I want to prove this property of adjoint operators.
Let $A:D(A)\subset \mathcal{H}\rightarrow \mathcal{H}$ and $B:D(B)\rightarrow \mathcal{H}$ two densely defined operators where $A\subset B$, i.e. B is the extension of A. Then $B^*\subset A^*$.
Here’s my proof: $B$ is the extension of $A$, so $Ax=Bx$ for every $x\in D(A)$. We need to prove that $D(B^*) \subset D(A^*)$ and that $A^*x = B^*x$ for every $x\in D(B^*)$. Let $x\in D(B^*)$, hence for every $y\in D(A)$ $$<x,Ay>= <x,By>=<B^*x,y>$$ Hence $A^* \equiv B^*$. Is it correct?
But how about $y\in D(B)\backslash D(A)$?