with $\sigma_p(T)=K_1$ and $\sigma(T)=K_2$ if and only if (i) $K_2$ is a nonempty closed and bounded set and (ii) $K_1\subseteq K_2$ is an F σ set (a countable union of closed sets). First, these conditions on $K_1,K_2$ are necessary. By standard properties of bounded operators, $\sigma(T)=K_2$ must be closed, bounded and