions and methods of their calculation with emphasis on heatkernel asymptotics and their application in various branches of modern physics.Following up on the classic [213] of F. Hirzebruch and D. Zagier on interrela-tions between the index theorem and elementary number theory, the comprehensive[376] of S. Scott covers the theory of traces and determinants on Banach algebrasof operators on vector bundles over closed manifolds, with emphasis on various al-gebras of pseudo-differential operators. He gives a series of calculations that givethe flavor of the subject in tractable cases, and relates these calculations to Pois-son and Selberg trace formulas. There is an impressive nonstandard proof of thelocal Atiyah-Singer index theorem, using resolvent expansions in place of the usualheat equation techniques. A wealth of radically new ideas of (partly yet unproven)geometric use of instantons are given in [269] of P.B. Kronheimer and T.S.Mrowka. Very inspiring is [226] of E.P. Hsu on stochastic analysis on manifolds.It gives a reformulation of the heat equation proof of the index theorem in termsof Wiener process asymptotics. Basically, that is what we should have after A.Einstein’s famous 1905-discovery of the basis of heat conduction in diffusion. Thedetails are interesting, though, in particular because they open a window to discreteanalysis. A taste of the recent revival of D-branes and other exotic instantons instring theory can be gained from [164] of H. Ghorbani, D. Musso and A. Lerda.Indications can be found in the review [361] of F. Sannino about, how stronglycoupled theories of gauge theoretic physics result in perceiving a composite universeand other new physics awaiting to be discovered. In the mathematically rigorousand richly illustrated [355], N. Reshetikhin explains why and how topologicalinvariants by necessity appear in various quantizations of gauge theories.

The Question of Originality: Seeking a Balance between Mathemat-ical Heritage and Innovation. Parts I-III and the two appendices teach whatmathematicians today consider general knowledge about the index theorem as oneof the great achievements of 20th century mathematics. But, actually, there aretwo novelties included which even not all experts may be aware of: The first noveltyappears when rounding up our comprehensive presentation of the topology of thespace of Fredholm operators: we do not halt with the Atiyah-Jänich Theorem andthe construction of the index bundle, but also confront the student with a thoroughpresentation of the various definitions of determinant line bundles. This is to re-mind the student that index theory is not a more or less closed collection of resultsbut a philosophy of regularization, of deformation invariance and of visionary crossconnections within mathematics and between its various branches.

A second novelty in the first three Parts is the emphasis on global constructions,e.g., in introducing and using the concept of pseudo-differential operators.

PREFACE xxi

Apart from these two innovations, the student can feel protected in the firstthree Parts against any originality.

Basically, Part IV follows the same line. Happily we could also avoid excessiveoriginality in the chapters dealing with instantons and the Donaldson-Kronheimer-Seiberg-Witten results about the geometry of moduli spaces of connections. Therewe also summarize, refer, define, explain great lines and details like in the first threeParts, though emphasizing variational aspects based on [59].

However, the core of Part IV is different. It consists of an original, full, quitelengthy (in parts almost unbearably meticulous) proof of the Local Index Theoremfor twisted Dirac operators in Chapter 17 and its applications to standard geometricoperators. That long Chapter is thought as a new contribution to the ongoing searchfor a deeper understanding of the index theorem and the “best” approach to it.

Clearly, a student looking for the most general formulation of the index theo-rem and a proof apt for wide generalizations should concentrate on our Part III,the so-called Embedding (or K-theoretic) Proof. However, a student wanting totrace the germs of index calculations back in the geometry of the considered stan-dard operators (all arising from various decompositions of the algebra of exteriordifferential forms) should consult Section 17.5 with a full proof of the Local IndexFormula for twisted Dirac operators on spin manifolds (all terms will be explained)and Section 17.6, where we derive the Index Theorem for Standard Geometric Op-erators. These geometric index theorems are by far less general than Part III’sembedding proof, but they are more geometric, and we hold, also more geomet-ric than the usual heat equation proofs of the index theorem. Not striving forgreatest generality, we obtain index formulas for the standard elliptic geometricoperators and their twists. The standard elliptic geometric operators include thesignature operator d + δ : (1 + ∗) Ω∗ (M) = Ω+(M) → Ω−(M) = (1− ∗) Ω∗ (M),the Euler-Dirac operator d + δ : Ωev (M) → Ωodd (M), and the Dolbeault-Diracoperator

√2(∂̄ + ∂̄∗

): Ω−,ev (M)→ Ω+,odd (M) (all symbols will be defined). The

index formula obtained for the above operators yields the Hirzebruch SignatureTheorem, the Chern-Gauss-Bonnet Theorem, and the Hirzebruch-Riemann-RochTheorem, respectively. While these operators generally are not globally twistedDirac operators, locally they are expressible in terms of chiral halves of twistedDirac operators. That applies also to the Yang-Mills operator. Thus, even if theunderlying Riemannian manifold M (assumed to be oriented and of even dimen-sion) does not admit a spin structure, we may still use the Local Index Theoremfor twisted Dirac operators to compute the index density and hence the index ofthese operators. While it is possible to carry this out separately for each of thegeometric operators, basically all of these theorems are consequences of one singleindex theorem for generalized Dirac operators on Clifford module bundles (all tobe defined). Using the Local Index Theorem for twisted Dirac operators, we provethis index theorem first (our Theorem 17.59), and then we apply it to obtain thegeometric index theorems, yielding the general Atiyah-Singer Index Theorem forpractically all geometrically defined operators.

Style and Notations. To present the rich world of index theory, we havechosen two different styles. We write all definitions, theorems, and proofs as conciseas possible to free the reader from dispensable side information. Where possible,we begin the introduction of a new concept with a simple but generic example ora review of the local theory, immediately followed by the corresponding global or

xxii PREFACE

general concept. That is one half of the book, so to speak the odd numbered pages.The other half of the book consists of exercises (often with extended hints) andhistorical reviews, motivations, perspectives, examples. We wrote those sections ina more open web-like style. Important definitions, notions, concepts are in boldface. Background information is in small between the signs I and J. In remarksand notes, leading terms are in italics.

The reader will notice our bias towards elder literature when more recent ref-erences would not add substantially more value. This is due not so much to theage of the authors (both born before the middle of the last century) but ratherto the common pride of mathematicians belonging to a community where biblio-graphic impact factors and research indices should rather be calculated in citationsafter some decades of years than in numbers of recently appeared, cited and soonforgotten publications.

There is also a distinction, due to Harald Bohr and disseminated by BørgeJessen, between expansive and consolidating periods of each individual science.While physics and biology had consolidating periods in the first half of the lastcentury and suffer now of the rapid change of ever new single and dispersed results,mathematics has had and still has good decades of consolidation and of long-timevalidity of key results. To the present authors, there is no reason to hide ourindifference to changing fashions.

