Let $A$ be a Hermitian linear operator on the space of $n\times n$ complex matrices. Let's suppose $A$ is "non-negative" in the sense that it preserves the cone of non-negative definite (Hermitian) matrices, and irreducible (in a suitable sense). Can I expect a Perron-Frobenius type theorem for $A$, namely that the spectral norm of $A$ is a simple eigenvalue and the corresponding eigenvector is positive (not necessarily unique)?
I'm having some trouble tracking down a suitable reference for this. Any help is appreciated.
