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On Stability and Weak-Star Stability of \(\varepsilon \)-Isometries

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Abstract

Let X, Y be two Banach spaces, \(f:X\rightarrow Y\) be an \(\varepsilon \)-isometry with \(f(0)=0\) for some \(\varepsilon \ge 0\), and let \(Y_f\equiv \overline{\mathrm{span}}f(X)\). In this paper, we first introduce a notion of \(w^*\)-stability of an \(\varepsilon \)-isometry f. Then we show that stability of f implies its \(w^*\)-stability; the two notions of stability and \(w^*\)-stability coincide whenever X is a dual space and they are not equivalent in general. Making use of a recent sharp weak stability estimate of f, we then improve some known results.

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Acknowledgements

The authors would like to thank Professors Xiaoping Xue and Tao Qian for their helpful conversations on this paper. They also thank the functional analysis seminar of Xiamen University, where many discussions were made.

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  1. School of Mathematical Sciences, Xiamen University, Xiamen, 361005, People’s Republic of China

    Xiaoling Chen, Lixin Cheng & Wen Zhang

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  1. Xiaoling Chen
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  2. Lixin Cheng
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  3. Wen Zhang
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Correspondence to Xiaoling Chen.

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Xiaoling Chen, Lixin Cheng, Wen Zhang: support in partial by NSFC, Grant Nos. 11731010 and 12071388.

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Chen, X., Cheng, L. & Zhang, W. On Stability and Weak-Star Stability of \(\varepsilon \)-Isometries. Results Math 76, 64 (2021). https://doi.org/10.1007/s00025-021-01374-w

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