Introduction to Hilbert Space and the Theory of Spectral Multiplicity Second Edition

by Halmos, Paul R.

• ISBN13: 9780486817330
• ISBN10: 0486817334
• Edition: 2nd
• Format: Paperback
• Copyright: 2017-12-13
• Publisher: Dover Publications

Summary

This concise introductory treatment consists of three chapters: The Geometry of Hilbert Space, The Algebra of Operators, and The Analysis of Spectral Measures. Author Paul R. Halmos notes in the Preface that his motivation in writing this text was to make available to a wider audience the results of the third chapter, the so-called multiplicity theory. The theory as he presents it deals with arbitrary spectral measures, including the multiplicity theory of normal operators on a not necessarily separable Hilbert space. His explication covers, as another useful special case, the multiplicity theory of unitary representations of locally compact abelian groups.
Suitable for advanced undergraduates and graduate students in mathematics, this volume's sole prerequisite is a background in measure theory. The distinguished mathematician E. R. Lorch praised the book in the Bulletin of the American Mathematical Society as "an exposition which is always fresh, proofs which are sophisticated, and a choice of subject matter which is certainly timely."

Author Biography

Hungarian-born Paul R. Halmos (1916–2006) is widely regarded as a top-notch expositor of mathematics. He taught at the University of Chicago and the University of Michigan as well as other universities and made significant contributions to several areas of mathematics, including mathematical logic, probability theory, ergodic theory, and functional analysis.

Table of Contents

Preface

0. Prerequisites and Notation

 

CHAPTER I: The Geometry of Hilbert Space

1. Linear Functionals

2. Bilinear Functionals

3. Quadratic Forms

4. Inner Product and Norm

5. The Inequalities of Bessel and Schwarz

6. Hilbert Space

7. Infinite Sums

8. Conditions for Summability

9. Examples of Hilbert Spaces

10. Subspaces

11. Vectors in and out of Subspaces

12. Orthogonal Complements

13. Vector Sums

14. Bases

15. A Non-closed Vector Sum

16. Dimension

17. Boundedness

18. Bounded Bilinear Functionals

 

CHAPTER II: The Algebra of Operators

19. Operators

20. Examples of Operators

21. Inverses

22. Adjoints

23. Invariance

24. Hermitian Operators

25. Normal and Unitary Operators

26. Projections

27. Projections and Subspaces

28. Sums of Projections

29. Products and Differences of Projections

30. Infima and Suprema of Projections

31. The Spectrum of an Operator

32. Compactness of Spectra

33. Transforms of Spectra

34. The Spectrum of a Hermitian Operator

35. Spectral Heuristics

36. Spectral Measures

37. Spectral Integrals

38. Regular Spectral Measures

39. Real and Complex Spectral Measures

40. Complex Spectral Integrals

41. Description of the Spectral Subspaces

42. Characterization of the Spectral Subspaces

43. The Spectral Theorem for Hermitian Operators

44. The Spectral Theorem for Normal Operators

 

CHAPTER III: The Analysis of Spectral Measures

45. The Problem of Unitary Equivalence

46. Multiplicity Functions in Finite-dimensional Spaces

47. Measures

48. Boolean Operations on Measures

49. Multiplicity Functions

50. The Canonical Example of a Spectral Measure

51. Finite-dimensional Spectral Measures

52. Simple Finite-dimensional Spectral Measures

53. The Commutator of a Set of Projections

54. Pairs of Commutators

55. Columns

56. Rows

57. Cycles

58. Separable Projections

59. Characterizations of Rows

60. Cycles and Rows

61. The Existence of Rows

62. Orthogonal Systems

63. The Power of a Maximal Orthogonal System

64. Multiplicities

65. Measures from Vectors

66. Subspaces from Measures

67. The Multiplicity Function of a Spectral Measure

68. Conclusion

References

Bibliography