---
abstract: 'We provide complete characterizations, on Banach spaces with cotype 2, of those linear operators which happen to be weakly mixing or strongly mixing transformations with respect to some nondegenerate Gaussian measure. These characterizations involve two families of small subsets of the circle: the countable sets, and the so-called *sets of uniqueness* for Fourier-Stieltjes series. The most interesting part, the sufficient conditions for weak and strong mixing, is valid on an arbitrary (complex, separable) Fréchet space.'
address:
- 'Clermont Université, Université Blaise Pascal, Laboratoire de Mathématiques, BP 10448, F-63000 Clermont-Ferrand - CNRS, UMR 6620, Laboratoire de Mathématiques, F-63177 Aubière.'
- 'Laboratoire de Mathématiques de Lens, Université d’Artois, Rue Jean Souvraz S. P. 18, 62307 Lens.'
author:
- Frédéric Bayart
- Étienne Matheron
title: |
    Mixing operators\
    and small subsets of the circle
---

Introduction
============

A basic problem in topological dynamics is to determine whether a given continuous map $T:X\to X$ acting on a topological space $X$ admits an ergodic probability measure. One may also ask for stronger ergodicity properties such as weak mixing or strong mixing, and put additional constraints on the measure $\mu$, for example that $\mu$ should have no discrete part, or that it should belong 