Restricted isometries in finite-dimensional $\ell^p$-spaces
For $d,n\in\mathbb{Z}_{\geq1}$ and $p\in[1,\infty)\setminus\{2\}$, we characterize all linear maps $s\colon\ell_d^p\to\ell_n^p$ for which there is $E\subseteq X_d$ such that $\|s(ξ)\|_p=\|ξ|_E\|_p$ for every $ξ\in\ell_d^p$. We call these maps \emph{restricted isometries}. Our Lamperti-type theorem gives a unique description of each such operator using a set transformation, a positive amplitude function, and a unimodular phase function. In particular, it includes noninjective and nonsurjective operators, even when the dimensions differ. We further characterize restricted isometries between finite $\ell^p$-spaces with arbitrary positive atomic weights. Applications include formulas counting set transformations and uniform restricted isometries, as well as a geometric description of all restricted isometries with fixed column supports.