residual spectra of adjoint operator.

Question

I saw an interesting claim saying "If $λ$ were in the residual spectrum of $T$, then $\overline{λ}$ would be in the point spectrum of $T^*$. Is there a short proof for this claim? recall point spectrum is the set of $\lambda$ such that $\lambda I-T$ is not one-to-one and residual spectrum is the set of $\lambda$ such that $\lambda I-T$ is one-to-one but its range is not dense in $H$.

Answer 1

If $\text{Ran}(\lambda I - T)$ is not dense in $H$, take $v$ in its orthogonal complement. Thus for all $x \in H$, $ 0 = \langle (\lambda I - T) x, v \rangle = \langle x, (\overline{\lambda} I - T^*) v \rangle$, and this implies $(\overline{\lambda} - T^*) v = 0$.