Basic Operator Theory

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.