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 $L^{2}(\gamma_{\infty})$, then ${\mathscr L}$ has bounded holomorphic functional calculus on $L^{r}(\gamma_{\infty})$, $1<r<\infty$, in any sector of angle $\theta>\theta^{*}_{r}$, where $\gamma_{\infty}$ is the associated invariant measure and $\theta^{*}_{r}$ the sectoriality angle of ${\mathscr L}$ on $L^{r}(\gamma_{\infty})$. The angle $\theta^{*}_{r}$ is optimal. In particular our result applies to any nondegenerate finite dimensional Ornstein-Uhlenbeck operator, with dimension-free estimates.