Hilbertian Jamison sequences and rigid dynamical systems

TL;DR

This paper characterizes Hilbertian Jamison sequences and explores their connection to rigidity sequences in weakly mixing dynamical systems, providing new insights into the spectral properties and rigidity phenomena in linear and measure-preserving systems.

Contribution

It offers a complete characterization of Hilbertian Jamison sequences and establishes new conditions linking these sequences to rigidity in dynamical systems.

Findings

Complete characterization of Hilbertian Jamison sequences

Conditions for sequences to be rigidity sequences in weakly mixing systems

Examples of systems that are both weakly mixing and uniformly rigid

Abstract

A strictly increasing sequence (n_k) of positive integers is said to be a Hilbertian Jamison sequence if for any bounded operator T on a separable Hilbert space such that the supremum over k of the norms ||T^{n_k}|| is finite, 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 (n_k) 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.