I don't quite follow your notation, but I'll answer what I think you might be asking.
A C0 or strongly continuous semigroup of operators Tt on a Banach space X is one such that Ttxtox in norm as tto0, i.e. ||Ttx−x||Xto0. In other words, TttoI in the strong operator topology.
A weakly continuous semigroup Tt has Ttxtox weakly as tto0, i.e. f(Ttx)tof(x) for each finX∗. In other words, TttoI in the weak operator topology.
In fact, these two conditions are equivalent. This appears as Theorem 1.6 of K.-J. Engel and R. Nagel, A Short Course on Operator Semigroups.
So this is why you never hear anyone talking about weakly continuous semigroups.
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