1 Banach Spaces

1 The Banach Space of Continuous Functions

2 Abstract Banach Spaces

3 The Conjugate Space of Continuous Linear Functionals

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

5 Weak Topologies on Banach Spaces

6 The Alaoglu Theorem

7 The Hahn-Banach Theorem

8 The Conjugate Space of C（［0,1］）

9 The Open Mapping Theorem

10 The Lebesgue Spaces：L1 and L∞

11 The Hardy Spaces：H1 and H∞

Notes

Exercises

2 Banach Algebras

1 The Banach Algebra of Confinuous Functions

2 Abstract Banach Algebras

3 Abstract Index in a Banach Algebra

4 The Space of Multiplicative Linear Functions

5 The Gelfand Transform

6 The Gelfand-Mazur Theorem

7 The Gelfand Theorem for Commutative Banach Algebras

8 The Spectral Radius Formula

9 The Stone-Weierstrass Theorem

10 The Generalized Stone-Weierstrass Theorem

11 The Disk Algebra

12 The Algebra of Functions with Absolutely Convergent Fourier Series

13 The Algebra of Bounded Measurable Functions

Notes

Exercises

3 Geometry of Hilbert Space

1 Inner Product Spaces

2 The Cauchy-Schwarz Inequality

3 The Pythagorean Theorem

4 Hilbert Spaces

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

6 The Riesz Representation Theorem

7 The Existence of Orthonormal Bases

8 The Dimension of Hilbert Spaces

Notes

Exercises

4 Operators on Hilbert Space and C-Algebras

1 The Adioint Operator

2 Normal and Self-adjoint Operators

3 Projections and Subspaces

4 Multiplication Operators and Maximal Abelian Algebras

5 The Bilateral Shift Operator

6 C-Algebras

7 The Gelfand-Naimark Theorem

8 The Spectral Theorem

9 The Functional Calculus

10 The Square Root of Positive Operators

11 The Unilateral Shift Operator

12 The Polar Decomposition

13 Weak and Strong Operator Topologies

14 W-Algebras

15 Isomorphisms of L∞-Spaces

16 Normal Operators with Cyclic Vectors

17 Maximal Abelian W-Algebras

18 Homomorphisms of C-Algebras

19 The Extended Functional Calculus

20 The Fuglede Theorem

Notes

Exercises

5 Compact Operators,Fredholm Operators,and Index Theory

1 The Ideals of Finite Rank and Compact Operators

2 Approximation of Compact Operators

3 Examples of Compact Operators：Integral Operators

4 The Calkin Algebra and Fredholm Operators

5 Atkinson's Theorem

6 The Index of Fredholm Operators

7 The Fredholm Altemative

8 Volterra Integral Operators

9 Connectedness of the Unitary Group in a W-Algebra

10 Characterization of Index

11 Quotient C-Algebras

12 Representations of the C-Algebra of Compact Operators

Notes

Exercises

6 The Hardy Spaces

1 The Hardy Spaces：H1,H2,and H∞

2 Reducing Subspaces of Unitary Operators

3 Beurling's Theorem

4 The F.and M.Riesz Theorem

5 The Maximal Ideal Space of H∞

6 The Inner-Outer Factorization of Functions in H2

7 The Modulus of Outer Functions

8 The Conjugates of H1 and L∞／H∞0

9 The Closedness of H∞＋C

10 Approximation by Quotients of Inner Functions

11 The Gleason-Whitney Theorem

12 Subalgebras between H∞ and L∞

13 Abstract Harmonic Extensions

14 The Maximal Ideal Space of H∞＋C

15 The Invertibility of Functions in H∞＋C

Notes

Exercises

7 Toeplitz Operators

1 Toeplitz Operators

2 The Spectral Inclusion Theorem

3 The Symbol Map

4 The Spectrum of Self-adjoint Toeplitz Operators

5 The Spectrum of Analytic Toeplitz Operators

6 The C-Algebra Generated by the Unilateral Shift

7 The Invertibility of Toeplitz Operators with Continuous Symbol

8 The Invertibility of Unimodular Toeplitz Operators and Prediction Theory

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

10 The Connectedness of the Essential Spectrum

11 Localization to the Center of a C-Algebra

12 Locality of Fredholmness for Toeplitz Operators

Notes

Exercises

References

Index