The aim of this Note is to provide a rigorous mathematical treatment of a new spectral problem, coming from a linear stability analysis in fluid-structure interaction. This eigenproblem involves the linearized incompressible Navier-Stokes equations coupled with those of a reduced structure, by means of specific transpiration boundary conditions. We prove that the eigensolutions of this spectral problem can be obtained from the characteristic values of a compact operator acting on a Hilbert space