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2 Bulletin of the Polish Academy of Sciences. Mathematics

2 Kisyński J.

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Equicontinuity and Convergent Sequences in the Spaces O′C and OM

Kisyński J.

Bulletin of the Polish Academy of Sciences. Mathematics

2011

Vol. 59, no 3

223--235

Characterizations of equicontinuity and convergent sequences are given for the space O′C(Rn) of rapidly decreasing distributions and the space OM(Rn) of slowly increasing infinitely differentiable functions.

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+].