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

In this paper, we study the extension problems of norm-additive maps between the positive unit spheres of \(\ell _q(\ell _p)\) ${\ell }_q({\ell }_p)$ , \(1\le p,q\le \infty \) $1\le p,q\le \infty$ . Let \(S_{\ell _q(\ell _p)}^+=\{x\in \ell _q(\ell _p): x\ge 0;\Vert x\Vert =1\}\) $S_{{\ell }_q({\ell }_p)}^{+}=\{x\in {\ell }_q({\ell }_p):x\ge 0;\Vert x\Vert =1\}$ be the positive unit sphere of \(\ell _q(\ell _p)\) ${\ell }_q({\ell }_p)$ , \(f:S_{\ell _q(\ell _p)}^+\rightarrow S_{\ell _q(\ell _p)}^+\) $f:S_{{\ell }_q({\ell }_p)}^{+}\to S_{{\ell }_q({\ell }_p)}^{+}$ be a bijective norm-additive map (preserving norm of sums), i.e., \(\begin{aligned} \Vert f(x)+f(y)\Vert =\Vert x+y\Vert ,\;\mathrm{for\;all\;}x,y\in S_{\ell _q(\ell _p)}^+. \end{aligned}\) $$\begin{array}{c}\hfill \Vert f(x)+f(y)\Vert =\Vert x+y\Vert ,\mathrm{for}\mathrm{all}x,y\in S_{{\ell }_q({\ell }_p)}^{+}.\end{array}$$ In the cases when \(1<p,q\le \infty \) $1<p,q\le \infty$ or \(1<p<\infty ,\;q=1\) $1<p<\infty ,q=1$ , we show that f can be extended to a linear surjective isometry from \(\ell _q(\ell _p)\) ${\ell }_q({\ell }_p)$ onto itself. Counter examples for the remaining cases when \(p=1,\;1\le q\le \infty \) $p=1,1\le q\le \infty$ or \(p=\infty ,\;q=1\) $p=\infty ,q=1$ are also presented.