Isometric classification of Sobolev spaces on graphs

M. I. Ostrovskii¹

¹ Department of Mathematics and Computer Science St. John's University 8000 Utopia Parkway Queens, NY 11439, U.S.A.

Colloquium Mathematicum, Tome 109 (2007) no. 2, pp. 287-295.

Résumé

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 $\mathcal{G}\times\mathbb{Z}_2$.

Détail

DOI : 10.4064/cm109-2-10

Keywords: isometric sobolev spaces finite graphs characterized characterization implies following analogue banach stone theorem valid sobolev spaces connected graphs exponent which even integer isometric corresponding graphs isomorphic corollary shown each finite group mathcal each which even integer there exists mathbb subspace subset ell whose group isometries direct product mathcal times mathbb