Operator ergodic theorems with M\"obius "weights"

Motivated by Sarnak's conjecture in topological dynamics for the Möbius function $\mu$, we study, for a power-bounded $T$ on a Banach space $E$, the weak convergence $$ (*) \qquad \qquad \frac1N\sum_{n=1}^N \mu(n)T^nv \to 0 \text{ weakly } \forall v\in E. $$
For that, we introduce a notion of dynamical entropy for operators, which we denote $h^*_{top}(T)$, and show that if Sarnak's conjecture is true, then $h^*_{top}(T)=0$ implies the desired convergence (*). We conclude an equivalent operator formulation of Sarnak's conjecture.
For several classes of operators we prove that (*) holds, and that $h^*_{top}(T)=0$.