On Schatten restricted norms

Abstract

We consider norms on a complex separable Hilbert space such that ⟨aξ,ξ⟩≤‖ξ‖2≤⟨bξ,ξ⟩ for positive invertible operators a and b that differ by an operator in the Schatten class. We prove that these norms have unitarizable isometry groups, our proof uses a generalization of a fixed point theorem for isometric actions on positive invertible operators. As a result, if their isometry groups do not leave any finite dimensional subspace invariant, then the norms must be Hilbertian. That is, if a Hilbertian norm is changed to a close non-Hilbertian norm, then the isometry group does leave a finite dimensional subspace invariant. The approach involves metric geometric arguments related to the canonical action on the non-positively curved space of positive invertible Schatten perturbations of the identity.