Let $E$ be a Banach Lattice such that $E$ is a $M$-space. Assume that $T:E\to E$ is a positive bounded non-compact irreducible linear operator with positive spectral radius. And define \begin{align*} T’: E’ &\to E\\ \phi&\mapsto [v\mapsto \phi(Tv)], \end{align*} where $E’$ is the dual of $E$.
Under the above assumptions, I would like to know: Is it possible to guarantee that $T’$ has a positive eigenvector? If not, is there any “mild assumption” that can be added to this setting to ensure the existence of a positive eigenvector of $T’$?