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

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."

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.

Preface0. Prerequisites and Notation CHAPTER I: The Geometry of Hilbert Space1. Linear Functionals2. Bilinear Functionals3. Quadratic Forms4. Inner Product and Norm5. The Inequalities of Bessel and Schwarz6. Hilbert Space7. Infinite Sums8. Conditions for Summability9. Examples of Hilbert Spaces10. Subspaces11. Vectors in and out of Subspaces12. Orthogonal Complements13. Vector Sums14. Bases15. A Non-closed Vector Sum16. Dimension17. Boundedness18. Bounded Bilinear Functionals CHAPTER II: The Algebra of Operators19. Operators20. Examples of Operators21. Inverses22. Adjoints23. Invariance24. Hermitian Operators25. Normal and Unitary Operators26. Projections27. Projections and Subspaces28. Sums of Projections29. Products and Differences of Projections30. Infima and Suprema of Projections31. The Spectrum of an Operator32. Compactness of Spectra33. Transforms of Spectra34. The Spectrum of a Hermitian Operator35. Spectral Heuristics36. Spectral Measures37. Spectral Integrals38. Regular Spectral Measures39. Real and Complex Spectral Measures40. Complex Spectral Integrals41. Description of the Spectral Subspaces42. Characterization of the Spectral Subspaces43. The Spectral Theorem for Hermitian Operators44. The Spectral Theorem for Normal Operators CHAPTER III: The Analysis of Spectral Measures45. The Problem of Unitary Equivalence46. Multiplicity Functions in Finite-dimensional Spaces47. Measures48. Boolean Operations on Measures49. Multiplicity Functions50. The Canonical Example of a Spectral Measure51. Finite-dimensional Spectral Measures52. Simple Finite-dimensional Spectral Measures53. The Commutator of a Set of Projections54. Pairs of Commutators55. Columns56. Rows57. Cycles58. Separable Projections59. Characterizations of Rows60. Cycles and Rows61. The Existence of Rows62. Orthogonal Systems63. The Power of a Maximal Orthogonal System64. Multiplicities65. Measures from Vectors66. Subspaces from Measures67. The Multiplicity Function of a Spectral Measure68. ConclusionReferencesBibliography