reference for weak*-semigroup

Let $X$ a dual Bancah space (there exists a Banach space $Y$ such that $X=Y'$). A weak* semigroup on $X$ is a semigroup $(T_t)_{\geq 0}$ on $X$ such that, for all $x\in X$, we have $T_tx\xrightarrow[t \to 0^+]{}x$ in the weak* topology.

I know a lot of books about $C_0$-semigroups but not about weak* semigroups.

Do you know a good place to read about weak* semigroups and their generators? A book would Be perfect.
• In your first line, presumably you mean convergence in w*-topology as $t$ tends to 0, for each $x$? – Yemon Choi Sep 2 '10 at 21:36
• Even that wouldn't make sense, unless $X$ is the dual of something. – Nate Eldredge Sep 2 '10 at 22:34
• Thank you very much for your answers. However, I forgot a part of the statement. I am sorry. – BigBill Sep 3 '10 at 6:44
• If the $T_t$ are all weak$^*$ continuous, you are just looking at the dual semigroup to a strongly continuous semigroup on the predual. – Bill Johnson Sep 3 '10 at 18:41

Echoing the remark of @Bill Johnson, one possibility is van Neerven's book on adjoint semigroups.

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.

• Thank you very much for your answer. However, I forgot a part of the statement. I am sorry. – BigBill Sep 3 '10 at 6:45