Operators commuting with mixing sequences

Abstract

Let $(X,\mathcal{F} ,\mu)$ be a probability space and let $L^{2}(X,0)$ be the collection of all $f \in L^{2}(X)$ with zero integrals. A collection $\mathcal{A}$ of linear operators on $L^{2}(X)$ is said to satisfy the Gaussian-distribution property (G.D.P.) if $L^{2}(X,0)$ is invariant under $\mathcal{A}$ and there exists a constant $C \lt \infty$ such that the following condition holds:

Whenever $T_{1},\ldots,T_{k}$ are finitely many operators in $\mathcal{A}$, and $f$ is a function in $L_{2}$ with zero integral, then, for any required degree of approximation, there is another $L_{2}$-function $g$ with $||g||_{2} \leq C ||f||_{2}$, such that all the inner products $(\mathrm{Re} T_{i}g, \mathrm{Re} T_{j}g)$ are approximately equal to the corresponding inner products $(\mathrm{Re} T_{i}f, \mathrm{Re} T_{j}f)$ for all $1 \leq i,j \leq k$ and such that the joint distribution of the functions $\mathrm{Re} T_{1}g,\ldots,\mathrm{Re} T_{k}g$ is approximately Gaussian.

It has been proved that if $(S_{n})^{\infty}_{1}$ is a sequence of uniformly bounded linear operators on $L^{2}(X)$ that satisfies the Bourgain's infinite entropy condition and the G.D.P., then there exists an $h \in L^{2}(X)$ such that $\liminf S_{n}\,h$ fails to exist $\mu$-a.e., as a finite limit on $X$.

The purpose of this paper is to provide sufficient conditions for a collection $\mathcal{A}$ of linear operators on $L^{2}(X)$ to satisfy the G.D.P.