Lp-spectrum of degenerate hypoelliptic Ornstein-Uhlenbeck operators

We describe the spectrum of degenerate hypoelliptic Ornstein-Uhlenbeck operators A = ∑ i , j = 1 n q i j D i j + ∑ i , j = 1 n b i j x j D i in L p ( R n ) , 1 ≤ p + ∞ , and in C 0 ( R n ) . 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 L p spaces.