1. Introduction. A few years ago, 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 Puhong in [8], More recently, ElMennaoui and Engel gave a more elemen-tary approach to the same result in [1]. It follows clearly from this that one can also easily obtain complete characterizations for compact semigroups (see [8]) and also for eventu-ally norm continuous ones (see [1]) in a similar fashion.
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