Spectrum proofs

Let $T$ be a densely defined closed unbounded operator on a Hilbert space $H$. Show that if $\lambda$ is a point in the residual spectrum of $T$, then $\bar{\lambda}$ is in the point spectrum of the adjoint $T^*$ of $T$. Hence deduce that the residual spectrum of any unbounded self-adjoint operator $T$ on $H$ is empty.

Solution 1:

If $\lambda$ is in the residual spectrum of $T$, then $K=\overline{(T-\lambda I)H}$ is a proper subspace of $H$. Then $$ \{0\}\ne K^\perp=\ker(T^*-\overline\lambda I). $$ So $\overline\lambda$ is an eigenvalue of $T^*$.

When $T$ is selfadjoint, we get that if a point is in the residual spectrum then it is an eigenvalue, which is a contradiction.