Mixing operators and small subsets of the circle

TL;DR

This paper characterizes weakly and strongly mixing linear operators on Banach spaces with cotype 2 using small subsets of the circle, such as countable sets and sets of uniqueness, with results applicable to general Fréchet spaces.

Contribution

It provides the first complete characterization of mixing operators in terms of small circle subsets on Banach spaces with cotype 2, extending to arbitrary Fréchet spaces.

Findings

Characterizations involve countable sets and sets of uniqueness for Fourier-Stieltjes series.

Sufficient conditions for mixing are valid on arbitrary complex separable Fréchet spaces.

Results connect operator mixing properties with harmonic analysis on the circle.

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, i.e. the sufficient conditions for weak and strong mixing, is valid on an arbitrary (complex, separable) Fr\'echet space.