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 $\mathcal{L}$ 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.

10 pages

References in corpus (1)

• Riesz transforms of a general Ornstein--Uhlenbeck semigroup

Cited by in corpus (1)

• Riesz transforms of a general Ornstein--Uhlenbeck semigroup