Acknowledgments and Dedication. Due to circumstance, the responsibleauthor had to finish the book alone, though on the basis of extended drafts forall chapters which had been worked out jointly with David Bleecker. First ofall, he thanks his wife Sussi Booß-Bavnbek for her encouragement, support andlove. The responsible author acknowledges the hints and help he received from I.Avramidi (Socorro), G. Chen (Hangzhou), G. Esposito (Napoli), K. Furutani(Tokyo), V.L. Hansen (Copenhagen), B. Himpel (Aarhus), P. Kirk (Blooming-ton), T. Kori (Tokyo), M. Lesch, B. Sauer and B. Vertman (Bonn), H.J.Munkholm (Odense), L. Nicolaescu (Notre Dame), M. Pflaum (Boulder), S.Scott (London), R.T. Seeley (Newton), G. Su (Tianjin), K. Uchiyama (Tokyo)and C. Zhu (Tianjin) in the critical phase of the final re-shuffling and updating. Inparticular, he thanks Roskilde University for an extraordinary leave for that work;the Università degli Studi di Napoli Federico II and the Istituto Nazionale di FisicaNucleare (INFN, Sezione di Napoli) for their hospitality during his stay there; andP.C. Anagnostopoulos (Carlisle), B.J. Bianchini (Somerville), H. Larsenand J. Larsen (Roskilde) and M. Lesch and B. Sauer (Bonn) for generouslyproviding useful LaTeX macros, hacks and environments and substantial help withthe final lay-out and promotion. A.E. Olsen (Roskilde) checked the bibliographicdata and assembled the Index of Names/Authors and Notations.

Both authors agreed to dedicate this book to their teachers, to the memory ofS.-S. Chern (thesis adviser of DB) on the occasion of his centenary in October 2011and to the memory of F. Hirzebruch (thesis adviser of BBB) who appreciatedthe announced dedication intended for his 85th birthday in October 2012, whichhe did not live to celebrate. I take the liberty to change the dedication. This bookis dedicated to David.

Bernhelm Booß-Bavnbek, Roskilde (Denmark), December 2012

If we do not succeed in solving amathematical problem, the reasonfrequently is our failure to recog-nize the more general standpointfrom which the problem before usappears only as a single link in achain of related problems.

David Hilbert, 1900

Part I

Operators with Index andHomotopy Theory

1

CHAPTER 1

Fredholm Operators

Synopsis. Hierarchy of Mathematical Objects. The Concept of a Bounded Fredholm

Operator in Hilbert Space. The Index. Forward and Backward Shift Operators. Alge-

braic Properties. Operators of Finite Rank. The Snake Lemma of Homological Algebra.

Product Formula. Operators of Finite Rank and the Fredholm Integral Equation. The

Spectra of Bounded Linear Operators (Terminology and Basic Properties).

1. Hierarchy of Mathematical Objects

“In the hierarchy of branches of mathematics, certain points are recog-nizable where there is a definite transition from one level of abstractionto a higher level. The first level of mathematical abstraction leads us tothe concept of the individual numbers, as indicated for example by theArabic numerals, without as yet any undetermined symbol representingsome unspecified number. This is the stage of elementary arithmetic; inalgebra we use undetermined literal symbols, but consider only individ-ual specified combinations of these symbols. The next stage is that ofanalysis, and its fundamental notion is that of the arbitrary dependenceof one number on another or of several others – the function. Still moresophisticated is that branch of mathematics in which the elementaryconcept is that of the transformation of one function into another, or,as it is also known, the operator.”

I Thus N. Wiener characterized the hierarchy of mathematical objects [443, p.1].Very roughly we can say: Classical questions of analysis are aimed mainly at investiga-tions within the third or fourth level. This is true for real and complex analysis, as well asthe functional analysis of differential operators with its focus on existence and uniquenesstheorems, regularity of solutions, asymptotic or boundary behavior which are of partic-ular interest here. Thereby research progresses naturally to operators of more complexcomposition and greater generality without usually changing the concerns in principle; thework remains directed mainly towards qualitative results.

In contrast it was topologists, as Michael Atiyah variously noted, who turned sys-tematically towards quantitative questions in their topological investigations of algebraicmanifolds, their determination of quantitative measures of qualitative behavior, the def-inition of global topological invariants, the computation of intersection numbers and di-mensions. In this way, they again broadly broke through the rigid separation of the“hierarchical levels” and specifically investigated relations between these levels, mainlyof the second and third level (algebraic surface – set of zeros of an algebraic function)with the first, but also of the fourth level (Laplace operators on Riemannian manifolds,Cauchy-Riemann operators, Hodge theory) with the first.

This last direction, starting with the work of Wilhelm Blaschke and WilliamV. D. Hodge, continuing with Kunihiko Kodaira, Shiing-Shen Chern and DonaldSpencer, with Henri Cartan and Jean Pierre Serre, with Friedrich Hirzebruch,Michael Atiyah and others, can perhaps be best described with the key word differential

2

1.2. THE CONCEPT OF FREDHOLM OPERATOR 3

topology or analysis on manifolds. Both its relation with and distinction from analysisproper is that (from [19, p.57])

“Roughly speaking we might say that the analysts were dealing withcomplicated operators and simple spaces (or were only asking simplequestions), while the algebraic geometers and topologists were onlydealing with simple operators but were studying rather general man-ifolds and asking more refined questions.”

We can read, e.g. in [92, pp.278-283], elaborated in [94] and the literature givenin [326], to what degree the contrast between quantitative and qualitative questions andmethods must be considered a driving force in the development of mathematics beyondthe realm sketched above.

