Lp-spectrum of degenerate hypoelliptic Ornstein-Uhlenbeck operators

We describe the spectrum of degenerate hypoelliptic Ornstein-Uhlenbeck operators A=∑i,j=1nqijDij+∑i,j=1nbijxjDi in Lp(Rn), 1≤p<+∞, and in C0(Rn). We show that the spectrum of A is the sum of (−∞,0] and the spectrum of the drift term. Our result gives a complete picture of the spectral properties of Ornstein-Uhlenbeck operators in Lp spaces.