Phase-isometries between two \(\ell^p(\Gamma , H)\)-type spaces (Q2198300)

A map \(f\) between two normed spaces \(X\) and \(Y\) is called a phase-isometry if the functional equation \[\{\|f(x)+f(y)\|,\|f(x)-f(y)\|\}=\{\|x+y\|,\|x-y\|\}\] holds for all \(x,y\in X\). The authors prove the following result, in the spirit of the classical theorem in [\textit{S. Mazur} and \textit{S. Ulam}, C. R. Acad. Sci., Paris 194, 946--948 (1932; Zbl 0004.02103)]: Theorem. If \(H\) and \(K\) are real inner product spaces, \(\Gamma\) and \(\Delta\) are nonempty sets, \(1\leq p<\infty\) and \(f:\ell^p(\Gamma,H) \rightarrow \ell^p(\Delta,K)\) is a surjective phase-isometry, then there is a function \(\varepsilon:\ell^p(\Gamma,H)\rightarrow \{-1,1\}\) such that \(\varepsilon f\) is a linear isometry. The case \(H=K=\mathbb{R}\) has been treated before in [\textit{X. Huang} and \textit{D. Tan}, Publ. Math. 92, No. 3--4, 411--418 (2018; Zbl 1399.39052)]. See also [\textit{X. Huang} and \textit{X. Jin}, Bull. Korean Math. Soc. 56, No. 6, 1377--1384 (2019; Zbl 1442.46008)] for a result on the extension of phase-isometries between the unit spheres of \(\ell^p(\Gamma)\) and \(\ell^p(\Delta)\).