  The aim of the present paper is twofold:(1) We carry on with developing an
abstract method for deriving decay estimates on the semigroup associated to
non-symmetric operators in Banach spaces as introduced in [10]. We extend the
method so as to consider the shrinkage of the functional space. Roughly
speaking, we consider a class of operators writing as a dissipative part plus a
mild perturbation, and we prove that if the associated semigroup satisfies a
decay estimate in some reference space then it satisfies the same decay
estimate in another-smaller or larger-Banach space under the condition that a
certain iterate of the "mild perturba- tion" part of the operator combined with
the dissipative part of the semigroup maps the larger space to the smaller
space in a bounded way. The cornerstone of our approach is a factorization
argument, reminiscent of the Dyson series.(2) We apply this method to the
kinetic Fokker-Planck equation when the spatial domain is either the torus with
periodic boundary conditions, or the whole space with a confinement potential.
We then obtain spectral gap es- timates for the associated semigroup for
various metrics, including Lebesgue norms, negative Sobolev norms, and the
Monge-Kantorovich-Wasserstein distance W\_1.
