GENERATORS OF SEMIGROUPS AND WEAK SOLUTIONS OF FIRST-ORDER DIFFERENTIAL OPERATOR EQUATIONS
Let U(t) (t > 0) be a strongly continuous semigroup in a Banach space X and kerU = {0}. We construct a locally convex space Q such that X can be densely embedded in Q and the semigroup U(t) can be continuously extended to a semigroup Ũ(t) of class C0 in Q . We describe weak solutions of the equation y'(t) = Ay(t), where A is a linear operator in X and A  Ũ'(0).