  We provide a complete elaboration of the $L^2$-Hilbert space hypocoercivity
theorem for the degenerate Langevin dynamics via studying the longtime behavior
of the strongly continuous contraction semigroup solving the associated
Kolmogorov (backward) equation as an abstract Cauchy problem. This
hypocoercivity result is proven in previous works before by Dolbeault, Mouhot
and Schmeiser in the corresponding dual Fokker-Planck framework, but without
including domain issues of the appearing operators. In our elaboration, we
include the domain issues and additionally compute the rate of convergence in
dependence of the damping coefficient. Important statements for the complete
elaboration are the m-dissipativity results for the Langevin operator
established by Conrad and the first named author of this article as well as the
essential selfadjointness results for generalized Schr\"{o}dinger operators by
Wielens or Bogachev, Krylov and R\"{o}ckner. We emphasize that the chosen
Kolmogorov approach is natural. Indeed, techniques from the theory of
(generalized) Dirichlet forms imply a stochastic representation of the Langevin
semigroup as the transition kernel of diffusion process which provides a
martingale solution to the Langevin equation. Hence an interesting connection
between the theory of hypocoercivity and the theory of (generalized) Dirichlet
forms is established besides.
