A new characterization of $B$-bounded semigroups with application to implicit evolution equations

Abstract

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 $\left(A,B\right)$ of operators if they generate a $B$-bounded semigroup ${\left(Y\left(t\right)\right)}_{t\ge 0}$. Then we determine the simplest further property of the couple $\left(A,B\right)$ which can assure the existence of a $C_0$-semigroup ${\left(T\left(t\right)\right)}_{t\ge 0}$ such that for all $t\ge 0,f\in D\left(B\right)$ we can write $Y\left(t\right)f=T\left(t\right)Bf$. 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.