Define for $f\in L_1(\R^+)$ and $x\in E$: $T(f)x\colon=\int_0^\infty f(t)P_tx\,dt$. Then $T:L_1(\R^+)\rar L(X)$ and $\ker T$ is an ideal in the convolution algebra $L_1(\R^+)$.
By Fubini we conclude that for all $f\in\ker T$ and $g\in L_1(\R^+)$:
\begin{eqnarray*}
T(f*g)x
&=&\int_0^\infty\int_0^tg(s)f(t-s)P_tx\,ds\,dt\\
&=&\int_0^\infty\int_s^\infty g(s)f(t-s)P_tx\,dt\,ds
=\int_0^\infty g(s)P_s\int_0^\infty f(t)P_tx\,ds\,dt
=0
\end{eqnarray*}