books - reference for weak*-semigroup

I don't quite follow your notation, but I'll answer what I think you might be asking.

A $C_0$ or strongly continuous semigroup of operators $T_t$ on a Banach space $X$ is one such that $T_t x to x$ in norm as $t to 0$, i.e. $||T_t x - x||_X to 0$. In other words, $T_t to I$ in the strong operator topology.

A weakly continuous semigroup $T_t$ has $T_t x to x$ weakly as $t to 0$, i.e. $f(T_t x) to f(x)$ for each $f in X^*$. In other words, $T_t to I$ 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.