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 *. 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 e-isometry f: X → X, there exists a surjective linear isometry S:X → X such that f-S is uniformly bounded by 2ε on X.

Stability Characterizations of ∈ -isometries on Certain Banach Spaces

Suppose that X, Y are two real Banach Spaces. We know that for a standard ∈-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 ∈-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 $$L(f) equiv overline {span} [f(x)]$$ contains a complemented linear isometric copy of X; Moreover, if X = Y, then for every ∈-isometry f : X → X, there exists a surjective linear isometry S : X → X such that f − S is uniformly bounded by 2∈ on X.