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Let

  • $E$ be a normed $\mathbb R$-vector space
  • $C_0(E)$ denote the space of continuous functions vanishing at infinity
  • $(T(t))_{t\ge0}$ be a contractive nonnegative semigroup on $C_0(E)$ with $$T_tf(x)\xrightarrow{t\to0}f(x)\;\;\;\text{for all }x\in E\text{ and }f\in C_0(E)\tag1.$$

Why can we conclude from $(1)$ that $(T(t))_{t\ge0}$ is strongly continuous?

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  • $\begingroup$ I don't see where you have defined (1) . $\endgroup$ – DisintegratingByParts Nov 8 '18 at 17:14
  • $\begingroup$ @DisintegratingByParts $(1)$ is $$T_tf(x)\xrightarrow{t\to0}f(x)\;\;\;\text{for all }x\in E\text{ and }f\in C_0(E)\tag1.$$ $\endgroup$ – 0xbadf00d Nov 8 '18 at 17:25
  • $\begingroup$ Did you check Ethier-Kurtz? Pretty sure that this is in there... $\endgroup$ – saz Nov 8 '18 at 17:49
  • $\begingroup$ @0xbadf00d : I see now. I don't like the MathJax tag system because it appears way off to the right for me, and I don't notice it. Thanks. $\endgroup$ – DisintegratingByParts Nov 8 '18 at 21:33
  • $\begingroup$ @saz Ethier and Kurtz are defining a Feller semigroup to be $C^0$, but there is a proof in Kallenberg. However, he's proving the strong continuity using the Yosida approximation which in turn requires the existence of the generator and that it is densely-defined and closed. He's proving the existence by the properties of the Feller semigroup given in the question. I would like to do it the other way around: If we could prove the strong continuity directly, nothing of what he proves would be special anymore, since all of that are properties which are fulfilled by any $C^0$-semigroup. $\endgroup$ – 0xbadf00d Nov 8 '18 at 23:14

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