Spectral radius of $T^*$ is an eigenvalue of $T^*$

Question

I need to prove the following result from Diemling's book "Nonlinear Functional Analysis"

Where K is a cone, ie, a convex closed set such that $\lambda K \subseteq K$ for all $\lambda \geq 0$ and $C \cap (-C) = \{0\}$.

I am realy stuck in proving this result with hypothesis (b), even with the hint, as I didn't manage to prove the equality of the norms and that $X_0\cap K = \emptyset$. Can you help me in these two steps?