Hilbertian Jamison sequences and rigid dynamical systems

A strictly increasing sequence (nk)k⩾0 of positive integers is said to be a Hilbertian Jamison sequence if for any bounded operator T on a separable Hilbert space such that supk⩾0‖Tnk‖<+∞, the set of eigenvalues of modulus 1 of T is at most countable. We first give a complete characterization of such sequences. We then turn to the study of rigidity sequences (nk)k⩾0 for weakly mixing dynamical systems on measure spaces, and give various conditions, some of which are closely related to the Jamison condition, for a sequence to be a rigidity sequence. We obtain on our way a complete characterization of topological rigidity and uniform rigidity sequences for linear dynamical systems, and we construct in this framework examples of dynamical systems which are both weakly mixing in the measure-theoretic sense and uniformly rigid.