Hostname: page-component-848d4c4894-x24gv Total loading time: 0 Render date: 2024-04-30T12:11:41.517Z Has data issue: false hasContentIssue false
  • English
  • Français

On the spectral theorem for normal operators

Published online by Cambridge University Press:  24 October 2008

R. G. Douglas  and
Carl Pearcy
R. G. Douglas
Affiliation:
University of Michigan
Carl Pearcy
Affiliation:
University of Michigan
Get access Rights & Permissions [Opens in a new window]

Extract

It has been known for some time that one can construct a proof of the spectral theorem for a normal operator on a Hilbert space by applying the Gelfand representation theorem to the Abelian von Neumann algebra generated by the normal operator, and using the fact that the maximal ideal space of an Abelian von Neumann algebra is extremely disconnected. This, in fact, is the spirit of the monograph (8). On the other hand, it is difficult to find in print accounts of the spectral theorem from this viewpoint and, in particular, the treatment in (8) uses a considerable amount of measure theory and does not have the proof of the spectral theorem as its main objective.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 68 , Issue 2 , September 1970 , pp. 393 - 400
Copyright
Copyright © Cambridge Philosophical Society 1970

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Brown, A. and Pearcy, C.Operators on Hilbert space (graduate textbook in preparation).Google Scholar
(2)Dixmier, J.Sur certains espaces considered par M. H. Stone, Summa Brasil Math. 2 (1951), 151–182.Google Scholar
(3)Douglas, R. G.Topics in operator theory (in preparation).Google Scholar
(4)Halmos, P. R.Measure theory (Van Nostrand, New York, 1950).CrossRefGoogle Scholar
(5)Halmos, P. R.Introduction to Hilbert space and the theory of multiplicity (Chelsea, New York, 1951).Google Scholar
(6)Kelley, J. L.General topology (Van Nostrand, New York, 1955).Google Scholar
(7)Naimark, M. A.Normed rings (Hafner Publishing Company, Inc., New York, 1964).Google Scholar
(8)Segal, I. E. Decomposition of operator algebras I, II. Memoirs Amer. Math. Soc. no. 9 (New York, 1951).Google Scholar
(9)Stone, M. H.Boundedness properties in function lattices. Canad. J. Math. 1 (1949), 176–186.CrossRefGoogle Scholar