In 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 Co-semigroup when B = I. We also show that

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... result represents a generalization of Banasiak's result. AMS subject classification: 47 D 06 system by means of the one-parameter family of linear operators (Y(t) = B exp(tA))j>o. Such a family was called a B-bounded semigroup. The original definition of this new class of evolution operators was introduced by Belleni Morante in (5] and generalized by himself in [6] . A further generalization has been given by Banasiak in [2] . Banasiak's definition reads as follows. Definition 1.1. Let X and Z be Banach spaces and suppose the following: L. Arlotti: Dipartimento di Ingegneria Civile, via delle Scienze 208,