Orbits of linear operators and Banach space geometry

2012/04/10 by Jean-Matthieu Augé, Augé, Jean-Matthieu
Mathematics · #47A16 #FOS: Mathematics #Functional Analysis (math.FA) #Primary 47A05 #Secondary 28A05 #math.FA #msc:28A05 #msc:47A05 #msc:47A16

paper · pdf · doi:10.48550/arxiv.1204.2046

16 pages

arxiv created 2012/04/10 · arxiv updated 2012/04/11

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 ∈ X such that ‖Tⁿx‖ \geqslant a_n ‖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.

Related

• Operators Whose Conjugation Orbits Satisfy Polynomial Growth Conditions
• Operator symbols. II
• Linear structures in the set of non-norm-attaining operators on Banach spaces
• Rank-one perturbations and norm-attaining operators
• Norm attaining operators and variational principle