On extension of isometries between the positive unit spheres of \((\sum \ell _p)_{\ell _q}\)

In this paper, we study the problem of extending surjective isometries between the positive unit spheres of two \((\sum \ell _p)_{\ell _q}\) ${(\sum {\ell }_p)}_{{\ell }_q}$ -spaces for \(1\le p,q\le \infty \) $1\le p,q\le \infty$ . Let \(S_{(\sum \ell _p)_{\ell _q}}^+=\left\{ x\in (\sum \ell _p)_{\ell _q}: x\ge 0;\Vert x\Vert =1\right\} \) $S_{{(\sum {\ell }_p)}_{{\ell }_q}}^{+}=\left\{x,\in ,{(\sum {\ell }_p)}_{{\ell }_q},:,x,\ge ,0,;,\Vert x\Vert ,=,1\right\}$ be the positive unit sphere of \((\sum \ell _p)_{\ell _q}\) ${(\sum {\ell }_p)}_{{\ell }_q}$ and \(f:S_{(\sum \ell _p)_{\ell _q}}^+\rightarrow S_{(\sum \ell _p)_{\ell _q}}^+\) $f:S_{{(\sum {\ell }_p)}_{{\ell }_q}}^{+}\to S_{{(\sum {\ell }_p)}_{{\ell }_q}}^{+}$ be an isometry, i.e., \(\begin{aligned} \Vert f(x)-f(y)\Vert =\Vert x-y\Vert ,\;\mathrm{for\;all\;}x,y\in S_{(\sum \ell _p)_{\ell _q}}^+. \end{aligned}\) $$\begin{array}{c}\hfill \Vert f(x)-f(y)\Vert =\Vert x-y\Vert ,\mathrm{for}\mathrm{all}x,y\in S_{{(\sum {\ell }_p)}_{{\ell }_q}}^{+}.\end{array}$$ We prove that every surjective isometry \(f: S_{(\sum \ell _p)_{\ell _q}}^+\rightarrow S_{(\sum \ell _p)_{\ell _q}}^+\) $f:S_{{(\sum {\ell }_p)}_{{\ell }_q}}^{+}\to S_{{(\sum {\ell }_p)}_{{\ell }_q}}^{+}$ can be extended to a linear surjective isometry from \((\sum \ell _p)_{\ell _q}\) ${(\sum {\ell }_p)}_{{\ell }_q}$ onto itself. This conclusion also holds for the case of \((\sum \ell _p)_{c_0}\) ${(\sum {\ell }_p)}_{c_0}$ , where \(1\le p\le \infty \) $1\le p\le \infty$ .