Basic Operator Theory

Basic Operator Theory by Israel Gohberg, Seymour Goldberg.

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 t...

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Bibliographic Details
Main Authors: Gohberg, Israel (Author), Goldberg, Seymour (Author)
Corporate Author: SpringerLink (Online service)
Format: eBook
Language:English
Published: Boston, MA : Birkhäuser Boston : Imprint: Birkhäuser, 2001.
Edition:1st ed. 2001.
Series:Springer eBook Collection.
Subjects:
Operator theory.
Functional analysis.
Applied mathematics.
Engineering mathematics.
Electronic resources (E-books)
Online Access:Click to view e-book
Holy Cross Note:Loaded electronically.
Electronic access restricted to members of the Holy Cross Community.
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
  • Exercises III
  • 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. 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.

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