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1 Banach Spaces 1

1 The Banach Space of Continuous Functions 2

2 Abstract Banach Spaces 2

3 The Conjugate Space of Continuous Linear Functionals 5

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

5 Weak Topologies on Banach Spaces 8

6 The Alaoglu Theorem 9

7 The Hahn-Banach Theorem 10

8 The Conjugate Space of C（［0,1］） 12

9 The Open Mapping Theorem 21

10 The Lebesgue Spaces：L1 and L∞ 23

11 The Hardy Spaces：H1 and H∞ 25

Notes 26

Exercises 26

2 Banach Algebras 30

1 The Banach Algebra of Confinuous Functions 30

2 Abstract Banach Algebras 31

3 Abstract Index in a Banach Algebra 33

4 The Space of Multiplicative Linear Functions 36

5 The Gelfand Transform 37

6 The Gelfand-Mazur Theorem 39

7 The Gelfand Theorem for Commutative Banach Algebras 41

8 The Spectral Radius Formula 42

9 The Stone-Weierstrass Theorem 43

10 The Generalized Stone-Weierstrass Theorem 43

11 The Disk Algebra 47

12 The Algebra of Functions with Absolutely Convergent Fourier Series 50

13 The Algebra of Bounded Measurable Functions 52

Notes 53

Exercises 53

3 Geometry of Hilbert Space 58

1 Inner Product Spaces 58

2 The Cauchy-Schwarz Inequality 59

3 The Pythagorean Theorem 60

4 Hilbert Spaces 61

5 Examples of Hilbert Spaces：Cn,l2,L2,and H2 61

6 The Riesz Representation Theorem 66

7 The Existence of Orthonormal Bases 69

8 The Dimension of Hilbert Spaces 70

Notes 71

Exercises 71

4 Operators on Hilbert Space and C-Algebras 74

1 The Adioint Operator 75

2 Normal and Self-adjoint Operators 77

3 Projections and Subspaces 78

4 Multiplication Operators and Maximal Abelian Algebras 80

5 The Bilateral Shift Operator 82

6 C-Algebras 83

7 The Gelfand-Naimark Theorem 84

8 The Spectral Theorem 85

9 The Functional Calculus 85

10 The Square Root of Positive Operators 86

11 The Unilateral Shift Operator 87

12 The Polar Decomposition 88

13 Weak and Strong Operator Topologies 91

14 W-Algebras 92

15 Isomorphisms of L∞-Spaces 94

16 Normal Operators with Cyclic Vectors 95

17 Maximal Abelian W-Algebras 97

18 Homomorphisms of C-Algebras 100

19 The Extended Functional Calculus 101

20 The Fuglede Theorem 103

Notes 104

Exercises 104

5 Compact Operators,Fredholm Operators,and Index Theory 108

1 The Ideals of Finite Rank and Compact Operators 108

2 Approximation of Compact Operators 110

3 Examples of Compact Operators：Integral Operators 112

4 The Calkin Algebra and Fredholm Operators 113

5 Atkinson's Theorem 114

6 The Index of Fredholm Operators 115

7 The Fredholm Altemative 116

8 Volterra Integral Operators 118

9 Connectedness of the Unitary Group in a W-Algebra 119

10 Characterization of Index 123

11 Quotient C-Algebras 124

12 Representations of the C-Algebra of Compact Operators 126

Notes 129

Exercises 129

6 The Hardy Spaces 133

1 The Hardy Spaces：H1,H2,and H∞ 133

2 Reducing Subspaces of Unitary Operators 135

3 Beurling's Theorem 136

4 The F.and M.Riesz Theorem 137

5 The Maximal Ideal Space of H∞ 138

6 The Inner-Outer Factorization of Functions in H2 141

7 The Modulus of Outer Functions 141

8 The Conjugates of H1 and L∞／H∞0 144

9 The Closedness of H∞＋C 145

10 Approximation by Quotients of Inner Functions 145

11 The Gleason-Whitney Theorem 146

12 Subalgebras between H∞ and L∞ 147

13 Abstract Harmonic Extensions 149

14 The Maximal Ideal Space of H∞＋C 150

15 The Invertibility of Functions in H∞＋C 152

Notes 153

Exercises 153

7 Toeplitz Operators 158

1 Toeplitz Operators 158

2 The Spectral Inclusion Theorem 160

3 The Symbol Map 160

4 The Spectrum of Self-adjoint Toeplitz Operators 163

5 The Spectrum of Analytic Toeplitz Operators 164

6 The C-Algebra Generated by the Unilateral Shift 164

7 The Invertibility of Toeplitz Operators with Continuous Symbol 166

8 The Invertibility of Unimodular Toeplitz Operators and Prediction Theory 167

9 The Spectrum of Toeplitz Operators with Symbol in H∞＋C 169

10 The Connectedness of the Essential Spectrum 174

11 Localization to the Center of a C-Algebra 177

12 Locality of Fredholmness for Toeplitz Operators 178

Notes 178

Exercises 181

References 185

Index 191