Actually, in the 1920’s already, mathematicians such as Fritz Noether and TorstenCarleman had developed the purely functional analytic concept of the index of an op-erator in connection with integral equations, and had determined its essential properties.But “although its (the theory of Fredholm operators) construction did not require thedevelopment of significantly different means, it developed very slowly and required theefforts of very many mathematicians” [177, p.185]. And although Soviet mathematicianssuch as Ilja N. Vekua had hit upon the index of elliptic differential equations at thebeginning of the 1950’s, we find no reference to these applications in the quoted principalwork on Fredholm operators. In 1960 Israel M. Gelfand published a programmaticarticle asking for a systematic study of elliptic differential equations from this quantita-tive point of view. He took as a starting point the theory of Fredholm operators with itstheorem on the homotopy invariance of the index (see below). Only after the subsequentwork of Michail S. Agranovich, Alexander S. Dynin, Aisik I. Volpert, and finallyof Michael Atiyah, Raoul Bott, Klaus Jänich and Isadore M. Singer, did it be-come clear that the theory of Fredholm operators is indeed fundamental for numerousquantitative computations, and a genuine link connecting the higher “hierarchical levels”with the lowest one, the numbers. J

2. The Concept of Fredholm Operator

Let H be a (separable) complex Hilbert space, and let B denote the Banachalgebra (e.g., [332, p.128], [358, p.228], [365, p.201]) of bounded linear operatorsT : H → H with the operator norm

‖T‖ := sup {‖Tu‖ : ‖u‖ ≤ 1}

4 1. FREDHOLM OPERATORS

Remark 1.2. a) We can analogously define Fredholm operators T : H → H ′,where H and H ′ are Hilbert spaces, Banach spaces, or general topological vectorspaces. In this case, we use the notation B(H,H ′) and F(H,H ′), correspondingto B and F above. However, in order to counteract a proliferation of notation andsymbols in this section, we will deal with a single Hilbert space H and its operatorsas far as possible. The general case H 6= H ′ does not require new arguments at thispoint. Later, however, we shall apply the theory of Fredholm operators to ellipticdifferential and pseudo-differential operators where a strict distinction between Hand H ′ (namely, the compact embedding of the domain H into H ′, see Chapter 9)becomes decisive.b) All results are valid for Banach spaces and a large part for Fréchet spaces also.For details see for example [347, p.182-318]. We will not use any of these but willbe able to restrict ourselves entirely to the theory of Hilbert space whose treatmentis in parts far simpler.c) Motivated by analysis on symmetric spaces – with transformation group G –operators have been studied whose index is not a number but an element of therepresentation ring R(G) generated by the characters of finite-dimensional repre-sentations of G [44, p.519f] or, still more generally, is a distribution on G [23,p.9-17]. We will not treat this largely analogous theory, nor the generalization ofthe Fredholm theory to the discrete situation of von Neumann algebras as it hasbeen carried out – with real-valued index in [90, 1968/1969].

Exercise 1.3. Let L2(Z+) denote the space of sequences c = (c0, c1, c2, . . .) ofcomplex numbers with square-summable absolute values; i.e.,

∞∑n=0

|cn|2

1.3. ALGEBRAIC PROPERTIES. OPERATORS OF FINITE RANK. THE SNAKE LEMMA 5

3. Algebraic Properties. Operators of Finite Rank. The Snake Lemma

Exercise 1.4. For finite-dimensional vector spaces the notion of Fredholmoperator is empty, since then every linear map is a Fredholm operator. Moreover,the index no longer depends on the explicit form of the map, but only on thedimensions of the vector spaces between which it operates. More precisely, showthat every linear map T : H → H ′ where H and H ′ are finite-dimensional vectorspaces has index given by

indexT = dimH − dimH ′.

[Hint: One first recalls the vector-space isomorphism H/Ker(T )∼−→ Im(T ) and

then (since H and H ′ are finite-dimensional) obtains the well-known identity fromlinear algebra

dimH − dim KerT = dimH ′ − dim CokerT.]

[Warning: If we let the dimensions of H and H ′ go to∞, we obtain only the formulaindexT = ∞ −∞. Thus, we need an additional theory to give this difference aparticular value.]

Exercise 1.5. For two Fredholm operators F : H → H and G : H ′ → H ′consider the direct sum,

F ⊕G : H ⊕H ′ → H ⊕H ′.

Show that F ⊕G is a Fredholm operator with index(F ⊕G) = indexF + indexG.[Hint: First verify that Ker(F ⊕ G) = KerF ⊕ KerG, and the corresponding factfor Im(F ⊕G). Then show that

(H ⊕H ′) /(ImF ⊕ ImG) ∼= (H/ ImF )⊕(H ′/ ImG) .]

[Warning: When H = H ′, we can consider the sum (not direct) F + G : H → H,but in general this is not a Fredholm operator; e.g., we could set G := −F .]

Exercise 1.6. Show by algebraic means, that the composition G ◦ F of twoFredholm operators F : H → H ′ and G : H ′ → H ′′ is again a Fredholm operator.[Hint: Which inequality holds between

dim KerG ◦ F and dim KerF + dim KerG

and between

dim CokerG ◦ F and dim CokerF + dim CokerG ?]

[Warning: Why are these in general not equalities? Nevertheless, the chain ruleindexG ◦ F = indexF + indexG can be proved, since the inequalities cancel outwhen we form the difference. Sometimes in mathematics a result of interest appearsas an offshoot of the proof of a rather dull general statement. Such is the case whenwe obtain the chain rule for differentiable functions by showing that the compositeof differentiable maps is again differentiable. Here, however, the proof of the indexformula requires extra work, and we must either utilize the topological structure byfunctional analytic means (see Exercise 2.3, p.14) or refine the algebraic arguments.The latter will be done next.]


We recall a basic idea of diagram chasing :

Definition 1.7. Let H1, H2, . . . be a sequence (possibly finite) of vector spacesand let Tk : Hk → Hk+1 be a linear map for each k = 1, 2, . . . . We call

H1T1−→ H2

T2−→ H3T3−→ · · ·

an exact sequence, if for all k ImTk = KerTk+1.

In particular, the following is a list of equivalences and implications (where 0denotes the 0-dimensional vector space)

0 −→ H1T1−→ H2 exact ⇔ KerT1 = 0 (T1 injective),

H1T1−→ H2 −→ 0 exact ⇔ ImT1 = H2 (T1 surjective),

0 −→ H1T1−→ H2 −→ 0 exact ⇔ T1 : H1

∼−→ H2 is an isomorphism,

0 −→ H1T1−→ H2

T2−→ H3 −→ 0 exact ⇔

{T1 : H1

∼−→ T1(H1) andT2 :

H2T1(H1)

∼−→ H3 .

In the last case, H2 ∼= H1 ⊕H3, but not canonically so.

Remark 1.8. By definition, the sequence

0 −→ KerT1 ↪→ H1T1−→ H2 −→ CokerT1 −→ 0

of linear maps between vector spaces is exact. If H1, H2 are Hilbert spaces and T1bounded, the sequence becomes an exact sequence of bounded mappings betweenHilbert spaces if and only if T1 has closed range (see also the Hint to Exercise 2.1).

The following theorem is a key device in Homological Algebra for various kindsof decomposition and additivity theorems, see also Remark 1.11.

Theorem 1.9 (Snake Lemma). Assume that the following diagram of vectorspaces and linear maps is commutative (i.e., jF = F ′i and qF ′ = F ′′p) with exacthorizontal sequences and Fredholm operators for vertical maps.

0 → H1i→ H ′1

p→ H ′′1 → 0↓ F ↓ F ′ ↓ F ′′

0 → H2j→ H ′2

q→ H ′′2 → 0.Then we have indexF − indexF ′ + indexF ′′ = 0.

