Sur le spectre des opérateurs rigides

Abstract

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

in French