Comparing Krein-Rutman theorem and Perron–Frobenius theorem

Krein–Rutman theorem is a generalization of Perron–Frobenius theorem, I know that things could be more subtle in infinite dimension, yet there's an important result in Perron–Frobenius that's missing in Krein-Rutman and I don't quite understand.

In Perron–Frobenius theorem, we know that for a irreducible non-negative matrix, its positive eigenvector is unique(up to scaling), corresponding to its largest eigenvalue. the analog for positive eigenfunction is not stated in Krein–Rutman theorem. So is it possible that we have a positive operator that has two positive eigenfunctions corresponding to two distinct eigenvalues?

If it helps to narrow thing down, I'm interested in integral operators in $L_2(R)$ space.

• FYI, symmetry is not needed in Perron-Frobenius. Which makes more unexpected the fact that the dominating eigenvalue is real. – Federico Poloni Jun 17 '14 at 9:50
• Yes, I've always found that striking too! – leo monsaingeon Jun 17 '14 at 11:00
• @FedericoPoloni thank you for pointing out this, I've edited it. – Sylvan Jun 17 '14 at 19:22

Assuming that the positive cone $C\subset X$ under consideration is solid (i-e has non empty interior) and that your operator $T:X\to X$ is compact and strongly positive (i-e maps the positive cone $C$ into its interior $\overset{\circ}{C}$), then the following stronger conclusion holds: the spectral radius is a simple eigenvalue associated with a strictly positive eigenvector $v\in \overset{\circ}{C}$, and there is no other eigenvalue associated with (non necessarily strictly) positive eigenvectors.
• Showing that the cone of positive functions has non-empty interior can be very tricky sometimes (or not even true) but you don't need it. It is enough to assume that $\overline{K-K}=X$ (and that holds for positive functions..), see here. – username Jun 17 '14 at 9:11