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...

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Bibliographic Details
Published in:Acta mathematica scientia 2019-07, Vol.39 (4), p.1163-1172
Main Authors: Dai, Duanxu, Zheng, Bentuo
Format: Article
Language:English
Subjects:
Analysis
Mathematics
Mathematics and Statistics
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Summary: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.
ISSN:0252-9602
1572-9087
DOI:10.1007/s10473-019-0418-9