Introduction to Spectral Theory in Hilbert Space

Introduction to Spectral Theory in Hilbert Space

North-Holland Series in Applied Mathematics and Mechanics

1st Edition - January 1, 1969

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  • Editors: H. A. Lauwerier, W. T. Koiter
  • eBook ISBN: 9781483164175

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North-Holland Series in Applied Mathematics and Mechanics, Volume 6: Introduction to Spectral Theory in Hilbert Space focuses on the mechanics, principles, and approaches involved in spectral theory in Hilbert space. The publication first elaborates on the concept and specific geometry of Hilbert space and bounded linear operators. Discussions focus on projection and adjoint operators, bilinear forms, bounded linear mappings, isomorphisms, orthogonal subspaces, base, subspaces, finite dimensional Euclidean space, and normed linear spaces. The text then takes a look at the general theory of linear operators and spectral analysis of compact linear operators, including spectral decomposition of a compact selfadjoint operator, weakly convergent sequences, spectrum of a compact linear operator, and eigenvalues of a linear operator. The manuscript ponders on the spectral analysis of bounded linear operators and unbounded selfadjoint operators. Topics include spectral decomposition of an unbounded selfadjoint operator and bounded normal operator, functions of a unitary operator, step functions of a bounded selfadjoint operator, polynomials in a bounded operator, and order relation for bounded selfadjoint operators. The publication is a valuable source of data for mathematicians and researchers interested in spectral theory in Hilbert space.

Table of Contents

  • Chapter I. The Concept of Hilbert Space

    § 1. Finite-Dimensional Euclidean Space

    § 2. Inner Product Spaces

    § 3. Normed Linear Spaces

    § 4. The Hilbert Space 2

    § 5. L2 Hilbert Spaces

    Chapter II. Specific Geometry of Hilbert Space

    § 6. Subspaces 36

    § 7. Orthogonal Subspaces

    § 8. Bases

    § 9. Polynomial Bases in 𝔏2 Spaces

    § 10. Isomorphisms

    Chapter III. Bounded Linear Operators

    § 11. Bounded Linear Mappings

    § 12. Linear Operators

    § 13. Bilinear Forms

    § 14. Adjoint Operators

    § 15. Projection Operators

    § 16. The Fourier-Plancherel Operator

    Chapter IV. General Theory of Linear Operators

    §17. Adjoint Operators (General Case)

    § 18. Differentiation Operators in L2 Spaces

    § 19. Multiplication Operators in L2 Spaces

    § 20. Closed Linear Operators

    § 21. Invariant Subspaces of a Linear Operator

    § 22. Eigenvalues of a Linear Operator

    § 23. The Spectrum of a Linear Operator

    § 24. The spectrum of a Selfadjoint Operator

    Chapter V. Spectral Analysis of Compact Linear Operators

    § 25. Compact Linear Operators

    § 26. Weakly Converging Sequences

    § 27. The Spectrum of a Compact Linear Operator

    § 28. The Spectral Decomposition of a Compact Selfadjoint Operator

    § 29. Fredholm Integral Equations

    Chapter VI. Spectral Analysis of Bounded Linear Operators

    § 30. The Order Relation for Bounded Selfadjoint Operators

    § 31. Polynomials in a Bounded Linear Operator

    § 32. Continuous Functions of a Bounded Selfadjoint Operator

    § 33. Step Functions of a Bounded Selfadjoint Operator

    § 34. The Spectral Decomposition of a Bounded Selfadjoint Operator

    § 35. Functions of a Unitary Operator

    § 36. The Spectral Decomposition of a Unitary Operator

    § 37. The Spectral Decomposition of a Bounded Normal Operator

    Chapter VII. Spectral Analysis of Unbounded Selfadjoint Operators

    § 38. The Cayley Transform

    § 39. The Spectral Decomposition of an Unbounded Selfadjoint Operator

    Appendix A. The Graph of a Linear Operator

    B. Riemann-Stieltjes and Lebesgue Integration

    B 1. Weierstrass' Approximation Theorem

    B 2. Riemann-Stieltjes Integration

    B 3. Lebesgue Measurable Sets

    B 4. Lebesgue Measure

    B 5. Lebesgue Measurable Functions

    B 6. Lebesgue Integrable Functions

    B 7. Properties of the Lebesgue Integral

    B 8. Fubini's Theorem

    B 9. Absolutely Continuous Functions

    B 10. Differentiation Under the Integral Sign


    Index of Symbols

    Subject Index

Product details

  • No. of pages: 362
  • Language: English
  • Copyright: © North Holland 1969
  • Published: January 1, 1969
  • Imprint: North Holland
  • eBook ISBN: 9781483164175

About the Editors

H. A. Lauwerier

W. T. Koiter

About the Author

Gilbert Helmberg

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