Spectral properties of integral operators in problems of interface dynamic and metastability

In this paper we study some integral operators that are obtained by linearizations of a non local evolution equation for a non conserved order parameter which describes the phase of a fluid. We prove a Perron-Frobenius theorem by showing that there is an isolated, simple, maximal eigenvalue larger than 1 with a positive eigenvector and that the rest of the spectrum is strictly inside the unit ball. Such properties are responsible for the existence of invariant, attractive unstable one dimensional manifolds under the full, non linear evolution. This part of the analysis and the application to interface dynamics and metastability will be carried out in separate papers.