Author | Gohberg, Israel. author |
---|---|
Title | Basic Operator Theory [electronic resource] / by Israel Gohberg, Seymour Goldberg |
Imprint | Boston, MA : Birkhรคuser Boston, 2001 |
Connect to | http://dx.doi.org/10.1007/978-1-4612-5985-5 |
Descript | XIII, 304 p. online resource |
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 -- Exercises III -- IV. Spectral Theory of Integral Operators -- 1. Hilbert-Schmidt theorem -- 2. Preliminaries for Mercerโs theorem -- 3. Mercerโs 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. Fredholm theory 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