# Show that a semigroup is strongly continuous on the domain of its generator

Let $$(\kappa_t)_{t\ge0}$$ be a semigroup of linear, conservative, contractive and nonnegative operators on $$C_0(E)$$ with $$(\kappa_tf)(x)\xrightarrow{t\to0}f(x)\tag1\;\;\;\text{for all }x\in\mathbb R$$ for all $$f\in C_0(\mathbb R)$$. Let $$(\mathcal D(A),A)$$ denote the generator of $$(\kappa_t)_{t\ge0}$$. Suppose we know that $$\kappa_tf-f=\int_0^t\kappa_s Af\:{\rm d}s\;\;\;\text{for all }f\in\mathcal D(A)\tag2.$$

How can we conclude that $$(\kappa_t)_{t\ge0}$$ is strongly continuous on $$\mathcal D(A)$$?

Clearly, $$(2)$$ again yields $$(1)$$, but I don't see why the convergence is uniform with respect to $$x$$.

• $C_{0}(\mathbb{R})$ denotes the space of continuous functions vanishing at infinity, right? Nov 24, 2018 at 10:27
• @sharpe Yes, that's correct. Nov 24, 2018 at 11:26
• $(2)$ gives $$\|\kappa_t f-f\|_{\infty} \leq t \|Af\|_{\infty}$$ which immediately yields the desired convergence.
– saz
Nov 30, 2018 at 14:00
• @saz Is it possible to obtain the strong continuity without $(2)$? math.stackexchange.com/questions/3096864/… Feb 2, 2019 at 17:39