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 and each p which is not an even integer, there exists n ∈ ℕ and a subspace whose group of isometries is the direct product × ℤ₂.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm109-2-10,
author = {M. I. Ostrovskii},
title = {Isometric classification of Sobolev spaces on graphs},
journal = {Colloquium Mathematicae},
volume = {107},
year = {2007},
pages = {287-295},
zbl = {1127.46006},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm109-2-10}
}
M. I. Ostrovskii. Isometric classification of Sobolev spaces on graphs. Colloquium Mathematicae, Tome 107 (2007) pp. 287-295. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm109-2-10/