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On Schatten restricted norms


Author: Martin Miglioli
Journal: Proc. Amer. Math. Soc. 148 (2020), 5249-5259
MSC (2010): Primary 46B04, 47B10, 58B20; Secondary 22F50
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 \Vert\xi \Vert^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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Additional Information

Martin Miglioli
Affiliation: Instituto Argentino de Matemática-CONICET, Saavedra 15, Piso 3, (1083) Buenos Aires, Argentina
MR Author ID: 1060057
Email: martin.miglioli@gmail.com

DOI: https://doi.org/10.1090/proc/15179
Keywords: Busemann $p$-space, unitarization, Mazur's rotation problem, isometry groups
Received by editor(s): March 13, 2020
Received by editor(s) in revised form: April 21, 2020, May 3, 2020, and May 6, 2020
Published electronically: September 18, 2020
Additional Notes: The author was supported by IAM-CONICET, grants PIP 2010-0757 (CONICET) and PICT 2010-2478 (ANPCyT)
Communicated by: Stephen Dilworth
Article copyright: © Copyright 2020 American Mathematical Society