On Schatten restricted norms

Miglioli, Martin

doi:10.1090/proc/15179

On Schatten restricted norms
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by Martin Miglioli

Proc. Amer. Math. Soc. 148 (2020), 5249-5259

DOI: https://doi.org/10.1090/proc/15179

Published electronically: September 18, 2020

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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. As a result, if their isometry groups do not leave any finite dimensional subspace invariant, then the norms must be Hilbertian. 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. Our proof of the main result uses a generalization of a unitarization theorem which follows from the Bruhat-Tits fixed point theorem.

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