Consider $A: D(A) \rightarrow C_0(\mathbb{R}^3)$ where $C_0(\mathbb{R}^3)$ is the space of continuous functions vanishing at infinity and $C_C^1(\mathbb{R}^3)$ is the sub-space of $C^1$ functions with compact support. I know that A is densely defined and closable. I want to study the spectrum of such operator, in principle i should solve $(I-\lambda A)f=g$ with $f \in D(A), g \in C_0(\mathbb{R}^3) $, but shouldn't I use the closure of $A$ instead, since we have $\sigma(A)=\mathbb{C}$?
In case it is useful the exact operator is $Af=Lx \cdot \nabla f$ where
$$ L = \left( \begin{array}{ccc} 0& K & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array} \right). $$