Sur le spectre des op\'erateurs rigides

TL;DR

This paper constructs examples of rigid bounded operators on l^2 with specific spectra, including invertible ones whose inverses are not rigid, advancing understanding of spectral properties and rigidity in operator theory.

Contribution

It provides the first examples of rigid invertible operators with non-rigid inverses and characterizes their spectra.

Findings

Constructed rigid operators with spectra rom 0 to 1.

Provided examples of invertible rigid operators with non-rigid inverses.

Analyzed spectral properties related to rigidity.

Abstract

A bounded operator $u$ on $X$ is called rigid when there is an increasing sequence of positive integers $(n_k)_{k\geq 1}$, such that for every $x$ in $X$ we have $\lim_{k \rightarrow +\infty} u^{n_k} x = x$. For any $r$ in $[0,1]$, we construct a rigid bounded operator of $l^2$ the spectrum of which is $\{\lambda \in \mathbb C: r \leq | \lambda | \leq 1\}$. For $0 < r < 1$, it gives the first examples of rigid bounded invertible operators, such that their inverse is not rigid.