Suppose T is a bounded operator on a Hilbert space. Show that if λ is in the residual spectrum of T, then $\bar{λ}$ is in the point spectrum of the adjoint.
Here is what I think needs to be done. We know that $\langle Tu,v\rangle = \langle u,T^*v\rangle = \overline{\langle T^*v,u\rangle}$. Does that help with connecting $\lambda$ with $\overline{\lambda}$?
I will explain Selberg's solution : Since $R:=\overline{R(T-\lambda I)}$, i.e., range of $T-\lambda I$, is a closed subspace and $H\neq R$ (since $\lambda$ is a residual spectrum), then there exists $$ x_0\neq 0 \in V,\ H=R\oplus V $$
So $$ 0= \langle (T-\lambda I)x,x_0\rangle=\langle x, (T^\ast -\overline{\lambda } I ) x_0 \rangle $$
Since $x$ is arbitrary, $(T^\ast -\overline{\lambda } I ) x_0 =0$.