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.


Thanks in advance!



Comparing Krein-Rutman theorem and Perron–Frobenius theorem
Beware of Wikipedia! It is true that the infinite dimensional setting makes things slightly more delicate, but actually not so much.


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.


You can find an elementary proof [here](http://www.google.fr/url?sa=t&rct=j&q=&esrc=s&source=web&cd=2&sqi=2&ved=0CC0QFjAB&url=ftp%3A%2F%2Fftp.ma.utexas.edu%2Fpub%2Fpapers%2Fllave%2F.grad%2F5999_chap1-1.pdf&ei=9mSfU76_NtOb0AXu1IBQ&usg=AFQjCNH52OG0HM5lEFOoRmVCFhrvGvh4uQ&bvm=bv.68911936,d.bGQ&cad=rja) (theorem 1.2)


