Let $T : \mathcal{D} (T) \subset H \to H$ and $A: \mathcal{D} (A) \subset H \to H$ two densely defined operators and $H$ a complex Hilbert space. Prove $T^*+A^* \subseteq (T+A)^*$.
My solution is as follows.
We want to prove that $\mathcal{D} (T^*+A^* ) \subseteq \mathcal{D} ((T+A)^*)$. Take $f \in \mathcal{D} (T^*+A^* ) =\mathcal{D} (T^*) \cap \mathcal{D} (A^*) $ so $ f \in \mathcal{D} (T^*)$ and $f \in \mathcal{D} (A^*)$.
Then for every $g \in \mathcal{D} (T+A)=\mathcal{D} (T) \cap \mathcal{D} (A)$ we have $$\langle (T^*+A^*)f,g \rangle =\langle T^* f,g \rangle +\langle A^* f, g \rangle $$ $$=\langle f,Tg \rangle +\langle f,Ag \rangle =\langle f, (T+A)g \rangle =\langle (T+A)^* f,g \rangle $$ Hence $f \in \mathcal{D} ((T+A)^*)$
Is my solution correct?
I think I am missing something.