A universal theorem for stability of ε-isometries of Banach spaces

Abstract

Let X, Y be two Banach spaces, and f:X→Y be a standard ε-isometry for some ε≥0. In this paper, we show the following sharp weak stability inequality of f: for every x⁎∈X⁎ there exists ϕ∈Y⁎ with ‖ϕ‖=‖x⁎‖≡r such that|〈x⁎,x〉−〈ϕ,f(x)〉|≤2εrfor allx∈X. It is not only a sharp quantitative extension of Figiel's theorem, but it also unifies, generalizes and improves a series of known results about stability of ε-isometries. For example, if the mapping f satisfies C(f)≡co¯[f(X)∪−f(X)]=Y, then it is equivalent to the following sharp stability theorem: There is a linear surjective operator T:Y→X of norm one such that ‖Tf(x)−x‖≤2ε, for all x∈X; When the ε-isometry f is surjective, it is equivalent to Omladič–Šemrl's theorem: There is a surjective linear isometry U:X→Y so that‖f(x)−Ux‖≤2ε,for allx∈X.