Proof. We do the proof in two parts.1. Here we show that the following sequence is exact:

(1.1) 0 −→ KerF −→ KerF ′ −→ KerF ′′ −→−→ CokerF −→ CokerF ′ −→ CokerF ′′ −→ 0.

For this, we first explain how the individual maps are defined. By the commutativityof the diagram, the maps KerF → KerF ′ and KerF ′ → KerF ′′ are given by i andp; and the maps CokerF → CokerF ′ and CokerF ′ → CokerF ′′ are induced by jand q in the natural way (please check).

Also KerF ′′ → CokerF is well defined: Let u′′ ∈ KerF ′′; i.e., u′′ ∈ H ′′and F ′′u′′ = 0. Since p is surjective we can choose u′ ∈ H ′1 with pu′ = u′′. ThenF ′u′ ∈ Ker q, since qF ′u′ = F ′′pu′ = F ′′u′′ = 0. By exactness, we have Ker q = Im j

1.3. ALGEBRAIC PROPERTIES. OPERATORS OF FINITE RANK. THE SNAKE LEMMA 7

and a unique (by the injectivity of j) element u ∈ H2 with ju = F ′u′. We mapu′′ ∈ KerF ′′ to the class of u in H2/ ImF = CokerF . It remains to show that weget the same class for another choice of u′. Thus, let ũ′ ∈ H ′ be such that pũ′ = u′′(p is in general not injective; hence, possibly ũ′ 6= u′). As above, we have a ũ ∈ H2with jũ = F ′u′. We must now find u0 ∈ H1 with Fu0 = u − ũ. Then we aredone. For this, note that j(u − ũ) = ju − jũ = F ′u′ − F ′ũ′ = F ′(u′ − ũ′). Sincepu′ = pũ′ = u′′, we have u′ − ũ′ ∈ Ker p. By exactness, we have u0 ∈ H1 withiu0 = u

′ − ũ′, whence F ′iu0 = F ′(u′ − ũ′). The left side is jFu0 and the right sideis j(u− ũ) from above. Hence, Fu0 = u− ũ by the injectivity of j, as desired. Weintroduced the map KerF ′′ → CokerF in such great detail in order to demonstratewhat is typical for diagram chasing: it is straightforward, largely independent oftricks and ideas, readily reproduced, and hence somewhat monotonous. Thereforewe will forgo showing the exactness of (1.1) except at KerF ′, and leave the rest as

an exercise. The exactness of KerFı̃→ KerF ′ p̃→ KerF ′′, where ı̃ and p̃ are the

restrictions of i and p, means Im ı̃ = Ker p̃. Thus, we have two inclusions to show:

⊆: This is clear, since p ◦ i = 0 implies p̃ ◦ ı̃ = 0.⊇: If u′ ∈ Ker p̃, then u′ ∈ KerF ′ and pu′ = 0, and

by the exactness of H1 → H ′1 → H ′′1 , we have u ∈ H1 with iu = u′. It remains toshow that Fu = 0, but this is clear, since jFu = F ′iu = F ′u′ = 0 and j is injective.

Note. Before we go to part 2 of the proof, we pause for a moment: It is interestingthat for H ′ = H ⊕H ′′, i = j the inclusion, and p = q the projection, we recapturethe addition formula of Exercise 1.5. In this case, the exact sequence (1.1) thenbreaks, as shown there, into two parts

0→ KerF → KerF ′ → KerF ′′ → 0 and0→ CokerF → CokerF ′ → CokerF ′′ → 0.

In the general case, however, we no longer have

dim KerF ′ = dim KerF + dim KerF ′′ and

dim CokerF ′ = dim CokerF + dim CokerF ′′,

but instead we must consider the interaction (KerF ′′ → CokerF ). From a topo-logical standpoint, the concept of the index of a Fredholm operator is a special caseof the general concept of the Euler characteristic χ(C) of a complex

C :Tk+1−→ Ck

Tk−→ Ck−1Tk−1−→ Ck−2

Tk−2−→ Ck−3Tk−3−→ · · ·

of vector spaces and linear maps (with Tk ◦ Tk+1 = 0) with finite Betti numbers

bk := dim

(KerTk

ImTk+1

).

Here KerTk/ ImTk+1 is called the k-th homology space Hk(C) . Assuming thatall these numbers are finite, as well as the number of nonzero Betti numbers, wedefine

χ(C) :=∑k

(−1)k bk

whence indexF = χ(C) for

C : 0 −→ 0 −→ H F−→ H −→ 0 −→ 0,


where C2 = C1 = H and Ci = 0 otherwise; thus, H2(C) = KerF and H1(C) =CokerF . This is the reason why we can follow in the proof of Theorem 1.9 thewell-known topological arguments (see [185, p.100f] or [140, p.52f] ) yielding theaddition (or pasting) theorem χ(C)− χ(C ′) + χ(C ′/C) = 0. See also Section 13.4.In particular, (1.1) is only a special case of the long exact homology sequence

→ Hk+1(C ′/C)→ Hk(C)→ Hk(C ′)→ Hk(C ′/C)→ Hk−1(C)→ Hk−1(C ′)→,(1.2)

[185, p.57f] or [140, p.125-128]. This more general description would have theadvantage that we only need to prove exactness of (1.2) at three adjacent placeswith an argument independent of k rather than at six places as in our simplifiedapproach where we restricted ourselves to complexes of length two. But back toour proof:2. For each exact sequence

(1.3) 0→ A1 → A2 → · · · → Ar → 0,

of finite-dimensional vector spaces, we wish to derive the formular∑

k=1

(−1)k dim Ak = 0.

Notice first that for r sufficiently large (r > 3), the formula for the alternating sumfor (1.3) follows, once we know the formula holds for the exact (prove!) sequences

0 −→ A1 −→ A2 −→ Im(A1 → A2) −→ 0

and

0 −→ Im(A2 → A3) −→ A3 −→ · · · −→ Ar −→ 0.Since these sequences have length less than r, the formula is proved by induction,if we verify it for r = 1, 2, 3.

r = 1 : trivial, since then A1 ∼= 0.r = 2 : also clear, since then A1 ∼= A2 .r = 3 : clear, since 0→ A1 → A2 → A3 → 0 implies A3 ∼= A2/A1,

whence dimA3 = dimA2 − dimA1.(1.4)

Thus, part 2 is finished and combining it with part 1, the Snake Lemma is proved.Actually, we have proven much more, namely, whenever two of the three maps F,F ′, F ′′ have a finite index, the third has finite index given by the snake formula. �

Exercise 1.10. Combine Theorem 1.9 and Exercise 1.6, to show that

(1.5) indexG ◦ F = indexF + indexG.

