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CHAPTER Ⅰ ABSTRACT HILBERT SPACE AND ITS REALIZATIONS 1

1．The Concept of Space 1

2．Abstract Hilbert Space 2

3．Abstract Unitary Spaces 16

4．Linear Manifolds in Hilbert Space 18

5．Realizations of Abstract Hilbert Space 23

CHAPTER Ⅱ TRANSFORMATIONS IN HILBERT SPACE 33

1．Linear Transformations 33

2．Symmetric Transformations 49

3．Bounded Linear Transformations 53

4．Projections 70

5．Isometric and Unitary Transformations 76

6．Unitary Invariance 83

CHAPTER Ⅲ EXAMPLES OF LINEAR TRANSFORMATIONS 86

1．Infinite Matrices 86

2．Integral Operators 98

3．Differential Operators 112

4．Operators of Other Types 124

CHAPTER Ⅳ RESOLVENTS,SPECTRA,REDUCIBILITY 125

1．The Fundamental Problems 125

2．Resolvents and Spectra 128

8．Reducibility 150

CHAPTER Ⅴ SELF-ADJOINT TRANSFORMATIONS 155

1．Analytical Methods 155

2．Analytical Representation of the Resolvent 165

3．The Reducibility of the Resolvent 172

4．The Analytical Representation of a Self-Adjoint Transformation 180

5．The Spectrum of a Self-Adjoint Transformation 184

CHAPTER Ⅵ THE OPERATIONAL CALCULUS 198

1．The Radon-Stieltjes Integral 198

2．The Operational Calculus 221

CHAPTER Ⅶ THE UNITARY EQUIVALENCE OF SELF-ADJOINT TRANSFORMATIONS 242

1．Preparatory Theorems 242

2．Unitary Equivalence 247

3．Self-Adjoint Transformations with Simple Spectra 275

4．The Reducibility of Self-Adjoint Transformations 288

5．Reduction to Principal Axes 294

CHAPTER Ⅷ GENERAL TYPES OF LINEAR TRANSFORMATIONS 299

1．Permutability 299

2．Unitary Transformations 302

3．Normal Transformations 311

4．A Theorem on Factorization 331

CHAPTER Ⅸ SYMMETRIC TRANSFORMATIONS 334

1．The General Theory 334

2．Real Transformations 357

3．Approximation Theorems 365

CHAPTER Ⅹ APPLICATIONS 397

1．Integral Operators 397

2．Ordinary Differential Operators of the First Order 424

3．Ordinary Differential Operators of the Second Order 448

4．Jacobi Matrices and Allied Topics 530

Index 615