巴拿赫代数在算子理论中的应用【2025|PDF|Epub|mobi|kindle电子书版本百度云盘下载】

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1 Banach Spaces1

1 The Banach Space of Continuous Functions2

2 Abstract Banach Spaces2

3 The Conjugate Space of Continuous Linear Functionals5

4 Examples of Banach spaces：co,l1,and l∞6

5 Weak Topologies on Banach Spaces8

6 The Alaoglu Theorem9

7 The Hahn-Banach Theorem10

8 The Conjugate Space of C（［0,1］）12

9 The Open Mapping Theorem21

10 The Lebesgue Spaces：L1 and L∞23

11 The Hardy Spaces：H1 and H∞25

Notes26

Exercises26

2 Banach Algebras30

1 The Banach Algebra of Confinuous Functions30

2 Abstract Banach Algebras31

3 Abstract Index in a Banach Algebra33

4 The Space of Multiplicative Linear Functions36

5 The Gelfand Transform37

6 The Gelfand-Mazur Theorem39

7 The Gelfand Theorem for Commutative Banach Algebras41

8 The Spectral Radius Formula42

9 The Stone-Weierstrass Theorem43

10 The Generalized Stone-Weierstrass Theorem43

11 The Disk Algebra47

12 The Algebra of Functions with Absolutely Convergent Fourier Series50

13 The Algebra of Bounded Measurable Functions52

Notes53

Exercises53

3 Geometry of Hilbert Space58

1 Inner Product Spaces58

2 The Cauchy-Schwarz Inequality59

3 The Pythagorean Theorem60

4 Hilbert Spaces61

5 Examples of Hilbert Spaces：Cn,l2,L2,and H261

6 The Riesz Representation Theorem66

7 The Existence of Orthonormal Bases69

8 The Dimension of Hilbert Spaces70

Notes71

Exercises71

4 Operators on Hilbert Space and C-Algebras74

1 The Adioint Operator75

2 Normal and Self-adjoint Operators77

3 Projections and Subspaces78

4 Multiplication Operators and Maximal Abelian Algebras80

5 The Bilateral Shift Operator82

6 C-Algebras83

7 The Gelfand-Naimark Theorem84

8 The Spectral Theorem85

9 The Functional Calculus85

10 The Square Root of Positive Operators86

11 The Unilateral Shift Operator87

12 The Polar Decomposition88

13 Weak and Strong Operator Topologies91

14 W-Algebras92

15 Isomorphisms of L∞-Spaces94

16 Normal Operators with Cyclic Vectors95

17 Maximal Abelian W-Algebras97

18 Homomorphisms of C-Algebras100

19 The Extended Functional Calculus101

20 The Fuglede Theorem103

Notes104

Exercises104

5 Compact Operators,Fredholm Operators,and Index Theory108

1 The Ideals of Finite Rank and Compact Operators108

2 Approximation of Compact Operators110

3 Examples of Compact Operators：Integral Operators112

4 The Calkin Algebra and Fredholm Operators113

5 Atkinson's Theorem114

6 The Index of Fredholm Operators115

7 The Fredholm Altemative116

8 Volterra Integral Operators118

9 Connectedness of the Unitary Group in a W-Algebra119

10 Characterization of Index123

11 Quotient C-Algebras124

12 Representations of the C-Algebra of Compact Operators126

Notes129

Exercises129

6 The Hardy Spaces133

1 The Hardy Spaces：H1,H2,and H∞133

2 Reducing Subspaces of Unitary Operators135

3 Beurling's Theorem136

4 The F.and M.Riesz Theorem137

5 The Maximal Ideal Space of H∞138

6 The Inner-Outer Factorization of Functions in H2141

7 The Modulus of Outer Functions141

8 The Conjugates of H1 and L∞／H∞0144

9 The Closedness of H∞＋C145

10 Approximation by Quotients of Inner Functions145

11 The Gleason-Whitney Theorem146

12 Subalgebras between H∞ and L∞147

13 Abstract Harmonic Extensions149

14 The Maximal Ideal Space of H∞＋C150

15 The Invertibility of Functions in H∞＋C152

Notes153

Exercises153

7 Toeplitz Operators158

1 Toeplitz Operators158

2 The Spectral Inclusion Theorem160

3 The Symbol Map160

4 The Spectrum of Self-adjoint Toeplitz Operators163

5 The Spectrum of Analytic Toeplitz Operators164

6 The C-Algebra Generated by the Unilateral Shift164

7 The Invertibility of Toeplitz Operators with Continuous Symbol166

8 The Invertibility of Unimodular Toeplitz Operators and Prediction Theory167

9 The Spectrum of Toeplitz Operators with Symbol in H∞＋C169

10 The Connectedness of the Essential Spectrum174

11 Localization to the Center of a C-Algebra177

12 Locality of Fredholmness for Toeplitz Operators178

Notes178

Exercises181

References185

Index191