Publishing house / Journals and Serials / Colloquium Mathematicum / All issues

Abstract

Isometric Sobolev spaces on finite graphs are characterized. The characterization implies that the following analogue of the Banach–Stone theorem is valid: if two Sobolev spaces on $3$-connected graphs, with the exponent which is not an even integer, are isometric, then the corresponding graphs are isomorphic. As a corollary it is shown that for each finite group $\mathcal{G}$ and each $p$ which is not an even integer, there exists $n\in\mathbb{N}$ and a subspace $L\subset\ell_p^n$ whose group of isometries is the direct product $\mathcal{G}\times\mathbb{Z}_2$.