In the quarter century since the first edition of this book appeared, tremendous development has occurred in operator theory and the topics covered here. However, the new edition remains unchanged except that several mistakes and typographical errors have been corrected. Further, a brief report on the current state of the doubleasterisk, open, problems is given along with references. No attempt is made to describe other progress that has been made in the study of Toeplitz operators and related topics nor has the bibliography been updated.

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

……

4 Operators on Hilbert Space and C*-Algebras

5 Compact Operators,Fredholm Operators,and Index Theory

6 The Hardy Spaces

7 Toeplitz Operators

References

Index