On the Symmetrizations of ε-Isometries on Banach Spaces

Abstract

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
\(\mid\langle{x}^*,x\rangle-\langle\varphi,\Theta(x)\rangle\mid\;\leqslant\frac{3}{2}r\varepsilon\;\;\;{\rm{for\;all}}\;x\in{X}.\)

This result is used to prove new stability theorems for the symmetrization Θ of f.