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The Fourier–Plancherel transform as a spectral representation of a rigged Hilbert space

Published online by Cambridge University Press:  24 October 2008

B. Fishel
B. Fishel
Affiliation:
School of Mathematical Sciences, Queen Mary College, London E1 4NS
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Extract

In ‘Generalized Translation Operators…’ [3] algebras associated with a self-adjoint operator were investigated. Examples were given in the cases, inter alia, of the operators Mt and i d/dt in the space L2(ℝ, m) (m = Lebesgue measure). This paper shows that by suitably rigging the space the examples can be seen as natural generalizations of certain familiar algebras. The rigging enables us to introduce, rigorously, into L2(ℝ,m) the improper elements used by Akhiezer and Glazman [1] as cyclic vectors for Mt and i d/dt in order to identify the Fourier–Plancherel transform with a spectral representation of the space.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 101 , Issue 3 , May 1987 , pp. 567 - 573
Copyright
Copyright © Cambridge Philosophical Society 1987

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References

REFERENCES

[1]Akhiezer, N. I. and Glazman, I. M.. Theory of Linear Operators in Hilbert Space, vol. I (Pitman, 1981).Google Scholar
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