Let $L$ be a closed linear operator such that for some $\l\notin\Spec(L)$ the operator $U_\l$ is compact. Then $\im L$ is of finite codimension and $\ker L$ is finite dimensional.
We will utilize the fact that if $A$ is a compact linear operator, then $\im(1+A)$ is closed and of finite codimension and $\ker(1+A)$ is finite dimensional. Since $L(\l-L)^{-1}=-1+\l(\l-L)^{-1}$ we get
$$
\im L
=L(\dom L)
=L(\im(\l-L)^{-1})
=\im L(\l-L)^{-1}
=\im(-1+\l(\l-L)^{-1})
$$
Now $\l(\l-L)^{-1}$ is compact and therefore $\im(-1+\l(\l-L)^{-1})$ is closed.
As for the kernel of $L$ we have:
$$
\ker L
=\{(\l-L)^{-1}x: x\in\ker L(\l-L)^{-1}\}
=(\l-L)^{-1}(\ker(-1+\l(\l-L)^{-1}))
$$
and again by compactness of $\l(\l-L)^{-1}$ the kernel of $-1+\l(\l-L)^{-1}$ is finite dimensional.