We consider the one-parameter family of linear operators that A.
Belleni Morante recently introduced and called
$B$ -bounded semigroups. We first determine all the properties
possessed by a couple $(A,B)$ of operators if they generate a $B$ -bounded semigroup $(Y(t))_{t\geq 0}$ . Then we determine the simplest further property of the couple $(A,B)$ which can assure the existence of a $C_0$ -semigroup $(T(t))_{t\geq 0}$ such that for all $t\geq 0,f\in D(B)$ we can write $Y(t)f = T(t)Bf$ . Furthermore, we compare our result with the previous ones and
finally we show how our method allows to improve the theory developed by Banasiak for solving implicit evolution equations.