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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On Schatten restricted norms
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by Martin Miglioli PDF
Proc. Amer. Math. Soc. 148 (2020), 5249-5259 Request permission


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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Additional Information
  • Martin Miglioli
  • Affiliation: Instituto Argentino de Matemática-CONICET, Saavedra 15, Piso 3, (1083) Buenos Aires, Argentina
  • MR Author ID: 1060057
  • Email:
  • 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
  • © Copyright 2020 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 148 (2020), 5249-5259
  • MSC (2010): Primary 46B04, 47B10, 58B20; Secondary 22F50
  • DOI:
  • MathSciNet review: 4163837