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Spectral representation of local semigroups associated with Klein-Landau systems

Published online by Cambridge University Press:  24 October 2008

W. Ricker
W. Ricker
Affiliation:
Department of Pure Mathematics, University of Adelaide, Australia
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Extract

In their paper [5], Klein and Landau prove that given a symmetric ‘local semigroup’ of unbounded operators {T(t); t ≥ 0} on a Hilbert space, there exists a unique selfadjoint operator T such that T(t) is a restriction of e−tT, for each t ≥ 0. A similar representation theorem was proved earlier by Nussbaum [8]. The result of Klein and Landau was recently extended to the setting of reflexive Banach spaces by Kantorovitz ([4], theorem 2–3). More precisely, Kantorovitz presented necessary and sufficient conditions for a local semigroup of unbounded operators {T(t); t ≥ 0} to consist of restrictions of e−tT, t ≥ 0, for some unbounded spectral operator of scalar-type T with real spectrum (cf. [1] for the terminology).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 95 , Issue 1 , January 1984 , pp. 93 - 100
Copyright
Copyright © Cambridge Philosophical Society 1984

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References

REFERENCES

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