Let $A \in L(X,Y)$ be a bounded operator between Banach spaces. Then its dual operator $A' \in L(Y',X')$ has the same spectrum as $A$ by the closed range theorem.
Now, if we have an unbounded operator $A: D(A) \subset X \rightarrow X$, are there sufficient conditions that $A$ and $A': D(A') \subset X' \rightarrow X'$ have the same spectrum?
Thus, I am looking for something like "self-adjointness on Banach spaces for example". Although it is weaker, since I am not interested in real spectra in general.
Example: Think for example about the Laplacian $\Delta: W^{2,1} \subset L^1 \rightarrow L^1.$
Its dual operator would then be an operator on $L^{\infty}.$ Can we claim that the spectra of $\Delta$ on $L^1$ and $L^{\infty}$ agree? These are the applications I have in mind.