If $L$ is closed, then a subspace $F$ of $\dom L$ is a core iff $(z-L)F$ is dense for all $z\notin\Spec(L)$.
The mapping $(z-L)^{-1}:E\rar(\dom(L),\norm{.}_L)$ is bounded, for we have for all $x\in E$:
$$
\tnorm{L(z-L)^{-1}x}
=\tnorm{-x+z(z-L)^{-1}x}
\leq(1+\norm{U_z}|z|)\norm x~.
$$
Hence $(z-L)^{-1}:E\rar(\dom(L),\norm{.}_L)$ is an isomorphism - with inverse $z-L$. Hence $F$ is a core iff $(z-L)F$ is dense in $E$.