《LINEAR TRANSFORMATIONS IN HILBERT SPACE AND THEIR APPLICATIONS TO ANALYSIS》

CHAPTER ⅠABSTRACT HILBERT SPACE AND ITS REALIZATIONS1

1．The Concept of Space1

2．Abstract Hilbert Space2

3．Abstract Unitary Spaces16

4．Linear Manifolds in Hilbert Space18

5．Realizations of Abstract Hilbert Space23

CHAPTER ⅡTRANSFORMATIONS IN HILBERT SPACE33

1．Linear Transformations33

2．Symmetric Transformations49

3．Bounded Linear Transformations53

4．Projections70

5．Isometric and Unitary Transformations76

6．Unitary Invariance83

CHAPTER ⅢEXAMPLES OF LINEAR TRANSFORMATIONS86

1．Infinite Matrices86

2．Integral Operators98

3．Differential Operators112

4．Operators of Other Types124

CHAPTER ⅣRESOLVENTS,SPECTRA,REDUCIBILITY125

1．The Fundamental Problems125

2．Resolvents and Spectra128

8．Reducibility150

CHAPTER ⅤSELF-ADJOINT TRANSFORMATIONS155

1．Analytical Methods155

2．Analytical Representation of the Resolvent165

3．The Reducibility of the Resolvent172

4．The Analytical Representation of a Self-Adjoint Transformation180

5．The Spectrum of a Self-Adjoint Transformation184

CHAPTER ⅥTHE OPERATIONAL CALCULUS198

1．The Radon-Stieltjes Integral198

2．The Operational Calculus221

CHAPTER ⅦTHE UNITARY EQUIVALENCE OF SELF-ADJOINT TRANSFORMATIONS242

1．Preparatory Theorems242

2．Unitary Equivalence247

3．Self-Adjoint Transformations with Simple Spectra275

4．The Reducibility of Self-Adjoint Transformations288

5．Reduction to Principal Axes294

CHAPTER ⅧGENERAL TYPES OF LINEAR TRANSFORMATIONS299

1．Permutability299

2．Unitary Transformations302

3．Normal Transformations311

4．A Theorem on Factorization331

CHAPTER ⅨSYMMETRIC TRANSFORMATIONS334

1．The General Theory334

2．Real Transformations357

3．Approximation Theorems365

CHAPTER ⅩAPPLICATIONS397

1．Integral Operators397

2．Ordinary Differential Operators of the First Order424

3．Ordinary Differential Operators of the Second Order448

4．Jacobi Matrices and Allied Topics530

Index615