# Introduction to Spectral Theory in Hilbert Space

### North-Holland Series in Applied Mathematics and Mechanics

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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*ℓ*_{2}

§ 5.*L*_{2}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*L*_{2}Spaces

§ 19. Multiplication Operators in*L*_{2}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

Bibliography

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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