Basic Classes of Linear Operators / Edition 1

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This book provides an introduction to functional analysis with an emphasis on the theory of linear operators and its application to differential equations, integral equations, infinite systems of linear equations, approximation theory, and numerical analysis. As textbook designed for senior undergraduate and graduate students, it begins with the geometry of Hilbert spaces and proceeds to the theory of linear operators on these spaces including Banach spaces. Presented as a natural continuation of linear algebra, the book provides a firm foundation in operator theory which is an essential part of mathematical training for students of mathematics, engineering, and other technical sciences.

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Editorial Reviews

From the Publisher
"With a name like Gohberg–Goldberg–Kaashoek, it has got to be good. But let me count the ways. If you are interested in learning the basic theories of Hilbert and Banach spaces together with the well-known operators that act on them, this book is for you. It is intended for advanced undergraduate and beginning graduate students in mathematics.... The volume is actually a second edition of Basic Operator Theory by Gohberg and Goldberg.... Exercises and examples have been added.... The text is well written, clear, and readable.... A student at the specified level will have no difficulty understanding the presentation."

—SIAM Review

"The material is presented in a very clear way, emphasizing concepts over technical generalities. Another pleasant feature of the book is the large number of examples and exercises. In summary, the authors have successfully found an interesting way from the elementary level to some of the highlights of operator theory."

- Monatshefte für Mathematik

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Product Details

  • ISBN-13: 9783764369309
  • Publisher: Birkhauser Basel
  • Publication date: 1/12/2004
  • Edition description: 2003
  • Edition number: 1
  • Pages: 423
  • Product dimensions: 9.30 (w) x 6.60 (h) x 0.90 (d)

