On Schatten restricted norms

Abstract: We consider norms on a complex separable Hilbert space such that $\langle a\xi,\xi\rangle\leq|\xi|^2\leq\langle b\xi,\xi\rangle$ 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 the isometry group does not leave any finite dimensional subspace invariant, then the norm 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 of the group on the non-positively curved space of positive invertible Schatten perturbations of the identity .