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

TL;DR

This paper explores two classes of integer sets, Jamison and Kazhdan sets, inspired by operator theory and harmonic analysis, illustrating their properties, relationships, and examples within the group of integers.

Contribution

It provides a detailed analysis of Jamison and Kazhdan sets in a5, highlighting their properties, connections, and examples using basic operator theory and harmonic analysis tools.

Findings

Identifies links between Jamison and Kazhdan sets in a5.

Provides numerous examples illustrating properties of these sets.

Clarifies the role of these sets in operator theory and group properties.

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.