On semigroups of operators in locally convex spaces

The following problem is investigated in this note: Let T(t) }I >o be a semigroup of continuous linear operators in a locally convex space E. Under which conditions (e.g. measurability conditions) is it true that the functions t-*T(t)x are continuous for every xCE? The point of departure is the paper [6] by Singbal-Vedak. In [6], the following generalization of earlier results for Banach spaces by Dunford [1] and Hille [2] was proved: If { T(t) }I>o is a measurable semigroup such that { T(t) } a?25: is equicontinuous for any [a, 3]C(O, oo), then the functions t-*T(t)x are continuous. It is shown in Proposition 1 that a weaker equicontinuity condition, which is only required to hold locally at the origin 0, implies the conclusion. Proposition 2 treats the special case of semigroups in LF spaces. Another theorem by Singbal-Vedak [6], generalizing a result of Phillips [4] for Banach spaces, says that if E is a Frechet space and the semigroup is measurable, then t->T(t)x is continuous for every x&E. In Proposition 3 it is shown that the conclusion remains true when E is merely a metrizable locally convex space.