# Optimal angle of the holomorphic functional calculus for the classical Ornstein-Uhlenbeck operator on $L^p$

## TL;DR

This paper proves that the classical Ornstein-Uhlenbeck operator on L^p spaces has an optimal angle for its holomorphic functional calculus, using a simple proof of R-sectoriality and applying abstract calculus theory.

## Contribution

It provides a straightforward proof of the optimal angle for the holomorphic functional calculus of the Ornstein-Uhlenbeck operator, improving understanding of its spectral properties.

## Key findings

- The Ornstein-Uhlenbeck operator is R-sectorial of angle arcsin|1-2/p|.
- The operator admits a bounded H-infinity functional calculus with this optimal angle.
- The proof simplifies previous approaches to the spectral analysis of the operator.

## Abstract

We give a simple proof of the fact that the classical Ornstein-Uhlenbeck operator $L$ is R-sectorial of angle $arcsin|1-2/p|$ on $L^{p}(\mathbb{R}^{n},\exp(-|x|^2/2)dx)$ (for $1<p<\infty$). Applying the abstract holomorphic functional calculus theory of Kalton and Weis, this immediately gives a new proof of the fact that $L$ has a bounded $H^{\infty}$ functional calculus with this optimal angle.