Introduction to Spectral Theory in Hilbert Space

North-Holland Series in Applied Mathematics and Mechanics

1st Edition, Volume 6 - January 1, 1969
Author: Gilbert Helmberg
Editors: H. A. Lauwerier, W. T. Koiter
Language: English
eBook ISBN:
9 7 8 - 1 - 4 8 3 1 - 6 4 1 7 - 5

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… Read more

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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.

Chapter I. The Concept of Hilbert Space§ 1. Finite-Dimensional Euclidean Space§ 2. Inner Product Spaces§ 3. Normed Linear Spaces§ 4. The Hilbert Space ℓ₂§ 5. L₂ Hilbert SpacesChapter II. Specific Geometry of Hilbert Space§ 6. Subspaces 36§ 7. Orthogonal Subspaces§ 8. Bases§ 9. Polynomial Bases in 𝔏2 Spaces§ 10. IsomorphismsChapter III. Bounded Linear Operators§ 11. Bounded Linear Mappings§ 12. Linear Operators§ 13. Bilinear Forms§ 14. Adjoint Operators§ 15. Projection Operators§ 16. The Fourier-Plancherel OperatorChapter 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 OperatorChapter 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 EquationsChapter 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 OperatorChapter VII. Spectral Analysis of Unbounded Selfadjoint Operators§ 38. The Cayley Transform§ 39. The Spectral Decomposition of an Unbounded Selfadjoint OperatorAppendix A. The Graph of a Linear OperatorB. Riemann-Stieltjes and Lebesgue IntegrationB 1. Weierstrass' Approximation TheoremB 2. Riemann-Stieltjes IntegrationB 3. Lebesgue Measurable SetsB 4. Lebesgue MeasureB 5. Lebesgue Measurable FunctionsB 6. Lebesgue Integrable FunctionsB 7. Properties of the Lebesgue IntegralB 8. Fubini's TheoremB 9. Absolutely Continuous FunctionsB 10. Differentiation Under the Integral SignBibliographyIndex of SymbolsSubject Index