Abstract. It is known that the infinitesimal generator of every $C_0$-group of operators in a hereditarily indecomposable Banach space is necessarily a bounded operator [20]. For $C_0$-semigroups this is not the case in general. We present certain classes of $C_0$-semigroups whose infinitesimal generators are always bounded operators. It is also shown that the infinitesimal generator of any strongly continuous, non-quasianalytic cosine family in a hereditarily indecomposable Banach space is necessarily a bounded operator.

AMS Subject Classification (1991): 47D03, 47D09

Received July 16, 1997. (Registered under 3317/2009.)