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On Lie Semi-Groups

Published online by Cambridge University Press:  20 November 2018

R. P. Langlands
R. P. Langlands*
Affiliation:
Yale University
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Suppose we have a semi-group structure defined on

a subset of real Euclidean n-space, En, by (p, q) → F (p, q) = poq. In this note we shall be concerned with a representation T(.) of π as a semi-group of bounded linear operators on a Banach space 𝒳. More particularly, we suppose that postulates P1, P2, P3, P5 and P6 of chapter 25 of (2) are satisfied so that, by Theorem 25.3.1 of that book, there is a continuous function, f(.), defined on π such that f((ρ + σ)a) = f(ρa)o f(σa) for a ∈ π, ρ,σ ≥ 0; that the representation is strongly continuous in a neighbourhood of the origin and that T(0) = I.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 12 , 1960 , pp. 686 - 693
Copyright
Copyright © Canadian Mathematical Society 1960

References

1. Hille, E., Lie theory of semi-groups of linear transformations, Bull. Amer. Math. Soc, 56 (1950).Google Scholar
2. Hille, E. and Phillips, R.S., Functional analysis and semi-groups, Amer. Math. Soc. Coll. Publ., 31 (1957).Google Scholar
3. de Leeuw, K., On the adjoint semi-group and some problems in the theory of approximation, Math. Zeit., 75 (1950).Google Scholar