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 ‖Tf (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.

Publication Title

Acta Mathematica Scientia