[Hint: Consider the diagram

0 −−−−→ H i−−−−→ H ⊕H ′ p−−−−→ H ′ −−−−→ 0yF yG◦F⊕Id yG0 −−−−→ H ′ j−−−−→ H ′′ ⊕H ′ q−−−−→ H ′′ −−−−→ 0,

where iu := (u, Fu), jv := (Gv, v), p(u, v) := Fu− v, and q(w, v) := w−Gv.]

1.4. OPERATORS OF FINITE RANK AND THE FREDHOLM INTEGRAL EQUATION 9

Remark 1.11. Alternative proofs of the product formula (1.5) can be foundin many places. They may appear shorter. Arguing via the Snake Lemma is morelengthy, but it puts the product formula in the correct format of a topologicalcomposition or gluing formula

(1.6) τ(Φ1 ∪ Φ2) = τ(Φ1) ∗ τ(Φ2) ∗ ε(Φ1 ∩ Φ2),where we have in (1.5) the vanishing of the typical third term on the right side, theerror term ε(Φ1 ∩ Φ2), for τ := index; Φ1,Φ2 ∈ F ;∪ := ◦; and ∗ := +.

A stunning impression of the intricacies of simple looking product formulasmay be gained by checking the proof of the corresponding product formula for theindex of closed (not necessarily bounded) densely defined Fredholm operators, seeTheorem 2.45, p.43f.

4. Operators of Finite Rank and the Fredholm Integral Equation

Exercise 1.12. Show that for any operator K : H → H of finite rank (i.e.,dimK(H) < ∞), the sum Id +K is a Fredholm operator and index(Id +K) = 0.Here, Id : H → H is the identity.[Hint: Set h := ImK and recall Theorem 1.9 for the diagram

0 // hi //

(Id +K)h

��

Hp//

Id +K

��

H/h //

(Id +K)H/h

��

0

0 // hj// H

q// H/h // 0.

To see that the vertical maps are well defined, we only need (Id +K)(h) ⊆ h. Thecommutativity of the diagram and the exactness of the rows are clear. Since onecan show Ker(Id +K) ⊆ h and dim Coker(Id +K) ≤ dimh, the Snake Formula givesus the result once we show

(1.7) index(Id +K)h = 0

and

(1.8) index(Id +K)H/h = 0.

But (1.7) is clear from Exercise 1.4 and (1.8) is trivial because(Id +K)H/h = IdH/h.

Note that one could deduce that Id +K has finite index by using the observationat the end of the proof of Theorem 1.9.]

Remark 1.13. One may be bothered by the way in which the proposed solutionproduces the result so directly from the Snake Formula by means of a trick. As amatter of fact, index(Id +K) can be computed in a pedestrian fashion by reductionto a system of n linear equations with n unknowns where n := dimh. To do thisone verifies that every operator K of finite rank has the form

Ku =

n∑i=1

〈u, ui〉 vi

with fixed u1, ..., un, v1, ..., vn ∈ H (note that every continuous linear functional isof the form 〈·, u0〉). Whether this direct approach, as detailed for example in [365,Theorem 4.9](see also [332, p.110f]), is in fact more transparent than the deviceused with the Snake Formula, depends a little on the perspective. While in the


first approach the key point (namely the use of Exercise 1.4 for equation (1.7)) issingled out and separated clearly in the remaining formal argument, we find in thesecond more constructive approach rather a fusion of the nucleus with its packaging.However, the use of the fairly nontrivial Riesz-Fischer Lemma is unnecessary in thecase where K is given in the desired explicit form, as in the following example.

Exercise 1.14. Consider the Fredholm integral equation of the secondkind

u(x) +

∫ ba

G(x, y)u(y) dy = h(x)

with degenerate (product-) weight function (or integral kernel)

G(x, y) =

n∑i=1

fi(x)gi(y)

with fixed a < b real and fi, gi square integrable on [a, b]. Prove the Fredholmalternative: Either there is a unique solution u ∈ L2[a, b] for every given right sideh ∈ L2[a, b], or the homogeneous equation (h = 0) has a solution which does notvanish identically. Moreover, the number of linearly independent solutions of thehomogeneous equation equals the number of linear conditions one needs to imposeon h in order that the inhomogeneous equation be solvable.[Hint: Consider the operator Id +K on the Hilbert space L2[a, b], where Ku =∑〈u, gi〉 fi, and apply Exercise 1.12. For the interpretation of the dimension of the

cokernel, see Exercise 2.1b below.]

5. The Spectra of Bounded Linear Operators: Basic Concepts

We close this chapter with a concept that belongs to the border region betweenalgebraic and analytic notions, the spectrum of a bounded linear operator. Forthe corresponding definitions and elementary properties in the more general (and,actually, different) case of not necessarily bounded linear operators we refer toDefinition 2.59 and Exercise 2.60, p.50.

Remark 1.15. The following comments correspond to the sets of Table 1.1.For wanted arguments we refer to [332, Section 4.1]:1. Res(T ) is open in C. We can argue as in Exercise 3.6, p.65 where one has toprove that the group of units in any Banach algebra is open.2. Spec(T ) is closed and bounded (i.e., compact) in C. Actually, one has Spec(T ) ⊂{λ : |λ| ≤ ‖T‖}.3. Res(T ) ⊆ Fred(T ), and Fred(T ) is the union of at most a countable number ofopen, connected components.4. Spece(T ) ⊆ Spec(T ) , and Spece(T ) = Spec(π(T )), where π : B → B/K is theprojection.5. Specp(T ) consists of isolated points of Spec(T ). Specp(T ) contains the limitpoints of Spec(T ) which are Fredholm points.6. Specc(T ) ⊆ Spece(T ).7. Spec(T ) = Specp(T ) ∪ Specc(T ) ∪ Specr(T ), a union of disjoint sets.

