Geometrical significance of the Löwner-Heinz inequality
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- by E. Andruchow, G. Corach and D. Stojanoff PDF
- Proc. Amer. Math. Soc. 128 (2000), 1031-1037 Request permission
Abstract:
It is proven that the Löwner-Heinz inequality ${\|A^{t}B^{t}\|\le \|AB\|^{t}}$, valid for all positive invertible operators ${A, B}$ on the Hilbert space ${\mathcal H }$ and ${t\in [0,1]}$, has equivalent forms related to the Finsler structure of the space of positive invertible elements of ${\mathcal L (\mathcal H )}$ or, more generally, of a unital ${C^{*}}$-algebra. In particular, the Löwner-Heinz inequality is equivalent to some type of “nonpositive curvature" property of that space.References
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Additional Information
- E. Andruchow
- Affiliation: Instituto de Ciencias, Universidad Nacional de General Sarmiento, Roca 850, 1663-San Miguel, Argentina
- MR Author ID: 26110
- Email: eandruch@mate.dm.uba.ar
- G. Corach
- Affiliation: Departamento de Matemática, Facultad de Ciencias Exactas, Ciudad Universitaria, 1428-Buenos Aires, Argentina
- Email: gcorach@mate.dm.uba.ar
- D. Stojanoff
- Affiliation: Instituto Argentino de Matemática, Saavedra 15, 1083-Buenos Aires, Argentina
- Email: demetrio@mate.dm.uba.ar
- Received by editor(s): May 29, 1997
- Received by editor(s) in revised form: May 18, 1998
- Published electronically: July 28, 1999
- Additional Notes: The authors were partially supported by UBACYT EX 261, PIP CONICET 4463/96 and PICT 2259 ANPCYT (Argentina)
- Communicated by: David R. Larson
- © Copyright 2000 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 128 (2000), 1031-1037
- MSC (1991): Primary 46L05, 58B20
- DOI: https://doi.org/10.1090/S0002-9939-99-05085-6
- MathSciNet review: 1636922
Dedicated: Dedicated to Mischa Cotlar, with affection and admiration, on his 86th anniversary