Bounded holomorphic functional calculus for nonsymmetric Ornstein-Uhlenbeck operators

Abstract

We study bounded holomorphic functional calculus for nonsymmetric infinite dimensional Ornstein-Uhlenbeck operators ${\mathscr L}$. We prove that if $-{\mathscr L}$ generates an analytic semigroup on $L^{2}(γ_{\infty})$, then ${\mathscr L}$ has bounded holomorphic functional calculus on $L^{r}(γ_{\infty})$, $1 θ^{*}_{r}$, where $γ_{\infty}$ is the associated invariant measure and $θ^{*}_{r}$ the sectoriality angle of ${\mathscr L}$ on $L^{r}(γ_{\infty})$. The angle $θ^{*}_{r}$ is optimal. In particular our result applies to any nondegenerate finite dimensional Ornstein-Uhlenbeck operator, with dimension-free estimates.