Special symmetries of Banach spaces isomorphic to Hilbert spaces

TL;DR

This paper characterizes Hilbert spaces among Banach spaces using symmetry properties related to isometry groups, showing that certain transitivity conditions imply the space is actually a Hilbert space.

Contribution

It introduces new symmetry-based criteria that distinguish Hilbert spaces within the class of Banach spaces, focusing on transitivity under specific isometry subgroups.

Findings

Convex-transitivity under finite-dimensional isometric perturbations implies the space is a Hilbert space.

Characterization of Hilbert spaces via symmetry and transitivity properties.

New criteria for identifying Hilbert spaces among Banach spaces.

Abstract

In this paper Hilbert spaces are characterized among Banach spaces in terms of transitivity with respect to nicely behaved subgroups of the isometry group. For example, the following result is typical here: 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.