Sets of integers determined by operator-theoretical properties: Jamison and Kazhdan sets in the group $\mathbb{Z}$

Abstract

The aim of this partly expository paper is to present and discuss two classes of sets of integers (Jamison and Kazhdan sets) whose definition and/or properties are determined or inspired by operator-theoretical properties. Jamison sets first appeared in the study of the relationship between the growth of the sequence of norms of iterates of a bounded linear operator on a separable Banach space and the size of its unimodular point spectrum. Kazhdan subsets of $\mathbb{Z}$ are particular cases of Kazhdan sets in general topological groups, which are especially important as they appear in the definition of Property (T). This paper is also intended as a companion to the authors' paper [C.Badea, S.Grivaux, Kazhdan sets in groups and equidistribution properties, \emph{J. Funct. Anal.} \textbf{273} (2017), p. 1931 -- 1969], which undertakes a study of Kazhdan subsets of some classical groups without Property (T). We present here in detail the case of the group $\mathbb{Z}$, which is one of the most natural examples of groups without Property (T), and which may be useful to build an intuition of some of the main results of [C.Badea, S.Grivaux, Kazhdan sets in groups and equidistribution properties, op. cit.]. Also, the proofs in the case of the group $\mathbb{Z}$ rely solely on tools from basic operator theory and harmonic analysis. Some crucial links between Jamison and Kazhdan sets in $\mathbb{Z}$ are exhibited, and many examples are given.