Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

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

 
 

 

Geometrical significance
of the Löwner-Heinz inequality


Authors: E. Andruchow, G. Corach and D. Stojanoff
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
Published electronically: July 28, 1999
MathSciNet review: 1636922
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • 1. W. Ballmann, M. Gromov, V. Schroeder, ``Manifolds of non positive curvature", Birkhauser, Boston-Basel-Stutgart, 1985. MR 87h:53050
  • 2. G. Corach, ``Operator inequalities, geodesics and interpolation", Functional Analysis and Operator Theory, Banach Center Publications, Vol. 30, Polish Academy of Sciences, Warszawa, 1994, pp. 101-115. MR 95d:58009
  • 3. G. Corach, H. Porta, L. Recht, ``A geometric interpretation of Segal's inequality $|e^{X+Y}|\le|e^{X/2}e^Ye^{X/2}|$", Proc. Amer. Math. Soc. 115 (1992), 229-231. MR 92h:46105
  • 4. G. Corach, H. Porta, L. Recht, ``The geometry of the space of selfadjoint invertible elements in a $C^{*}$-algebras", Integral Equations Operator Theory 16 (1993), 333-359. MR 94d:58010
  • 5. G. Corach, H. Porta, L. Recht, ``Geodesics and operator means in the space of positive operators", Internat. J. Math. 4 (1993), 193-202. MR 94c:46114
  • 6. G. Corach, H. Porta, L. Recht, ``Convexity of the geodesic distance on spaces of positive operators", Illinois J. Math. 38 (1994), 87-94. MR 94i:58010
  • 7. H.O. Cordes, ``Spectral theory of linear differential operators and comparison algebras", London Mathematical Society Lecture Notes Series 76, Cambridge University Press, 1987. MR 88g:35144
  • 8. M. Fujii, T. Furuta, R. Nakamoto, ``Norm inequalities in the Corach-Porta-Recht theory and operator means", Illinois J. Math. 40 (1996), 527-534. MR 97i:47026
  • 9. T. Furuta, ``Norm inequalities equivalent to Löwner-Heinz theorem", Rev. Math. Phys. 1 (1989), 135-137. MR 91b:47028
  • 10. M. Gromov, ``Structures métriques pour les variétés riemanniennes", CEDIC/Fernand Nathan, Paris, 1981. MR 85e:53051
  • 11. E. Heinz, ``Beiträge zur Storungstheorie der Spektralzerlegung", Math. Ann. 123 (1951), 415-438. MR 13:471f
  • 12. C. Liverani, M.P. Wojtkowski, ``Generalization of the Hilbert metric to the space of positive definite matrices", Pacific J. Math. 166 (1994), 339-355. MR 95m:53099
  • 13. K. Löwner, ``Über monotone Matrix funktionen", Math. Z. 38 (1934), 177-216.
  • 14. R. Nussbaum, ``Hilbert's projective metric and iterated nonlinear maps", Mem. Amer. Math. Soc. 391 (1988). MR 89m:47046
  • 15. R. Nussbaum, ``Finsler structures for the part metric and Hilbert's projective metric and applications to ordinary differential equations", Diff. Integral Equations 7 (1994), 1649-1707. MR 95b:58010
  • 16. I. Segal, ``Notes toward the construction of non linear relativistic quantum fields III", Bull. Amer. Math. Soc. 75 (1969), 1390-1395. MR 40:5217
  • 17. A.C. Thompson, ``On certain contraction mappings in a partially ordered vector space", Proc. Amer. Math. Soc. 14 (1963), 438-443. MR 26:6727
  • 18. E. Vesentini, ``Invariant metrics on convex cones", Ann. Sc. Norm. Sup. Pisa (Ser. 4) 3 (1976), 671-696. MR 55:6206

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 46L05, 58B20

Retrieve articles in all journals with MSC (1991): 46L05, 58B20


Additional Information

E. Andruchow
Affiliation: Instituto de Ciencias, Universidad Nacional de General Sarmiento, Roca 850, 1663-San Miguel, Argentina
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

DOI: https://doi.org/10.1090/S0002-9939-99-05085-6
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)
Dedicated: Dedicated to Mischa Cotlar, with affection and admiration, on his 86th anniversary
Communicated by: David R. Larson
Article copyright: © Copyright 2000 American Mathematical Society

American Mathematical Society