Let $X$ be a hilbert space and $T\in L(X)$
Show that:
(i) $\sigma_c(T^*)=(\sigma(T))^*$
(ii) $\sigma_r(T)=((\sigma_p(T^*))^*)$\ $\sigma_p(T)$
(i): $"\subset"$
Let $\lambda\in\sigma_c(T^*)$ then $T^*-\lambda:X\rightarrow R(T^*-\lambda)$ is injective with $R(T^*-\lambda)\neq X$ and $\overline{R(T^*-\lambda)}=X$
We have $T^*-\lambda=(T-\overline{\lambda})^*$ and we need that $T-\overline{\lambda}$ is injective too with dense in X but not equal to X image.
Can someone help me? Also with the others inclusions.