On the entangled ergodic theorem

TL;DR

This paper proves the strong convergence of entangled ergodic averages in Banach spaces under weak compactness conditions, providing a spectral analysis-based formula for the limits.

Contribution

It establishes the convergence of entangled ergodic averages for general Banach spaces without restrictions on the partition, extending previous results.

Findings

Strong convergence of entangled ergodic averages proven

Convergence holds under weak compactness assumptions

Spectral analysis formula for the limit provided

Abstract

We study the convergence of the so-called entangled ergodic averages $\frac{1}{N^k}\sum_{n_1,...,n_k=1}^{N}T_m^{n_{\alpha(m)}}A_{m-1}T_{m-1}^{n_{\alpha(m-1)}}A_{m-2}...A_1T_1^{n_{\alpha(1)}},$ where $k\leq m$ and $\alpha:\{1,...,m\}\to\{1,...,k\}$ is a surjective map. We show that, on general Banach spaces and without any restriction on the partition $\alpha$, the above averages converge strongly as $N\to \infty$ under some quite weak compactness assumptions on the operators $T_j$ and $A_j$. A formula for the limit based on the spectral analysis of the operators $T_j$ and the continuous version of the result are presented as well.