Let $X$ be a real Banach space, $K\subset X$ a normal cone with nonempty interior. Let $T\in \mathcal {B}(X)$ a positive operator, i.e., $T(K)\subset K$, with spectral radius $r>0$. Can anyone tell me if there exists eigenfunctional $\phi \in X^{*}$ that satisfies $\phi(x)\ge 0, \forall x\in K$, such that $T^{*}\phi=r\phi$, where $T^{*}$ is the adjoint operator of $T$?
This is a problem from Nonlinear Functional Analysis by Klaus Deimling. There there exist the following hint: $X_0\cap int(K)=\emptyset$, where $X_{0}=\{ rx-Tx : x\in X\}$, before use Hahn-Banach theorem to find $\phi$. Any help will be appreciate. THANKS!