We say that a map \(f:S_X \rightarrow S_Y\) between the unit spheres of two Banach spaces X and Y is a phase-isometry if it satisfies \(\begin{aligned} \big \{\Vert f(x)+f(y)\Vert , \Vert f(x)-f(y)\Vert \}=\{\Vert x+y\Vert , \Vert x-y\Vert \big \}\quad (x,y\in S_X).\end{aligned}\) Given two arbitrary index sets \(\Gamma \) and \(\Delta \) , and real Hilbert spaces H and K with \(p\in [1, \infty ]\) , we show that every surjective phase-isometry between \(S_{\ell ^p(\Gamma ,H)}\) and \(S_{\ell ^p(\Delta , K)}\) can be extended to a surjective phase-isometry from \(\ell ^p(\Gamma ,H)\) onto \(\ell ^p(\Delta , K)\) , which is phase equivalent to a linear isometry.
