A new characterization ofB-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 calledB-bounded semigroups. We first determine all the properties possessed by a couple(A,B)of operators if they generate aB-bounded semigroup(Y(t))t≥0. Then we determine the simplest further property of the couple(A,B)which can assure the existence of aC0-semigroup(T(t))t≥0such that for allt≥0,f∈D(B)we can writeY(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.