Norm continuity and related notions for semigroups on Banach spaces ∗ Oscar Blasco and Josep Martinez Departamento de An´ alisis Matem´ atico, Universidad de Valencia, 46100 Burjassot (Valencia), Spain. Abstract We find some conditions on a c 0 -semigroup on a Banach space and its resolvent connected with the norm continuity of the semigroup. We use them to get characterizations of norm continuous, eventually norm continuous and eventually compact semigroups on Hilbert spaces in terms of the growth of the resolvent of their generator. 1 Introduction. Quite recently a characterization of norm continuous semigroups on Hilbert spaces in terms of the convergence to zero of the resolvent on vertical lines was achieved by Y. Puhong in [8]. Later O. ElMennaoui and K-J. Engel gave a simpler approach to the same result in [1]. It is clear that from this one can easily get also complete characterizations for compact semigroups (see [8]) and also for eventually norm continuous ones (see [1]) in a similar fashion. * 1980 Mathematics Subject Classification (1985 Revision). 47D05. Key words and phrases. Norm continuous semigroups. The authors have been partially supported by the Spanish DGICYT, Proyecto PB92-0699 and PB91-0331 respectively. 1