Orbits of linear operators and Banach space geometry

Jean-Matthieu Augé¹

¹ Université Bordeaux 1 351, Cours de la Libération F-33405 Talence Cedex, France

Studia Mathematica, Tome 212 (2012) no. 1, pp. 21-39

Résumé

Let $T$ be a bounded linear operator on a (real or complex) Banach space $X$. If $(a_n)$ is a sequence of non-negative numbers tending to 0, then the set of $x \in X$ such that $\|T^nx\| \geq a_n \|T^n\|$ for infinitely many $n$'s has a complement which is both $\sigma$-porous and Haar-null. We also compute (for some classical Banach space) optimal exponents $q>0$ such that for every non-nilpotent operator $T$, there exists $x \in X$ such that $(\|T^nx\|/\|T^n\|) \notin \ell^{q}(\mathbb{N})$, using techniques which involve the modulus of asymptotic uniform smoothness of $X$.

Détail

DOI : 10.4064/sm212-1-2

Keywords: bounded linear operator real complex banach space sequence non negative numbers tending set geq infinitely many has complement which sigma porous haar null compute classical banach space optimal exponents every non nilpotent operator there exists notin ell mathbb using techniques which involve modulus asymptotic uniform smoothness nbsp