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Stability of a Pair of Banach Spaces for ε-Isometries

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Abstract

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.

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Authors and Affiliations

  1. College of Mathematics and Computer Science, Quanzhou Normal University, Quanzhou, 362000, China

    Duanxu Dai  (戴端旭)

  2. Department of Mathematical Sciences, University of Memphis, Memphis, TN, 38152, USA

    Bentuo Zheng  (郑本拓)

Authors
  1. Duanxu Dai  (戴端旭)
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  2. Bentuo Zheng  (郑本拓)
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Corresponding author

Correspondence to Duanxu Dai  (戴端旭).

Additional information

Duanxu Dai is supported in part by NSFC (11601264, 11471270 and 11471271), the Fundamental Research Funds for the Central Universities (20720160037), the Outstanding Youth Scientific Research Personnel Training Program of Fujian Province, the High level Talents Innovation and Entrepreneurship Project of Quanzhou City (2017Z032), the Research Foundation of Quanzhou Normal University (2016YYKJ12), and the Natural Science Foundation of Fujian Province of China (2019J05103). Bentuo Zheng is supported in part by NSFC (11628102).

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Dai, D., Zheng, B. Stability of a Pair of Banach Spaces for ε-Isometries. Acta Math Sci 39, 1163–1172 (2019). https://doi.org/10.1007/s10473-019-0418-9

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  • DOI: https://doi.org/10.1007/s10473-019-0418-9

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