  We consider the Friedrichs-Faddeev model in the case where the kernel of the
potential operator is holomorphic in both arguments on a certain domain of
$\mathbb{C}$. For this model we, first, study the structure of the $T$- and
$S$-matrices on unphysical energy sheet(s). To this end, we derive
representations that explicitly express them in terms of these same operators
considered exclusively on the physical sheet. Furthermore, we allow the
Friedrichs-Faddeev Hamiltonian undergo a complex deformation (or even a complex
scaling/rotation if the model is associated with an infinite interval).
Isolated non-real eigenvalues of the deformed Hamiltonian are called the
deformation resonances. For a class of perturbation potentials with analytic
kernels, we prove that the deformation resonances do correspond to the
scattering matrix resonances, that is, they represent the poles of the
scattering matrix analytically continued to the respective unphysical sheet.
