# On the 1-dimensional complex Ornstein-Uhlenbeck operator

## TL;DR

This paper demonstrates that the 1-dimensional complex Ornstein-Uhlenbeck operator is a normal diffusion operator for certain parameter ranges, expanding understanding of its mathematical properties.

## Contribution

It establishes the normality of the complex Ornstein-Uhlenbeck operator for all fixed angles within a specific range, revealing new structural insights.

## Key findings

- The operator is a normal diffusion operator for heta in (-π/2, 0)∪(0, π/2).
- The operator is nonsymmetric but normal.
- Provides a mathematical characterization of the operator's properties.

## Abstract

We show that for any fixed $\theta\in(-\frac{\pi}{2},\,0)\cup (0,\,\frac{\pi}{2})$, the 1-dimensional complex Ornstein-Uhlenbeck operator \begin{equation*} \tilde{\mathcal{L}}_{\theta}= 4\cos\theta \frac{\partial^2}{\partial z\partial \bar{z}}-e^{\mi\theta} z \frac{\partial}{\partial z}-e^{-\mi\theta}\bar{z} \frac{\partial}{\partial \bar{z}}, \end{equation*} is a normal (but nonsymmetric) diffusion operator.