On isometric reflections in Banach spaces

Анотація

We obtain the following characterization of Hilbert spaces. Let E be a Banach space the unit sphere S of which has a hyperplane of symmetry. Then E is a Hilbert space iff any of the following two conditions is fulfilled:

a) the isometry group Iso E of E has a dense orbit in S;

b) the identity component G₀ of the group Iso E endowed with the strong operator topology acts topologically irreducible on E. Some related results on infinite dimensional Coxeter groups generated by isometric reflections are given which allow us to analyse the structure of isometry groups containing sufficiently many reflections.