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

We consider the one-parameter family of linear operators that A. Belleni Morante re-cently introduced and called B-bounded semigroups. We first determine all the proper-ties possessed by a couple (A,B) of operators if they generate a B-bounded semigroup (Y (t))t≥0. Then we determine the simplest further property of the couple (A,B) which can assure the existence of a C0-semigroup (T (t))t≥0 such that for all t ≥ 0,f ∈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. 1