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Isometric flows in Hilbert space

Published online by Cambridge University Press:  24 October 2008

Béla Sz.-Nagy
Béla Sz.-Nagy
Affiliation:
Szeged, Hungary
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Extract

1. Let {Vi}i≥0 be a weakly (hence also strongly) continuous semigroup of (linear) contraction operators on a Hilbert space H, i.e. |Vt| ≤ 1 ( t ≥ 0). Let Z and W denote the corresponding infinitesimal generator and cogenerator, i.e.

Z is in general non-bounded, but closed and densely defined, and W is a contraction operator (everywhere defined in H), such that 1 is not a proper value of W. Conversely, every contraction operator W not having the proper value 1 is the infinitesimal cogenerator of exactly one semigroup {Vi} of the above type; one has namely

in the sense of the functional calculus for contraction operators (4).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 60 , Issue 1 , January 1964 , pp. 45 - 49
Copyright
Copyright © Cambridge Philosophical Society 1964

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References

REFERENCES

(1)Cooper, J. L. B.One parameter semi-groups of isometric operators in Hilbert space. Ann. of Math. 48 (1947), 827–842CrossRefGoogle Scholar
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(3)Masani, P.Isometric flows on Hilbert spaces. Bull. American Math. Soc. 68 (1962), 624–632CrossRefGoogle Scholar
(4)Sz.-Nagy, B. and Foiaş, C.Sur les contractions de l'espace de Hilbert. III. Acta Sci. Math. Szeged, 19 (1958), 26–45Google Scholar