数学物理学报(英文版) ›› 2019, Vol. 39 ›› Issue (4): 1163-1172.doi: 10.1007/s10473-019-0418-9

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STABILITY OF A PAIR OF BANACH SPACES FOR ε-ISOMETRIES

戴端旭1, 郑本拓2   

  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
  • 收稿日期:2018-03-22 修回日期:2018-09-18 出版日期:2019-08-25 发布日期:2019-09-12
  • 通讯作者: Duanxu DAI E-mail:dduanxu@163.com
  • 作者简介:Bentuo ZHENG,E-mail:bzheng@memphis.edu
  • 基金资助:
    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).

STABILITY OF A PAIR OF BANACH SPACES FOR ε-ISOMETRIES

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).

摘要: 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.

关键词: Stability, ε-isometry, Figiel theorem, Banach space

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

中图分类号: 

  • 46B04