  The aim of this paper is to extend to the spaces L^2(R^d , (1+|v|)^2k dv) the
spectral study led in L^2(R^d , exp(|v|^2/2)dv) by R. Ellis and M. Pinsky on
the space inhomogeneous linearized Boltzmann operator for hard spheres. More
precisely, we look at the Fourier transform in the space variable of the
inhomogeneous operator and consider the dual Fourier variable as a fixed
parameter. We then perform a precise study of this operator for small
frequencies (by seeing it as a perturbation of the homogeneous one) and also
for large frequencies from spectral and semigroup point of views. Our approach
is based on perturbation theory for linear operators as well as enlargement
arguments from M.P. Gualdani, S. Mischler and C. Mouhot.
