Stability of isometries on Hilbert spaces

Abstract

Let X and Y be real Banach spaces and epsilon, pge 0. A mapping T between X and Y is called an (epsilon,p)-isometry if ||T(x)-T(y)|-|x-y| | le epsilon |x-y|^pfor x, y in X. Let H be a real Hilbert space and T:H to Han (epsilon, p)-isometry with T(0)=0. If p neq 1 is anonnegative number, then there exists a unique isometry I:Hto Hsuch that |T(x)-I(x)| le C(epsilon)(|x|^{(1+p)/2}+|x|^p)for all x in H, where C(epsilon)to 0 as epsilon to 0.