1.5. THE SPECTRA OF BOUNDED LINEAR OPERATORS: BASIC CONCEPTS 11

Table 1.1. The spectra of bounded linear operators T : H → H

Symbol Name Definition

1. Res(T ) resolvent set{z ∈ C : (T − z Id)−1 ∈ B

}2. Spec(T ) spectrum C\Res(T )3. Fred(T ) Fredholm points {z ∈ C : T − z Id ∈ F}4. Spece(T ) essential spectrum C \ Fred(T )

5. Specp(T )point spectrum

(eigenvalues){z ∈ C : Ker(T − z Id) 6= {0}}

6. Specc(T ) continuous spectrum

z ∈ C :

Ker(T − z Id) = {0} ,Im(T − z Id) H, andIm(T − z Id) = H

7. Specr(T ) residual spectrum

{z ∈ C :

{Ker(T − z Id) = {0} ,codim

(Im(T − z Id)

)> 0

}}

Example 1.16. (a) T := shift+ (see [234, 1970/1982, 5.3]) =⇒ Specp(T ) = ∅,Specr(T ) = {z ∈ C : |z| < 1} and Specc(T ) = Spece(T ) = {z ∈ C : |z| = 1} .(b) T compact and self-adjoint (see [234, 1970/1982, 6.2]) =⇒ Spec(T ) ⊆ R,Specp(T ) = {λn : n = 1, 2, . . .}, Specr(T ) = {0} and Spece(T ) = {0}.(c) T = Fourier transformation on L2(R) (see [134, 1972, p.98]) =⇒ Spec(T ) =Spece(T ) = Specp(T ) = {1, i,−1,−i}.

CHAPTER 2

Analytic Methods. Compact Operators

Synopsis. Adjoint and Self-Adjoint Operators — Recalling Fischer-Riesz. Dual

Characterization of Fredholm Operators. Compact Operators: Spectral Decomposition,

Why Compact Operators also are Called Completely Continuous, K as Two-Sided Ideal,Closure of Finite-Rank Operators, and Invariant under ∗. Classical Integral Operators.

Fredholm Alternative and Riesz Lemma. Sturm-Liouville Boundary Value Problems. Un-

bounded Operators: Comprehensive Study of Linear First Order Differential Operators

Over S1: Sobolev Space, Dirac Distribution, Normalized Integration Operator as Para-

metrix, The Index Theorem on the Circle for Systems. Closed Operators, Closed Ex-

tensions, Closed (not necessarily bounded) Fredholm Operators, Composition Rule, Sym-

metric and Self–Adjoint Operators, Formally Self-Adjoint and Essentially Self-Adjoint.

Spectral Theory. Metrics on the Space of Closed Operators. Trace Class and Hilbert-

Schmidt Operators.

1. Analytic Methods. The Adjoint Operator

I With Exercises 1.12 and 1.14, we have reached the limits of our so far purely al-gebraic reasoning where we could reduce everything to the elementary theory of solutionsof n linear equations in n unknowns. In fact, the limit process n → ∞ marks the emer-gence of functional analysis which went beyond the methods of linear algebra while beingmotivated by its questions and results. This occurred mainly in the study of integralequations.

In 1927, Ernst Hellinger and Otto Toeplitz stressed in their article Integralequations and equations in infinitely many unknowns in the Enzyklopädie der mathema-tischen Wissenschaften that “the essence of the theory of integral equations rests in theanalogy with analytic geometry and more generally in the passage from facts of alge-bra to facts of analysis” [201, p.1343]. They showed in a concise historical survey howthe awareness of these connections progressed in the centuries since Daniel Bernoulliinvestigated the oscillating string as a limit case of a system of n mass points:

“Orsus itaque sum has meditationes a corporibus duobus filo flexili indata distantia cohaerentibus; postea tria consideravi moxque quatuor, ettandem numerum eorum distantiasque qualescunque; cumque numerumcorporum infinitum facerem, vidi demum naturam oscillantis catenaesive aequalis sive inaequalis crassitiei sed ubique perfecte flexilis.”1

The passage to the limit means for the Fredholm integral equation of Exercise 1.14that more general nondegenerate weight functions G(x, y) are allowed which then can beapproximated by degenerate weights (e.g., polynomials). Such weights play a prominent

1Petropol. Comm. 6 (1732/33, ed. 1738), 108-122. Our translation: “In these considerations

I started with two bodies at a fixed distance and connected by an elastic string; next I considered

three then four and finally an arbitrary number with arbitrary distances between them; but onlywhen I made the number of bodies infinite did I fully comprehend the nature of an oscillating

elastic chain of equal or varying thickness.”

12

2.1. ANALYTIC METHODS. THE ADJOINT OPERATOR 13

role in applications, especially when dealing with differential equations with boundaryconditions, as we will see below. J

For the theory of Fredholm operators developed here we must analogouslyabandon the notion of an operator of finite rank and generalize it (to compactoperator, see below) whereby topological, i.e. continuity considerations, becomeessential in connection with the limit process. This will bring out the full force ofthe concept of Fredholm operator. We now turn to this topic.

Closed Image of Fredholm Operators. To begin with, the reader shouldderive the topological closedness of the image of a Fredholm operator from the purelyalgebraic property of finite codimension of the image.

Exercise 2.1. For F : H → H a Fredholm operator, prove:a)ImF is closed.b) There is an explicit criterion for deciding when an element of H lies in ImF .Namely, let n = dim CokerF ; then there are u1, ..., un ∈ H, such that for all w ∈ H,we have

w ∈ ImF ⇔ 〈w, u1〉 = · · · = 〈w, un〉 = 0.[Hint for a): As a vector subspace of H, naturally ImF is closed under additionand multiplication by complex numbers. However, here we are interested in topo-logical closure, namely that in passing to limit points we do not leave ImF . Thishas far-reaching consequences, since only closed subspaces inherit the completenessproperty of the ambient Hilbert space, and hence have orthonormal bases (= com-plete orthonormal vector systems, see e.g., [332, p.83] or [365, p.31 and Lemma11.9]). The trivial fact that finite-dimensional subspaces are closed can be exploited:Since CokerF = H/ ImF has finite dimension, we can find v1, ..., vn ∈ H whoseclasses in H/ ImF form a basis. The linear span h of v1, ..., vn is then an algebraiccomplement of ImF inH. Consider the map Φ: H⊕h→ H with Φ(u, v) := Fu+v.Since Φ is linear, surjective, and (by the boundedness of F ) continuous, we havethat Φ is open (according to the open mapping principle; e.g., see [332, p.53]). Itfollows that H \ F (H) = Φ(H ⊕ h \H ⊕ {0}) is open.[Hint for b): As a closed subspace of H, ImF is itself a Hilbert space possessinga countable orthonormal basis w1, w2, . . .. Now set ui := vi − Pvi, where vi areas above (i = 1, ..., n) and P : H → ImF is the projection Pu :=

∑∞j=1 〈u,wj〉wj .

Then {u1, ..., un} forms a basis for (ImF )⊥, the orthogonal complement of ImF inH.For aesthetic reasons, one can orthonormalize u1, ..., un by the Gram-Schmidt process(i.e., without loss of generality, assume they are orthonormal). Then u1, ..., un,w1, w2, w3, . . . is a countable orthonormal basis for H.]

