Orbits of linear operators and Banach space geometry

TL;DR

This paper investigates the behavior of orbits of bounded linear operators on Banach spaces, showing that certain sets of vectors with specific growth properties are small in a measure-theoretic and topological sense, and computes optimal exponents related to these properties.

Contribution

It introduces new results on the size and structure of sets of vectors with particular orbit growth rates, and determines optimal exponents for these growth conditions in classical Banach spaces.

Findings

The set of vectors with orbit growth exceeding a sequence tending to zero is small (porous and Haar-null).

Optimal exponents for orbit growth in classical spaces are computed.

Techniques involve the modulus of asymptotic uniform smoothness of Banach spaces.

Abstract

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 $\|T^nx\| \geqslant a_n \|T^n\|$ 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 $X$.