As I was studying the proof of the following theorem , I got stuck to the following implication:
$(0,\infty) \subset \rho(\Delta_D) \Rightarrow\sigma(\Delta_D) \subset \mathbb R (*)$
With the usual definition of spectrum I can see why:
$(0,\infty) \subset \rho(\Delta_D) \Rightarrow \sigma(\Delta_D) \subset \mathbb C \setminus (0,\infty)$
Hence, in order to deduce $(*)$ I should have used instead that: $\sigma(\Delta_D)=\mathbb R \setminus \rho(\Delta_D)$ but I don't get how this could be true here.
Any help is much appreciated!
Thanks in advance!