On B-bounded semigroups as a generalization of C-0-semigroups

In this paper we consider the one-parameter family of linear operators that A. Belleni Morante recently introduced and called B-bounded semigroups. Su ch a family was studied by A. Belleni Morante himself and by J. Banasiak. H ere we give a necessary and sufficient condition that a pair (A,B) of linea r operators be the generator of a B-bounded semigroup. Our procedure is con structive 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 gene ralization of Banasiak's result.