Let A be a self adjoint operator on a Hilbert space $H$ (possibly unbounded, densely defined with domain $D(A)$
) and let $F$ be a closed subspace of $H$.
My question is: If $D(A)\cap F$ is a dense subspace of $F$ and $A(D(A)\cap F) \subset F$ , the restriction of $A$ to $F$ becomes self adjoint ?
If this statement is true, we can find some eigenvalues embedded in continuous spectrum of $A$ easily, by checking out the descrete spectrums of $A|_F$. So I'm interested in wether this statement is true or not.