On the Symmetrizations of ε-Isometries on Banach Spaces

A weak stability bound for the symmetrization Θ = (f(·) − f(−·))/2 of a general e-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 \(\mid\langle{x}^*,x\rangle-\langle\varphi,\Theta(x)\rangle\mid\;\leqslant\frac{3}{2}r\varepsilon\;\;\;{\rm{for\;all}}\;x\in{X}.\)