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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 thatdoi:10.4171/zaa/936 fatcat:mq6ltireu5cghh5tqlbmcp3yje