Generalized Stability of Isometries

Abstract

Let X and Y be real Banach spaces and let ε,p≥0. A mapping f: X→Y is called an (ε,p)-isometry if |‖f(x)−f(y)‖−‖x−y‖|≤ε‖x−y‖p holds for all x,y∈X. A pair (X,Y) is p-stable with respect to isometries if there exists a function δ: [0,∞)→[0,∞) with limε→0δ(ε)=0 such that for every surjective (ε,p)-isometry f: X→Y there is a surjective isometry U: X→Y satisfying the estimate ‖f(x)−U(x)‖≤δ(ε)‖x‖p, x∈X. We show that every pair of Banach spaces (X,Y) is p-stable for 0≤p<1. The pair (R2,R2) is not 1-stable. When p>1 a superstability phenomenon occurs for finite-dimensional Banach spaces.