We study the superposition operator f on on the space ac 0 of sequences almost converging to zero. Conditions are derived for which f has a representation of the form f x = a+bx +g x, for all x ∈ ac 0 with a = f 0, b ∈ D(ac 0), g a superposition operator from ℓ∞ into I(ac 0), D(ac 0) = {z: zx ∈ ac 0 for all x ∈ ac 0}, and I(ac 0) the maximal ideal in ac 0. If f is generated by a function f of a real variable, then f is linear. We consider the conditions for which a bounded function f generates f acting on ac 0 and the conditions for which there exists a sequence y ∈ ac 0 such that y − f y ∈ ac 0. In terms of f, criteria for the boundedness, continuity, and sequential σ(ac 0ℓ1)-continuity of f on ac 0 are given. It is shown that the continuity of f is equivalent to the weak sequential continuity. Finally, properties of spaces D(ac 0) and I(ac 0) are studied, and in particular it is established that the inclusion I(ac 0) ⊕ {λe: λ ∈ ℝ} ⊂ D(ac 0) is proper, where e = (1, 1, …). By means of D(ac 0), a number of Banach-Mazur limit properties are derived.
@article{bwmeta1.element.doi-10_2478_s11533-011-0135-7,
author = {Egor Alekhno},
title = {Superposition operator on the space of sequences almost converging to zero},
journal = {Open Mathematics},
volume = {10},
year = {2012},
pages = {619-645},
zbl = {1256.47036},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-011-0135-7}
}
Egor Alekhno. Superposition operator on the space of sequences almost converging to zero. Open Mathematics, Tome 10 (2012) pp. 619-645. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-011-0135-7/
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