On $B$-Bounded Semigroups as a Generalization of $C_0$-Semigroups

Abstract

this paper we consider the one-parameter family of linear operators that A. Belleni Morante recently introduced and called $B$-bounded semigroups. Such a family was studied by A. Belleni Morante himself and by J. Banasiak. Here we give a necessary and sufficient condition that a pair ($A,B$) of linear operators be the generator of a $B$-bounded semigroup. Our procedure is constructive and is equivalent to the Yosida procedure for the construction of a $C_0$-semigroup when $B=I$. We also show that our result represents a generalization of Banasiak’s result.