Table of Contents

I Hilbert Spaces.- 1.1 Complex n-Space.- 1.2 The Hilbert Space—2.- 1.3 Definition of Hilbert Space and its Elementary Properties.- 1.4 Distance from a Point to a Finite Dimensional Space.- 1.5 The Gram Determinant.- 1.6 Incompatible Systems of Equations.- 1.7 Least Square Fit.- 1.8 Distance to a Convex Set and Projections onto Subspaces.- 1.9 Orthonormal Systems.- 1.10 Szegö Polynomials.- 1.11 Legendre Polynomials.- 1.12 Orthonormal Bases.- 1.13 Fourier Series.- 1.14 Completeness of the Legendre Polynomials.- 1.15 Bases for the Hilbert Space of Functions on a Square.- 1.16 Stability of Orthonormal Bases.- 1.17 Separable Spaces.- 1.18 Isometry of Hilbert Spaces.- 1.19 Example of a Non Separable Space.- Exercises.- II Bounded Linear Operators on Hilbert Spaces.- 2.1 Properties of Bounded Linear Operators.- 2.2 Examples of Bounded Linear Operators with Estimates of Norms.- 2.3 Continuity of a Linear Operator.- 2.4 Matrix Representations of Bounded Linear Operators.- 2.5 Bounded Linear Functionals.- 2.6 Operators of Finite Rank.- 2.7 Invertible Operators.- 2.8 Inversion of Operators by the Iterative Method.- 2.9 Infinite Systems of Linear Equations.- 2.10 Integral Equations of the Second Kind.- 2.11 Adjoint Operators.- 2.12 Self Adjoint Operators.- 2.13 Orthogonal Projections.- 2.14 Two Fundamental Theorems.- 2.15 Projections and One-Sided Invertibility of Operators.- 2.16 Compact Operators.- 2.17 The Projection Method for Inversion of Linear Operators.- 2.18 The Modified Projection Method.- 2.19 Invariant Subspaces.- 2.20 The Spectrum of an Operator.- Exercises.- III Laurent and Toeplitz Operators on Hilbert Spaces.- 3.1 Laurent Operators.- 3.2 Toeplitz Operators.- 3.3 Band Toeplitz operators.- 3.4 Toeplitz Operators with Continuous Symbols.- 3.5 Finite Section Method.- 3.6 The Finite Section Method for Laurent Operators.- Exercises.- IV Spectral Theory of Compact Self Adjoint Operators.- 4.1 Example of an Infinite Dimensional Generalization.- 4.2 The Problem of Existence of Eigenvalues and Eigenvectors.- 4.3 Eigenvalues and Eigenvectors of Operators of Finite Rank.- 4.4 Existence of Eigenvalues.- 4.5 Spectral Theorem.- 4.6 Basic Systems of Eigenvalues and Eigenvectors.- 4.7 Second Form of the Spectral Theorem.- 4.8 Formula for the Inverse Operator.- 4.9 Minimum-Maximum Properties of Eigenvalues.- Exercises.- V Spectral Theory of Integral Operators.- 5.1 Hilbert-Schmidt Theorem.- 5.2 Preliminaries for Mercer’s Theorem.- 5.3 Mercer’s Theorem.- 5.4 Trace Formula for Integral Operators.- Exercises.- VI Unbounded Operators on Hilbert Space.- 6.1 Closed Operators and First Examples.- 6.2 The Second Derivative as an Operator.- 6.3 The Graph Norm.- 6.4 Adjoint Operators.- 6.5 Sturm-Liouville Operators.- 6.6 Self Adjoint Operators with Compact Inverse.- Exercises.- VII Oscillations of an Elastic String.- 7.1 The Displacement Function.- 7.2 Basic Harmonic Oscillations.- 7.3 Harmonic Oscillations with an External Force.- VIII Operational Calculus with Applications.- 8.1 Functions of a Compact Self Adjoint Operator.- 8.2 Differential Equations in Hilbert Space.- 8.3 Infinite Systems of Differential Equations.- 8.4 Integro-Differential Equations.- Exercises.- IX Solving Linear Equations by Iterative Methods.- 9.1 The Main Theorem.- 9.2 Preliminaries for the Proof.- 9.3 Proof of the Main Theorem.- 9.4 Application to Integral Equations.- X Further Developments of the Spectral Theorem.- 10.1 Simultaneous Diagonalization.- 10.2 Compact Normal Operators.- 10.3 Unitary Operators.- 10.4 Singular Values.- 10.5 Trace Class and Hilbert Schmidt Operators.- Exercises.- XI Banach Spaces.- 11.1 Definitions and Examples.- 11.2 Finite Dimensional Normed Linear Spaces.- 11.3 Separable Banach Spaces and Schauder Bases.- 11.4 Conjugate Spaces.- 11.5 Hahn-Banach Theorem.- Exercises.- XII Linear Operators on a Banach Space.- 12.1 Description of Bounded Operators.- 12.2 Closed Linear Operators.- 12.3 Closed Graph Theorem.- 12.4 Applications of the Closed Graph Theorem.- 12.5 Complemented Subspaces and Projections.- 12.6 One-Sided Invertibility Revisited.- 12.7 The Projection Method Revisited.- 12.8 The Spectrum of an Operator.- 12.9 Volterra Integral Operator.- 12.10 Analytic Operator Valued Functions.- Exercises.- XIII Compact Operators on a Banach Space.- 13.1 Examples of Compact Operators.- 13.2 Decomposition of Operators of Finite Rank.- 13.3 Approximation by Operators of Finite Rank.- 13.4 First Results in Fredholm Theory.- 13.5 Conjugate Operators on a Banach Space.- 13.6 Spectrum of a Compact Operator.- 13.7 Applications.- Exercises.- XIV Poincaré Operators: Determinant and Trace.- 14.1 Determinant and Trace.- 14.2 Finite Rank Operators, Determinants and Traces.- 14.3 Theorems about the Poincaré Determinant.- 14.4 Determinants and Inversion of Operators.- 14.5 Trace and Determinant Formulas for Poincaré Operators.- Exercises.- XV Fredholm Operators.- 15.1 Definition and Examples.- 15.2 First Properties.- 15.3 Perturbations Small in Norm.- 15.4 Compact Perturbations.- 15.5 Unbounded Fredholm Operators.- Exercises.- XVI Toeplitz and Singular Integral Operators.- 16.1 Laurent Operators on—p(?).- 16.2 Toeplitz Operators on—p.- 16.3 An Illustrative Example.- 16.4 Applications to Pair Operators.- 16.5 The Finite Section Method Revisited.- 16.6 Singular Integral Operators on the Unit Circle.- Exercises.- XVII Non Linear Operators.- 17.1 Fixed Point Theorems.- 17.2 Applications of the Contraction Mapping Theorem.- 17.3 Generalizations.- Appendix 1: Countable sets and Separable Hilbert Spaces.- Suggested Reading.- References.- List of Symbols.

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