Stability Characterizations of ϵ-isometries on Certain Banach Spaces

Suppose that X, Y are two real Banach Spaces. We know that for a standard ϵ-isometry f : X → Y, the weak stability formula holds and by applying the formula we can induce a closed subspace N of Y*. In this paper, by using again the weak stability formula, we further show a sufficient and necessary condition for a standard ϵ-isometry to be stable in assuming that N is w*-closed in Y*. Making use of this result, we improve several known results including Figiel’s theorem in reflexive spaces. We also prove that if, in addition, the space Y is quasi-reflexive and hereditarily indecomposable, then L(f) ≡ span¯[f(X)] contains a complemented linear isometric copy of X; Moreover, if X = Y, then for every ϵ-isometry f : X → X, there exists a surjective linear isometry S : X → X such that f − S is uniformly bounded by 2ϵ on X.