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

Abstract

We study the orthogonality of the generalized eigenspaces of an Ornstein--Uhlenbeck operator L in RN, 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.