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An Invitation to Operator Theory

An Invitation to Operator Theory​

(Graduate Studies in Mathematics, V. 50)​

By Y. A. Abramovich, Charalambos D. Aliprantis​





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  • Publisher: American Mathematical Society ​
  • Number Of Pages: 530 ​
  • Publication Date: 2002-09-10 ​
  • Binding: Hardcover ​
Product Description


This book offers a comprehensive and reader-friendly exposition of the theory of linear operators on Banach spaces and Banach lattices using their topological and order structures and properties. Abramovich and Aliprantis give a unique presentation that includes many new and very recent developments in operator theory and also draws together results which are spread over the vast literature. For instance, invariant subspaces of positive operators and the Daugavet equation are presented in monograph form for the first time.
The authors keep the discussion self-contained and use exercises to achieve this goal. The book contains over 600 exercises to help students master the material developed in the text. The exercises are of varying degrees of difficulty and play an important and useful role in the exposition. They help to free the proofs of the main results of some technical details but provide students with accurate and complete accounts of how such details ought to be worked out. The exercises also contain a considerable amount of additional material that includes many well-known results whose proofs are not readily available elsewhere.
The companion volume, Problems in Operator Theory, also by Abramovich and Aliprantis, is available from the AMS as Volume 51 in the Graduate Studies in Mathematics series, and it contains complete solutions to all exercises in An Invitation to Operator Theory.
The solutions demonstrate explicitly technical details in the proofs of many results in operator theory, providing the reader with rigorous and complete accounts of such details. Finally, the book offers a considerable amount of additional material and further developments. By adding extra material to many exercises, the authors have managed to keep the presentation as self-contained as possible. The best way of learning mathematics is by doing mathematics, and the book Problems in Operator Theory will help achieve this goal.
Prerequisites to each book are the standard introductory graduate courses in real analysis, general topology, measure theory, and functional analysis. An Invitation to Operator Theory is suitable for graduate or advanced courses in operator theory, real analysis, integration theory, measure theory, function theory, and functional analysis. Problems in Operator Theory is a very useful supplementary text in the above areas. Both books will be of great interest to researchers and students in mathematics, as well as in physics, economics, finance, engineering, and other related areas, and will make an indispensable reference tool.​



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Basic Operator Theory

Basic Operator Theory ​

by Israel Gohberg, Seymour Goldberg ​



TABLE OF CONTENTS ​


INTRODUCTION xi



CHAPTER I. HILBERT SPACES 1


1. Complex n-space 1


2. The Hilbert space I2 3


3. Definition of Hilbert space and its elementary properties 5


4. Distance from a point to a finite dimensional subspace 10


5. The Gram determinant 12


6. Incompatible systems of equations 16


7. Least squares fit 17


8. Distance to a convex set and projections onto subspaces 19


9. Orthonormal systems 21


10. Legendre polynomials 22


11. Orthonormal Bases 25


12. Fourier series 28


13. Completeness of the Legendre polynomials 30


14. Bases for the Hilbert space of functions on a square 31


15. Stability of orthonormal bases 33


16. Separable spaces 34


17. Equivalence of Hilbert spaces 36


18. Example of a non separable space 37


EXERCISES I 38



CHAPTER II. BOUNDED LINEAR OPERATORS ON HILBERT SPACES 51


1. Properties of bounded linear operators 51


2. Examples of bounded linear operators with estimates of norms 53


3. Continuity of a linear operator 57


4. Matrix representations of bounded linear operators 58


5. Bounded linear functionals 60


6. Operators of finite rank 63


7. Invertible operators 65


8. Inversion of operators by the iterative method 70


9. Infinite systems of linear equations 72


10. Integral equations of the second kind 74


11. Adjoint operators 77


12. Self adjoint operators 80


13. Orthogonal projections 82


14. Compact operators 83


15. Invariant subspaces 88


EXERCISES II 91



CHAPTER III, SPECTRAL THEORY OF COMPACT SELF ADJOINT OPERATORS 105


1. Example of an infinite dimensional generalization 106


2. The problem of existence of eigenvalues and eigenvectors 106


3. Eigenvalues and eigenvectors of operators of finite rank 108


4. Theorem of existence of eigenvalues 110


5. Spectral theorem 113


6. Basic systems of eigenvalues and eigenvectors 115


7. Second form of the spectral theorem 118


8. Formula for the inverse operator 119


9. Minimum-Maximum properties of eigenvalues 121


EXERCISES III 125


CHAPTER IV. SPECTRAL THEORY OF INTEGRAL OPERATORS 131

1. Hilbert-Schmidt theorem 131


2. Preliminaries for Mercer's theorem 134

3 . Mercer's theorem 136

4. Trace formula for integral operators 138

5. Integral operators as inverses of differential operators 139

6. Sturm-Liouville systems 142

EXERCISES IV 148


CHAPTER V. OSCILLATIONS OF AN ELASTIC STRING 153


1. The displacement function 153

2. Basic harmonic oscillations 155

3. Harmonic oscillations with an external force . 157


CHAPTER VI. OPERATIONAL CALCULUS WITH APPLICATIONS 159

1. Functions of a compact self adjoint operator 159


2. Differential equations in Hilbert space 165

3. Infinite systems of differential equations ... 167

4. Integro-differential equations 168

EXERCISES VI 170


CHAPTER VII. SOLVING LINEAR EQUATIONS BY ITERATIVE METHODS 73

1. The ma in theorem 173

2. Preliminaries for the proof 174

3. Proof of the main theorem 177

4. Application to integral equations 179


CHAPTER VIII. FURTHER DEVELOPMENTS OF THE SPECTRAL THEOREM 181

1. Simultaneous diagonalization 181

2. Compact normal operators 182

3. Unitary operators 184

4. Characterizations of compact operators 187

EXERCISES VIII 189


CHAPTER IX. BANACH SPACES 193

1. Definitions and examples 194

2. Finite dimensional normed linear spaces 196

3. Separable Banach spaces and Schauder bases 200

4. Conjugate spaces 201

5. Hahn-Banach theorem 203

EXERCISES IX 206


CHAPTER X. LINEAR OPERATORS ON A BANACH SPACE 211

1. Description of bounded operators 211

2. An approximation scheme 214

3. Closed linear operators 219

4. Closed graph theorem and its applications 221

5. Complemented subspaces and projections 224

6. The spectrum of an operator 226

7. Volterra Integral Operator 229

8. Analytic operator valued functions 231

EXERCISES X 232


CHAPTER XI. COMPACT OPERATORS ON A BANACH SPACES 237

1. Examples of compact operators 237

2. Decomposition of operators of finite rank 240

3. Approximation by operators of finite rank 241

H. Fredholm theory of compact operators 242

5. Conjugate operators on a Banach space 245

6. Spectrum of a compact operator 248

7. Applications 251

EXERCISES XI 253


CHAPTER XII. NON LINEAR OPERATORS 255

1. Fixed point theorem 255

2. Applications of the contraction mapping theorem 256

3. Generalizations 261


APPENDIX 1. COUNTABLE SETS AND SEPARABLE HILBERT SPACES 265

APPENDIX 2. LEBES6UE INTEGRATION AND Lp SPACES 267

APPENDIX 3. PROOF OF THE HAHN-BANACH THEOREM 273

APPENDIX 4. PROOF OF THE CLOSED GRAPH THEOREM 277


SUGGESTED READING 280


REFERENCES 281
INDEX 282


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