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On Extension of Norm-Additive Maps Between the Positive Unit Spheres of \(\ell _q(\ell _p)\)

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Abstract

In this paper, we study the extension problems of norm-additive maps between the positive unit spheres of \(\ell _q(\ell _p)\), \(1\le p,q\le \infty \). Let \(S_{\ell _q(\ell _p)}^+=\{x\in \ell _q(\ell _p): x\ge 0;\Vert x\Vert =1\}\) be the positive unit sphere of \(\ell _q(\ell _p)\), \(f:S_{\ell _q(\ell _p)}^+\rightarrow S_{\ell _q(\ell _p)}^+\) be a bijective norm-additive map (preserving norm of sums), i.e.,

$$\begin{aligned} \Vert f(x)+f(y)\Vert =\Vert x+y\Vert ,\;\mathrm{for\;all\;}x,y\in S_{\ell _q(\ell _p)}^+. \end{aligned}$$

In the cases when \(1<p,q\le \infty \) or \(1<p<\infty ,\;q=1\), we show that f can be extended to a linear surjective isometry from \(\ell _q(\ell _p)\) onto itself. Counter examples for the remaining cases when \(p=1,\;1\le q\le \infty \) or \(p=\infty ,\;q=1\) are also presented.

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Acknowledgements

The authors would like to thank the referee and the editor for their constructive comments and helpful suggestions on this paper.

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  1. Hebei Key Laboratory of Physics and Energy Technology, School of Mathematics and Physics, North China Electric Power University, Baoding, 071003, People’s Republic of China

    Longfa Sun & Yinghua Sun

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Longfa Sun is supported by the National Natural Science Foundation of China ( Grant no. 12101234), the Natural Science Foundation of Hebei Province (Grant no. A2022502010), the Fundamental Research Funds for the Central Universities (Grant no. 2023MS164).

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Sun, L., Sun, Y. On Extension of Norm-Additive Maps Between the Positive Unit Spheres of \(\ell _q(\ell _p)\). Results Math 79, 126 (2024). https://doi.org/10.1007/s00025-024-02154-y

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