Orbits of linear operators and Banach space geometry

Jean-Matthieu Augé. "Orbits of linear operators and Banach space geometry." Studia Mathematica 212.1 (2012): 21-39. <http://eudml.org/doc/286636>.

@article{Jean2012,
abstract = {Let T be a bounded linear operator on a (real or complex) Banach space X. If (aₙ) is a sequence of non-negative numbers tending to 0, then the set of x ∈ X such that ||Tⁿx|| ≥ aₙ||Tⁿ|| for infinitely many n’s has a complement which is both σ-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 ∈ X such that $(||Tⁿx||/||Tⁿ||) ∉ ℓ^\{q\}(ℕ )$, using techniques which involve the modulus of asymptotic uniform smoothness of X.},
author = {Jean-Matthieu Augé},
journal = {Studia Mathematica},
keywords = {orbits of operators; compact operators; -porosity; Haar negligibility; asymptotic smoothness},
language = {eng},
number = {1},
pages = {21-39},
title = {Orbits of linear operators and Banach space geometry},
url = {http://eudml.org/doc/286636},
volume = {212},
year = {2012},
}

TY - JOUR
AU - Jean-Matthieu Augé
TI - Orbits of linear operators and Banach space geometry
JO - Studia Mathematica
PY - 2012
VL - 212
IS - 1
SP - 21
EP - 39
AB - Let T be a bounded linear operator on a (real or complex) Banach space X. If (aₙ) is a sequence of non-negative numbers tending to 0, then the set of x ∈ X such that ||Tⁿx|| ≥ aₙ||Tⁿ|| for infinitely many n’s has a complement which is both σ-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 ∈ X such that $(||Tⁿx||/||Tⁿ||) ∉ ℓ^{q}(ℕ )$, using techniques which involve the modulus of asymptotic uniform smoothness of X.
LA - eng
KW - orbits of operators; compact operators; -porosity; Haar negligibility; asymptotic smoothness
UR - http://eudml.org/doc/286636
ER -