On the entangled ergodic theorem

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.