Contents

Preface ix

Chapter 1. Unbounded Linear Operators 1

1. Introduction 1

2. Closed Linear Operators 5

3. Analytic Vector-Valued Functions 9

4. Spectral Theory 21

5. Poles of the Resolvent 35

Chapter 2. Fredholm Operators 41

1. Basic Properties 41

2. Spectral Theory for Fredholm Operators 44

3. Spectral Theory for Index Zero 58

4. Hilbert-Schmidt Operators 64

5. Quasi-Nilpotent Hilbert-Schmidt Operators 70

6. A Hilbert-Schmidt Completeness Theorem 78

Chapter 3. Introduction to the Spectral Theory of

Differential Operators 83

1. An Overview 83

2. Sobolev Spaces 87

3. The Characteristic Determinant and Eigenvalues 89

4. Algebraic Multiplicities 92

Chapter 4. Principal Part of a Differential Operator 97

1. The Principal Part T 97

2. The Characteristic Determinant of T 98

3. The Green's Function of XI - T 103

4. Alternate Representations 107

5. The Boundary Values: Case n = 2v 110

6. The Boundary Values: Case n = 2v — 1 118

7. The Eigenvalues: Case n = 2v 128

8. The Eigenvalues: Case n = 2v — 1 146

9. Completeness of the Generalized Eigenfunctions 181

Chapter 5. Projections and Generalized Eigenfunction Expansions 193

1. The Associated Projections: n — 2v 193

2. The Associated Projections: n = 2v — 1 201

3. Expansions in the Generalized Eigenfunctions 206