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