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Continuous sums of measures and continuous spectra

Published online by Cambridge University Press:  18 May 2009

S. Sankaran
S. Sankaran
Affiliation:
Queen Elizabeth College, London
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Von Neumann's definition of the continuous sum of Hilbert spaces led Segal [3] to define the continuous sum of measures on a measurable space. In this note we employ Segal's definition to investigate the measure structures associated with a self-adjoint transformation of pure point spectrum and a self-adjoint transformation of pure continuous spectrum. While these transformations, as operators on separable Hilbert spaces, are the antithesis of each other we show that in their measure structure one is a particular case of the other.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 7 , Issue 1 , January 1965 , pp. 9 - 14
Copyright
Copyright © Glasgow Mathematical Journal Trust 1965

References

REFERENCES

1.Halmos, P. R., Measure theory (New York, 1951).Google Scholar
2.Sankaran, S., Ordered decomposition of Hilbert spaces, J. London Math. Soc. 36 (1961), 97–107.CrossRefGoogle Scholar
3.Segal, I. E., Decomposition of operator algebras, J. Mem. Amer. Math. Soc., No. 9 (1951).Google Scholar
4.Stone, M. H., Linear transformations in Hilbert space (American Mathematical Society Colloquium Publications, Vol. 15, 1932).Google Scholar