Basic Operator Theory

rii application of linear operators on a Hilbert space. We begin with a chapter on the geometry of Hilbert space and then proceed to the spectral theory of compact self adjoint operators; operational calculus is next presented as a nat­ ural outgrowth of the spectral theory. The second part of the text concentrates on Banach spaces and linear operators acting on these spaces. It includes, for example, the three 'basic principles of linear analysis and the Riesz­ Fredholm theory of compact operators. Both parts contain plenty of applications. All chapters deal exclusively with linear problems, except for the last chapter which is an introduction to the theory of nonlinear operators. In addition to the standard topics in functional anal­ ysis, we have presented relatively recent results which appear, for example, in Chapter VII. In general, in writ­ ing this book, the authors were strongly influenced by re­ cent developments in operator theory which affected the choice of topics, proofs and exercises. One of the main features of this book is the large number of new exercises chosen to expand the reader's com­ prehension of the material, and to train him or her in the use of it. In the beginning portion of the book we offer a large selection of computational exercises; later, the proportion of exercises dealing with theoretical questions increases. We have, however, omitted exercises after Chap­ ters V, VII and XII due to the specialized nature of the subject matter.

Table of Contents:
I. Hilbert Spaces.- 1. Complex n-space.- 2. The Hubert space ?2.- 3. Definition of Hubert space and its elementary properties.- 4. Distance from a point to a finite dimensional subspace.- 5. The Gram determinant.- 6. Incompatible systems of equations.- 7. Least squares fit.- 8. Distance to a convex set and projections onto subspaces.- 9. Orthonormal systems.- 10. Legendre polynomials.- 11. Orthonormal Bases.- 12. Fourier series.- 13. Completeness of the Legendre polynomials.- 14. Bases for the Hubert space of functions on a square.- 15. Stability of orthonormal bases.- 16. Separable spaces.- 17. Equivalence of Hilbert spaces.- 18. Example of a non separable space.- Exercises I.- II. Bounded Linear Operators on Hilbert Spaces.- 1. Properties of bounded linear operators.- 2. Examples of bounded linear operators with estimates of norms.- 3. Continuity of a linear operator.- 4. Matrix representations of bounded linear operators.- 5. Bounded linear functionals.- 6. Operators of finite rank.- 7. Invertible operators.- 8. Inversion of operators by the iterative method.- 9. Infinite systems of linear equations.- 10. Integral equations of the second kind.- 11. Adjoint operators.- 12. Self adjoint operators.- 13. Orthogonal projections.- 14. Compact operators.- 15. Invariant subspaces.- Exercises II.- III. Spectral Theory of Compact Self Adjoint Operators.- 1. Example of an infinite dimensional generalization.- 2. The problem of existence of eigenvalues and eigenvectors.- 3. Eigenvalues and eigenvectors of operators of finite rank.- 4. Theorem of existence of eigenvalues.- 5. Spectral theorem.- 6. Basic systems of eigenvalues and eigenvectors.- 7. Second form of the spectral theorem.- 8. Formula for the inverse operator.- 9. Minimum-Maximum properties of eigenvalues.- ExercisesIII.- IV. Spectral Theory of Integral Operators.- 1. Hilbert-Schmidt theorem.- 2. Preliminaries for Mercer’s theorem.- 3. Mercer’s theorem.- 4. Trace formula for integral operators.- 5. Integral operators as inverses of differential operators.- 6. Sturm-Liouville systems.- Exercises IV.- V. Oscillations of an Elastic String.- 1. The displacement function.- 2. Basic harmonic oscillations.- 3. Harmonic oscillations with an external force.- VI. Operational Calculus with Applications.- 1. Functions of a compact self adjoint operator.- 2. Differential equations in Hubert space.- 3. Infinite systems of differential equations.- 3. Integro-differential equations.- Exercises VI.- VII. Solving Linear Equations by Iterative Methods.- 1. The main theorem.- 2. Preliminaries for the proof.- 3. Proof of the main theorem.- 4. Application to integral equations.- VIII. Further Developments of the Spectral Theorem.- 1. Simultaneous diagonalization.- 2. Compact normal operators.- 3. Unitary operators.- 4. Characterizations of compact operators.- Exercises VIII.- IX. Banach Spaces.- 1. Definitions and examples.- 2. Finite dimensional normed linear spaces.- 3. Separable Banach spaces and Schauder bases.- 4. Conjugate spaces.- 5. Hahn-Banach theorem.- Exercises IX.- X. Linear Operators on a Banach Space.- 1. Description of bounded operators.- 2. An approximation scheme.- 3. Closed linear operators.- 4. Closed graph theorem and its applications.- 5. Complemented subspaces and projections.- 6. The spectrum of an operator.- 7. Volterra Integral Operator.- 8. Analytic operator valued functions.- Exercises X.- XI. Compact Operators on a Banach Spaces.- 1. Examples of compact operators.- 2. Decomposition of operators of finite rank.- 3. Approximation by operators of finite rank.- 4. Fredholmtheory of compact operators.- 5. Conjugate operators on a Banach space.- 6. Spectrum of a compact operator.- 7. Applications.- Exercises XI.- XII. Non Linear Operators.- 1. Fixed point theorem.- 2. Applications of the contraction mapping theorem.- 3. Generalizations.- Appendix 1. Countable Sets and Separable Hilbert Spaces.- Appendix 3. Proof of the Hahn-Banach Theorem.- Appendix 4. Proof of the Closed Graph Theorem.- Suggested Reading.- References.