Prove directly the implication 5.$\Rar$2..
For $x\in\ker L$ we have: $P_tx=x$ and thus for all $\l > 0$:
$$
\l\int_0^\infty P_txe^{-\l t}\,dt
=x
\quad\mbox{i.e.}\quad
x\in F~.
$$
For $x\in\im L$, i.e. $x=Ly$, $y\in\dom L$, we have:
$$
\lim_{\l\to0}\tnorm{\l(\l-L)^{-1}x}
=\lim_{\l\to0}\l\tnorm{\l(\l-L)^{-1}y-y}
\leq\lim_{\l\to0}2\l\norm{y}
=0~.
$$
It follows that: $\ker L+\im L\sbe F$, i.e. $F=E$.