Operator ergodic theorems with Möbius "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 $$(\ast )\frac{1}{N}\sum _{n=1}^N\mu (n)T^{n}v\to 0\text{ weakly }\mathrm{\forall }v\in E\mathrm{.}$$ For that, we introduce a notion of dynamical entropy for operators, which we denote $h_{top}^{\ast }(T)$, and show that if Sarnak’s conjecture is true, then $h_{top}^{\ast }(T)=0$ 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 $h_{top}^{\ast }(T)=0$.