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Introduction to Hilbert Space and the Theory of Spectral Multiplicity: Second Edition
Paul R. Halmos, University of Santa Clara, CA
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AMS Chelsea Publishing
1957; 114 pp; hardcover
Volume: 82
Reprint/Revision History:
second AMS printing 2000
ISBN-10: 0-8218-1378-1
ISBN-13: 978-0-8218-1378-2
List Price: US$25 Member Price: US$22.50
Order Code: CHEL/82.H

A clear, readable introductory treatment of Hilbert Space. The multiplicity theory of continuous spectra is treated, for the first time in English, in full generality.

Reviews

"The main purpose of this book is to present the so-called multiplicity theory and the theory of unitary equivalence, for arbitrary spectral measures, in separable or not separable Hilbert space ... The approach to this theory, as presented by the author, has much claim to novelty. By a skillful permutation of the fundamental ideas of Wecken and Nakano and consistently referring to the simple situation in the finite-dimensional case, the author succeeds in presenting the theory in a clear and perspicuous form."

-- Mathematical Reviews

The Geometry of Hilbert Space
• 1. Linear functionals
• 2. Bilinear functionals
• 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 space
• 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
The Algebra of Operators
• 19. Operators
• 20. Examples of operators
• 21. Inverses
• 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
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
• 69. Conclusion
• References
• Bibliography