Skip to main content Accessibility help
×
Home
Hostname: page-component-56f9d74cfd-fpcrz Total loading time: 0.245 Render date: 2022-06-26T09:17:59.300Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "useNewApi": true }

On the spectral theorem for normal operators

Published online by Cambridge University Press:  24 October 2008

R. G. Douglas
Affiliation:
University of Michigan
Carl Pearcy
Affiliation:
University of Michigan

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
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

(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), 151182.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), 176186.CrossRefGoogle Scholar
3
Cited by

Save article to Kindle

To save this article to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

On the spectral theorem for normal operators
Available formats
×

Save article to Dropbox

To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

On the spectral theorem for normal operators
Available formats
×

Save article to Google Drive

To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

On the spectral theorem for normal operators
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *