We characterize Hilbert spaces among Banach spaces in terms of transitivity with respect to nicely behaved subgroups of the isometry group. For example, the following result is typical: If X is a real Banach space isomorphic to a Hilbert space and convex-transitive with respect to the isometric finite-dimensional perturbations of the identity, then X is already isometric to a Hilbert space.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm200-1-2,
author = {Jarno Talponen},
title = {Special symmetries of Banach spaces isomorphic to Hilbert spaces},
journal = {Studia Mathematica},
volume = {196},
year = {2010},
pages = {31-40},
zbl = {1214.46016},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm200-1-2}
}
Jarno Talponen. Special symmetries of Banach spaces isomorphic to Hilbert spaces. Studia Mathematica, Tome 196 (2010) pp. 31-40. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm200-1-2/