Preface to the Second Edition Preface to the First Edition Acknowledgments Symbols and Notation 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， l， 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: Ll and L 11 The Hardy Spaces: Hl and H Notes Exercises 2 Banach Algebras 1 The Banach Algebra of Continuous 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 Absolutey Convergent Fourier series 13 the Algebra of Bounded Measurable Functions Notes exercises 3 Geometry of Hilbert Space l 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 Existenee of Orthonormal Bases 8 The Dimension Of Hilbert Spaces Notes Exercises 4 Operators on Hilbert Space and C*-Algebras 1 The Adjoint Operator 2 Normal and Self-adjoint Operators 3 Proiections and Subspaces 4 Multiplication Operators and Maximal Abelian Algebras
5 rnle Bilateral Shift Operator 6 C*-Algebras 7 The Gelfand-Naimark Theorem 8 The Spectral Theorem 9 The Funcfional Calculus 10 The Square Root of Positive Operators 11 The Unilateral Shift Operator 12 The Po1ar Decomposition 13 Weak and Strong Operator Topologies 14 W*-Algebras 15 Isomorphisms of L-Spaces 16 Normal Operators with Cycfic 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 l 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 Frcdholm Operators 5 Atkinson's Theorem 6 The Index of Frcdholm Operators 7 The Fredholm Altemative 8 Volterra Integral Operators 9 Connectedness of the Unitary Group in a W*-Algebras 10 Characterization of Index 1l Quotient C*-Algebras 12 Representations of the C*-Algebra of Compact Operators Notes Exercises 6 The Hardy Spaces l The Hardy Spaces: Hl, 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/H0 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 Inverfibility of Toeplitz Operators witll 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 Essenfial Spectrum 11 Localization to the Centerofa C*-Algebra 12 Locality of Fredholmness for Toeplitz Operators Notes Exercises References Index
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