Bounded holomorphic functional calculus for nonsymmetric Ornstein-Uhlenbeck operators

TL;DR

This paper establishes optimal bounds for the bounded holomorphic functional calculus of nonsymmetric Ornstein-Uhlenbeck operators in infinite dimensions, extending known results to a broader class of operators.

Contribution

It proves that if the generator of an analytic semigroup is sectorial on L^2, then the operator has a bounded holomorphic functional calculus on L^r for all r, with optimal sector angles.

Findings

Results apply to finite-dimensional Ornstein-Uhlenbeck operators with dimension-free estimates.

The sectoriality angle is proven to be optimal.

The work extends functional calculus results to nonsymmetric operators in infinite dimensions.

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.