On the entangled ergodic theorem

Let $U$ be a unitary operator acting on the Hilbert space H, and $\alpha :\{1,\mathrm{.}\mathrm{.}\mathrm{.},m\}\mapsto \{1,\mathrm{.}\mathrm{.}\mathrm{.},k\}$ a partition of the set $\{1,\mathrm{.}\mathrm{.}\mathrm{.},m\}$. We show that the ergodic average $$\frac{1}{N^k}\sum _{n_1,\mathrm{.}\mathrm{.}\mathrm{.},n_k=0}^{N-1}{U}^{n_{\alpha (1)}}{A}_1{U}^{n_{\alpha (2)}}\mathrm{.}\mathrm{.}\mathrm{.}{U}^{n_{\alpha (m-1)}}{A}_{m-1}{U}^{n_{\alpha (m)}}$$ converges in the weak operator topology if the $A_j$ belong to the algebra of all the compact operators on H. We write esplicitely the formula for these ergodic averages in the case of pair–partitions. Some results without any restriction on the operators $A_j$ are also presented in the almost periodic case.