Spectral decompositions and 𝕃2-operator norms of toy hypocoercive semi-groups

For any a > 0, consider the hypocoercive generators y∂_x+a∂²_y-y∂_y and y∂_x-ax∂_y+∂²_y-y∂_y, respectively for (x,y) ε R/(2πZ)×R and (x,y) ε R×R. The goal of the paper is to obtain exactly the L²(μ_a)-operator norms of the corresponding Markov semi-group at any time, where μ_a is the associated invariant measure. The computations are based on the spectral decomposition of the generator and especially on the scalar products of the eigenvectors. The motivation comes from an attempt to find an alternative approach to classical ones developed to obtain hypocoercive bounds for kinetic models.