Orbits of linear operators and Banach space geometry

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 $\Vert T^{n}x\Vert ⩾a_n\Vert T^n\Vert$ 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 $(\Vert T^{n}x\Vert /\Vert T^n\Vert )\notin {\ell }^q(\mathbb{N})$, using techniques which involve the modulus of asymptotic uniform smoothness of $X$.

arXiv comment16 pages