On the entangled ergodic theorem

We study the convergence of the so-called entangled ergodic averages $\frac{1}{N^k}{\sum }_{n_1,\mathrm{.}\mathrm{.}\mathrm{.},n_k=1}^N{T}_m^{n_{\alpha (m)}}{A}_{m-1}{T}_{m-1}^{n_{\alpha (m-1)}}{A}_{m-2}\mathrm{.}\mathrm{.}\mathrm{.}{A}_1{T}_1^{n_{\alpha (1)}},$ where $k\le m$ and $\alpha :\{1,\mathrm{.}\mathrm{.}\mathrm{.},m\}\to \{1,\mathrm{.}\mathrm{.}\mathrm{.},k\}$ 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 $N\to \mathrm{\infty }$ 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.