POINT WISE CONVERGENCE AND SEMIGROUPS ACTING ON VECTOR-VALUED FUNCTIONS

A submarkovian C(0) semigroup (T(t))(t is an element of R+) acting on the scale of complex-valued functions L(p)(X, C) extends to a semigroup of operators on the scale of vector-valued function spaces L(p)(X, E), when E is a Banach space. It is known that, if f is an element of L(p)(X, C), where 1 < p < infinity, then T(t) f -> f pointwise almost everywhere. We show that the same holds when f is an element of L(p)(X, E).