Remark 2.2. Be aware that in spite of the strength of the open-mappingargument, it can not be applied to show that any subspace W of finite codimensiondimH/W < ∞ is closed. Of course, one could once again construct a boundedsurjective operator Φ: H ⊕W → H, say by Φ(u, v) := u + v. But, in general,H \W can not be obtained as the image of an open subset of H ⊕W by applyingΦ. Certainly H \W 6= Φ(H ⊕W \ H ⊕ {0}). Actually, the kernel Ker(f) of anyunbounded linear functional provides a counterexample. It is a space of codimension1, but it is not closed since closed Ker(f) would imply continuity of f in 0 and henceeverywhere.

14 2. ANALYTIC METHODS. COMPACT OPERATORS

Exercise 2.3. Once more, prove the chain rule indexG◦F = indexF+indexGfor Fredholm operators F : H → H ′ and G : H ′ → H ′′.[Hint: In place of the purely algebraic argument in Exercise 1.10, use Exercise 2.1ato first prove that the images are closed, and then use the technique of orthogonalcomplements in Exercise 2.1b.]

How trivial or nontrivial is it to prove that operators have closed ranges? Foroperators with finite rank and for surjective operators it is trivial, and for Fredholmoperators it was proved in Exercise 2.1a. Is it perhaps true that all bounded linearoperators have closed images? As the following counterexample explicitly shows,the answer is no. Moreover, we will see below that all compact operators withinfinite-dimensional image are counterexamples.

Exercise 2.4. For a Hilbert space H with orthonormal system e1, e2, e3, . . . ,consider the contraction operator

Au :=

∞∑j=1

1

j〈u, ej〉 ej .

Show that ImA is not closed.[Hint: Clearly A is linear and bounded (‖A‖ =?), and furthermore, we have thecriterion for Im(A)

v ∈ ImA⇔∞∑j=1

j 〈v, ej〉 ej ∈ H ⇔∞∑j=1

j2 |〈v, ej〉|2 1(e.g., for a = 1 + 1/n), but diverges for a = 1.) To finally prove that the sequenceactually converges to v0, observe that

‖v0 − vn‖2 =∞∑j=1

1

j2

(j1/n − 1

)2j j2/n

.

It is clear, that for each j

j1/n − 1j1/n

→ 0 as n→∞,

and then in particular 〈v0 − vn, ej〉 → 0. However, to show that ‖v0 − vn‖2 con-verges to 0 as n→∞, one must estimate more precisely. To do this, we exploit thefact that

∑∞j>j0

j−3 can be made smaller than any ε > 0 for j0 sufficiently large,

while on the other hand, choosing n sufficiently large (so large that (1 + ε)n ≥ j0),we have

j1/n − 1j1/n

< ε for j ≤ j0.]


The Adjoint Operator. We will draw further conclusions from the closureof the image of a Fredholm operator and to do this we introduce adjoint operators.The purpose is to eliminate the asymmetry between kernel and cokernel or, in otherwords, between the theory of the homogeneous equation (questions of uniquenessof solutions) and the theory of the inhomogeneous equation (questions of existenceof solutions). This is achieved by representing the cokernel of an operator as thekernel of a suitable adjoint operator.

Projective geometry deals with a comparable problem via duality: one thinks ofspace on the one hand as consisting of points, on the other as consisting of planes,and depending on the point of view, a straight line is the join of two points orthe intersection of two planes. Analytic geometry passes from a matrix (aij) to itstranspose(aji) or(aji) in the complex case to technically deal with dual statements.We can do the same successfully for operators (= infinite matrices). The basic toolis the following Representation Theorem with nice proofs in [332, Proposition 3.1.9]or [365, Theorem 2.1]. Originally, it was proven independently by Frigyes Rieszand Ernst Sigismund Fischer only for H := L2([a, b]).

Theorem 2.5 (E. Fischer, F. Riesz, 1907). Let H be a separable complexHilbert space. To each continuous linear mapping ϕ : H → C (called functional)there exists a unique element u ∈ H such that ϕv = 〈v, u〉 for all v ∈ H.

Exercise 2.6. Show that on the space B(H) of bounded linear operators of aHilbert space H, there is a natural isometric (anti-linear) involution

∗ : B(H)→ B(H)

which assigns to each T ∈ B(H) the adjoint operator T ∗ ∈ B(H) such that

〈u, T ∗v〉 = 〈Tu, v〉 for all u, v ∈ H.

[Hint: It is clear that T ∗v is well defined for each v ∈ H, since u 7→ 〈Tu, v〉 isa continuous linear functional on H; and so, by the preceding theorem, you canexpress the functional through a unique element of H, which you may denote byT ∗v. The linearity of T ∗ is clear by construction. While proving the continuity(i.e., boundedness) of T ∗, show more precisely that ‖T ∗‖ = ‖T‖ (i.e., that T 7→ T ∗is an isometry).]

Just as easily, we have the involution property T ∗∗ = T , the composition rule(T ◦R)∗ = R∗ ◦T ∗, and conjugate-linearity (aT +bR)∗ = āT ∗+ b̄R∗, where the barsdenote complex conjugation. Details can be found for example in [332, Theorem3.2.3], [365, Sections 3.2 and 11.2]. Observe that for T ∈ B(H,H ′), where H andH ′ may differ, the adjoint operator T ∗ is in B(H ′, H).

Theorem 2.7. For F ∈ F , the u1, ..., un in Exercise 2.1b form a basis ofKerF ∗, whence

ImF = (KerF ∗)⊥ and Coker(F ) = Ker(F ∗) .

Proof. First we note that u ∈ (ImF )⊥ exactly when

0 = 〈u,w〉 = 〈u, Fv〉 = 〈F ∗u, v〉 ,

for all w ∈ Im(F ) (i.e., for all v ∈ H); thus, (ImF )⊥ = KerF ∗. By again takingorthogonal complements, we have (KerF ∗)⊥ = (ImF )⊥⊥ = ImF , since ImF isclosed by Exercise 2.1a. �


Observe that the above argument remains valid for any bounded linear operatorwith closed range Such operators are also called normally-solvable operators. Inthis case, we have the criterion that the equation Fv = w is solvable exactly whenw⊥KerF ∗. This is the basic

Lemma 2.8 (Polar Lemma). Let H, H ′ be Hilbert spaces and T ∈ B(H,H ′).Then (ImT )⊥ = KerT ∗ . Moreover, if ImT is closed, we have ImT = (KerT ∗)⊥.

In the language of categories and functors (see e.g. [89, p.176]) this can bereformulated in the following, a bit exaggerated way:

Theorem 2.9. The functor H 7→ H, H ′ 7→ H ′, B(H,H ′) 3 T 7→ T ∗ ∈B(H ′, H) is a contravariant functor on the category of (separable) Hilbert spacesand bounded operators. It preserves the norm and is exact; i.e.,

HT−→ H ′ S−→ H ′′ R−→ exact =⇒ R

∗

−→ H ′′ S∗

−→ H ′ T∗

−→ H exact.

