Strongly mixing operators on Hilbert spaces and speed of mixing

TL;DR

This paper studies the rate at which certain operators on infinite-dimensional Hilbert spaces mix functions, showing that uniform mixing speed isn't possible and identifying classes with polynomial decay rates based on eigenvector regularity.

Contribution

It demonstrates the impossibility of uniform mixing speed for all functions and characterizes classes with specific polynomial decay rates linked to eigenvector regularity.

Findings

No uniform speed of mixing for all square-integrable functions.

Polynomial decay rate of correlations for regular functions.

Connection between eigenvector regularity and mixing speed.

Abstract

We investigate the subject of speed of mixing for operators on infinite dimensional Hilbert spaces which are strongly mixing with respect to a nondegenerate Gaussian measure. We prove that there is no way to find a uniform speed of mixing for all square-integrable functions. We give classes of regular functions for which the sequence of correlations decreases to zero with speed $n^{-\alpha}$ when the eigenvectors associated to unimodular eigenvalues of the operator are parametrized by an $\alpha$-H\"olderian $\mathbb{T}$-eigenvector field.