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

We study the orthogonality of the generalized eigenspaces of an Ornstein–Uhlenbeck operator ℒ in 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 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 ℒ may or may not be orthogonal.