Superposition operator on the space of sequences almost converging to zero

Abstract

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.