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