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Distribution semigroups and one parameter semigroups

Kisyński J.

Bulletin of the Polish Academy of Sciences. Mathematics

2002

Vol. 50, no 2

189-216

Let X be a Banach space, and S an L(X)-valued pre-distribution semigroup in the sense of P. C. Kunstmann [15]. Let X[infinity] be the space of [C^infinity] vectors for S. Assume that X[infinity is not equal to ]{0}. Equipped with a natural topology, X[infinity] is a Frechet space continuously imbedded into X. It is proved that there is a unique operator semigroup (St)[t belongs to R] +[is a subset of] L(X[infinity]) with [C^infinity] trajectories such that S(phi)x = [...] for every x [belongs to] X[infinity] and every [phi belongs to] D(R). Furthermore, it is shown that X[infinity] = [...], where [...] is the set of finite sums of elements of the form S(phi)x, x [belongs to] X, [phi belongs to] D(R), sup[phi is a subset of R+].