On Stability Of ε-Isometries Of Banach Spaces

Suppose that X,Y are two real Banach spaces,f:X→Y Y is a2 standard ε-isometry.This thesis is devoted to studying the stability properties of f.For the symmetrization θ =f(·)-f(-)/2 of a general ε-isometry f,we consider its weak stabil-ity formula and obtain the following sharp estimate:For every x*∈ X*,there existsφ∈Y*with ||φ|| =||x*|| ≡r so that|<x*,x>-<φ,θ(x)>|≤3/2rε for all x∈X.As its applications,we show that for every ε-isometry f,θ is a symmetric 3ε-isometry;and we prove a somewhat surprising stability result:if f is almost surjec-five,then there is a linear surjective isometry X → Y so that θ-U is uniformly bounded by 3/1 ε on X.By using again the weak stability formula,we further show a sufficient and neces-sary condition for the standard ε-isometry to be stable in assuming that N is*-closed in Y*,1.e.f is stable if and only if there exists a projection P:L(f)→⊥N so that Pf:X → ⊥N is a δ-isometry for some δ ≥ 0.If Y is a quasi-reflexive HI-space,and f:X → Y is a stable standard ε-isometry,then Y contains a complemented linear isometric copy of X;and if X is a quasi-reflexive HI-space,then for every standard self ε-isometry f:X→ X,there exists a surjective linear self-isometry U:X→ X so that f—U is uniformly bounded by 2ε on X.