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We study degenerate hypoelliptic Ornstein-Uhlenbeck operators in $L^2$ spaceswith respect to invariant measures. The purpose of this article is to show howrecent results on general quadratic operators apply to the study of degeneratehypoelliptic Ornstein-Uhlenbeck operators. We first show that some knownresults about the spectral and subelliptic properties of Ornstein-Uhlenbeckoperators may be directly recovered from the general analysis of quadraticoperators with zero singular spaces. We also provide new resolvent estimatesfor hypoelliptic Ornstein-Uhlenbeck operators. We show in particular that thespectrum of these non-selfadjoint operators may be very unstable under smallperturbations and that their resolvents can blow-up in norm far away from theirspectra. Furthermore, we establish sharp resolvent estimates in specificregions of the resolvent set which enable us to prove exponential return toequilibrium.

Some remarks on degenerate hypoelliptic Ornstein-Uhlenbeck operators