On the Symmetrizations of ε-Isometries on Banach Spaces

A weak stability bound for the symmetrization Θ = (f(·) − f(−·))/2 of a general ε-isometry f from a Banach space X to a Banach space Y is presented. As a corollary, the following somewhat surprising weak stability result is obtained: For every x* ∈ X*, there exists ϕ ∈ Y* with ‖ϕ‖ = ‖x*‖ ≔ r such that ∣⟨x∗,x⟩−⟨φ,Θ(x)⟩∣⩽32rεforallx∈X.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mid\langle{x}^*,x\rangle-\langle\varphi,\Theta(x)\rangle\mid\;\leqslant\frac{3}{2}r\varepsilon\;\;\;{\rm{for\;all}}\;x\in{X}.$$\end{document}