Proof. We show the exactness only at H ′, i.e., ImS∗ = KerT ∗. Since ST = 0,we have T ∗S∗ = 0, hence ImS∗ is contained in KerT ∗.

To show the opposite inclusion, we notice that ImT = KerS is closed in H ′

and ImS = KerR is closed in H ′′ . So we have a decomposition of

(2.1) H ′ = KerT ∗ ⊕KerS = (ImT )⊥ ⊕ ImTand

(2.2) H ′′ = ImS ⊕ (ImS)⊥

into pairs of mutually orthogonal closed subspaces. Notice also that

(2.3) S|KerT∗ : KerT ∗ −→ ImSis bounded, injective and surjective, hence its inverse is also bounded (though notnecessarily a Hilbert space isomorphism, i.e. not necessarily unitary).

Now let y ∈ H ′ with y ∈ KerT ∗ , i.e., 〈y, y′〉 = 0 for all y′ ∈ T (H). We considerthe mapping

[y] : H ′′ −→ C given by y′ + z′ 7→ 〈y′, y〉,where the splitting on the left side is according to (2.2) and the inner product onthe right side is taken in the Hilbert space H ′. By construction, the mapping [y] islinear and vanishes on the second factor of H ′′ . On the first factor it is continuousbecause of the homeomorphism of (2.3). Hence the functional [y] can be representedby an element of the Hilbert space H ′′ which we also will denote by [y]. So we have

(2.4) 〈y′ + y′′, S∗([y])〉 = 〈Sy′ + Sy′′, [y]〉 = 〈Sy′, [y]〉 = 〈y′, y〉 = 〈y′ + y′′, y〉for all elements y′ + y′′ ∈ H ′ with y′ ∈ (ImT )⊥ and y′′ ∈ ImT according to thedecomposition (2.1). Note that the second and the third inner product in (2.4)are taken in H ′′ and the other inner products in H ′ . Equation (2.4) shows thaty = S∗([y]). Thus KerT ∗ ⊆ ImS∗, and KerT ∗ = ImS∗, as desired. �

Theorem 2.10. A bounded linear operator F is a Fredholm operator, preciselywhen KerF and KerF ∗ are finite-dimensional and ImF is closed. In this case,

indexF = dim KerF − dim KerF ∗.Thus, in particular, indexF = 0 in case F is self-adjoint (i.e., F ∗ = F ).

Proof. Use Theorem 2.7 and Exercise 2.1a. �


Remark 2.11. a) Note that KerF ∗F = KerF : “⊇” is clear; for “⊆”, takeu ∈ KerF ∗F , and then

〈F ∗Fu, u〉 = 〈Fu, Fu〉 = 0,

and u ∈ KerF . If ImF is closed (e.g., if F ∈ F), then we also have

ImF ∗F = ImF ∗.

Here “⊆” is clear. To prove “⊇”, consider F ∗v for v ∈ H, and decompose vinto orthogonal components v = v′ + v′′ with v′ ∈ ImF and v′′ ∈ KerF ∗; thenF ∗v = F ∗v′. In this way, we then have represented the kernel and cokernel of anyFredholm operator F as the kernels of the self-adjoint operators F ∗F and FF ∗,respectively.b) The contraction operator A of Exercise 2.4 provides an example of a bounded,self-adjoint operator with KerA (= KerA∗) = {0} which is not a Fredholm opera-tor.

Corollary 2.12. Let H, H ′ be Hilbert spaces and F : H → H ′ a boundedFredholm operator. Then F ∗ : H ′ → H is a Fredholm operator and we have

KerF ∗ ∼= CokerF, CokerF ∗ ∼= KerF, and indexF ∗ = − indexF.

Proof. Consider the exact sequence

0 −→ KerF −→ H F−→ H ′ −→ CokerF −→ 0.

Then by Theorem 2.9, the sequence

0 −→ CokerF −→ H ′ F∗

−→ H −→ KerF −→ 0

is also exact, and the assertion follows. �

Positive operators. Another concept based on the scalar product is the no-tion of positive operators.

Definition 2.13. If C ∈ B := B(H) with 〈Cx, x〉 ≥ 0 for all x ∈ H, then Cis called a positive operator. We denote the (convex) set of such operators byB+.

Proposition 2.14. Any C ∈ B+is self-adjoint. Moreover, for any n ∈ N,Cn ∈ B+.

Proof. For all x ∈ H,

〈Cx, x〉 = 〈x,Cx〉 = 〈x,Cx〉 = 〈C∗x, x〉 =⇒ 〈(C − C∗)x, x〉 = 0.

For D := C − C∗ and all x, y ∈ H, we then have

〈D(x+ y) ,(x+ y)〉 = 0 =⇒ 〈Dx, y〉+ 〈Dy, x〉 = 0〈D(x+ iy) ,(x+ iy)〉 = 0 =⇒ −i 〈Dx, y〉+ i 〈Dy, x〉 = 0

}=⇒ 〈Dx, y〉 = 0.

So D = 0 and C = C∗. We now have〈C2x, x

〉= 〈Cx,Cx〉 ≥ 0 and for n ≥ 3,

〈Cnx, x〉 =〈Cn−2(Cx) , Cx

〉,

whence the positivity of Cn follows by induction. �


For real Hilbert spaces H, 〈Cx, x〉 ≥ 0 does not imply C = C∗ (e.g., 〈Ax, x〉 =0 ≥ 0 for any skew-symmetric A), but we assumed that H is complex.

Below, in Section 2.7 on trace class and Hilbert-Schmidt operators, we shallprove the fundamental Square Root Lemma (Theorem 2.66, p.54ff) for all operatorsbelonging to B+ by means of completely elementary arguments.

2. Compact Operators

So far we found that the space F of Fredholm operators is closed under compo-sition and passage to adjoints and particularly that all operators of the form Id +Tbelong to F when T is an operator of finite rank. We will increase this supply ofexamples, in passing to compact operators by taking limits. This, however, doesnot lead to Fredholm operators of nonzero index.

We begin with an exercise which emphasizes a simple topological property ofoperators of finite rank, more generally characterizes the finite-dimensional sub-spaces which are fundamental for the index concept, and prepares the introductionof compact operators.

Exercise 2.15. a) Every operator with finite rank maps the unit ball (or anybounded subset) of H to a relatively compact set.b) If H is finite-dimensional, then the closed unit ball BH := {u ∈ H : ‖u‖ ≤ 1}is compact.c) If H is infinite-dimensional, then BH is noncompact.[Hint for a)

index theory with applications to mathematics and physics...

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