On the entangled ergodic theorem

Let $U$ be a unitary operator acting on the Hilbert space H, and $\alpha:\{1,..., m\}\mapsto\{1,..., k\}$ a partition of the set $\{1,..., m\}$. We show that the ergodic average $$ \frac{1}{N^{k}}\sum_{n_{1},...,n_{k}=0}^{N-1} U^{n_{\alpha(1)}}A_{1}U^{n_{\alpha(2)}}... U^{n_{\alpha(m-1)}}A_{m-1}U^{n_{\alpha(m)}} $$ 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.