On the orthogonality of generalized eigenspaces for the Ornstein--Uhlenbeck operator

We study the orthogonality of the generalized eigenspaces of an Ornstein--Uhlenbeck operator $\mathcal L$ in $\mathbb{R}^N$, with drift given by a real matrix $B$ whose eigenvalues have negative real parts. If $B$ has only one eigenvalue, we prove that any two distinct generalized eigenspaces of $\mathcal L$ are orthogonal with respect to the invariant Gaussian measure. Then we show by means of two examples that if $B$ admits distinct eigenvalues, the generalized eigenspaces of $\mathcal L$ may or may not be orthogonal.