STABILITY OF A PAIR OF BANACH SPACES FOR ε-ISOMETRIES

A pair of Banach spaces(X, Y) is said to be stable if for every ε-isometry f :X → Y, there exist γ > 0 and a bounded linear operator T : L(f) → X with ||T|| ≤α such that ||T f(x)-x|| ≤γε for all x ∈ X, where L(f) is the closed linear span of f(X). In this article, we study the stability of a pair of Banach spaces(X, Y) when X is a C(K) space.This gives a new positive answer to Qian’s problem. Finally, we also obtain a nonlinear version for Qian’s problem.