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

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  • 出版年份:1932
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  • 页数:622 页
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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