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 (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 C 0 -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.

Causal Relations in Support of Implicit Evolution Equations

“…The concept of K-bounded semigroups (precisely, B-bounded semigroups but we had to change the name to avoid the conict of notation) was introduced by Belleni-Morante, [14,15]. The generation theorems can be found in [16,17,19,20]. The latter work also contains a comparison of K-bounded semigroups and C-semigroups.…”

Lumpability of linear evolution Equations in Banach spaces

“…Indeed, our aim is to study lumpability from the dynamical systems point of view, in order to obtain well-posed reduced dynamics. In passing, we mention the related notion of B-bounded semigroups [6,4,2] and the successive reflection method [7] for proving the existence of a semigroup. Before proceeding to operators on generic Banach spaces, it is instructive to look at the situation in finite-dimensional Euclidean spaces.…”