Acta mathematica scientia,Series B ›› 2019, Vol. 39 ›› Issue (4): 1163-1172.doi: 10.1007/s10473-019-0418-9

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Duanxu DAI1, Bentuo ZHENG2   

  1. 1. College of Mathematics and Computer Science, Quanzhou Normal University, Quanzhou 362000, China;
    2. Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, USA
  • Received:2018-03-22 Revised:2018-09-18 Online:2019-08-25 Published:2019-09-12
  • Supported by:
    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).

Abstract: A pair of Banach spaces (X, Y) is said to be stable if for every ε-isometry f:XY, there exist γ > 0 and a bounded linear operator T:L(f) → X with ||T|| ≤ α such that ||Tf(x)-x|| ≤ γε for all xX, 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.

Key words: Stability, ε-isometry, Figiel theorem, Banach space

CLC Number: 

  • 46B04