Operator ergodic theorems with Möbius "weights"

Abstract

Motivated by Sarnak's conjecture in topological dynamics for the Möbius function $\hat{I}¼$, we study, for a power-bounded $T$ on a Banach space $E$, the weak convergence $$(\ast )\frac{1}{N}\sum _{n=1}^N\hat{I}¼(n)T^{n}v\to 0\text{ weakly }\forall v\in E.$$ 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 $h_{top}^{\ast }(T)=0$.

Dedicated to the memory of Alexandra Bellow. 41 pages, 97 references, 4 appendices; Appendix C by C. Cuny. Comments, questions, and suggestions are welcome