How can we show that a Feller semigroup is strongly continuous?

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?

• I don't see where you have defined (1) . – DisintegratingByParts Nov 8 '18 at 17:14
• @DisintegratingByParts $(1)$ is $$T_tf(x)\xrightarrow{t\to0}f(x)\;\;\;\text{for all }x\in E\text{ and }f\in C_0(E)\tag1.$$ – 0xbadf00d Nov 8 '18 at 17:25
• Did you check Ethier-Kurtz? Pretty sure that this is in there... – saz Nov 8 '18 at 17:49
• @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. – DisintegratingByParts Nov 8 '18 at 21:33
• @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. – 0xbadf00d Nov 8 '18 at 23:14