Geometrical significance of the Löwner-Heinz inequality

Andruchow, E.; Corach, G.; Stojanoff, D.

doi:10.1090/S0002-9939-99-05085-6

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
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Abstract: It is proven that the Löwner-Heinz inequality ${\|A^{t}B^{t}\|\le \|AB\|^{t}}$ , valid for all positive invertible operators 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.

1. Werner Ballmann, Mikhael Gromov, and Viktor Schroeder, Manifolds of nonpositive curvature, Progress in Mathematics, vol. 61, Birkhäuser Boston, Inc., Boston, MA, 1985. MR 823981
2. Gustavo Corach, Operator inequalities, geodesics and interpolation, Functional analysis and operator theory (Warsaw, 1992) Banach Center Publ., vol. 30, Polish Acad. Sci. Inst. Math., Warsaw, 1994, pp. 101–115. MR 1285601
3. G. Corach, H. Porta, and L. Recht, A geometric interpretation of Segal’s inequality |𝑒^{𝑋+𝑌}|≤|𝑒^{𝑋/2}𝑒^{𝑌}𝑒^{𝑋/2}|, Proc. Amer. Math. Soc. 115 (1992), no. 1, 229–231. MR 1075945, https://doi.org/10.1090/S0002-9939-1992-1075945-8
4. Gustavo Corach, Horacio Porta, and Lázaro Recht, The geometry of the space of selfadjoint invertible elements in a 𝐶*-algebra, Integral Equations Operator Theory 16 (1993), no. 3, 333–359. MR 1209304, https://doi.org/10.1007/BF01204226
5. Gustavo Corach, Horacio Porta, and Lázaro Recht, Geodesics and operator means in the space of positive operators, Internat. J. Math. 4 (1993), no. 2, 193–202. MR 1217380, https://doi.org/10.1142/S0129167X9300011X
6. G. Corach, H. Porta, and L. Recht, Convexity of the geodesic distance on spaces of positive operators, Illinois J. Math. 38 (1994), no. 1, 87–94. MR 1245836
7. H. O. Cordes, Spectral theory of linear differential operators and comparison algebras, London Mathematical Society Lecture Note Series, vol. 76, Cambridge University Press, Cambridge, 1987. MR 890743
8. Masatoshi Fujii, Takayuki Furuta, and Ritsuo Nakamoto, Norm inequalities in the Corach-Porta-Recht theory and operator means, Illinois J. Math. 40 (1996), no. 4, 527–534. MR 1415016
9. Takayuki Furuta, Norm inequalities equivalent to Löwner-Heinz theorem, Rev. Math. Phys. 1 (1989), no. 1, 135–137. MR 1041534, https://doi.org/10.1142/S0129055X89000079
10. Mikhael Gromov, Structures métriques pour les variétés riemanniennes, Textes Mathématiques [Mathematical Texts], vol. 1, CEDIC, Paris, 1981 (French). Edited by J. Lafontaine and P. Pansu. MR 682063
11. E. Heinz, ``Beiträge zur Storungstheorie der Spektralzerlegung", Math. Ann. 123 (1951), 415-438. MR 13:471f
12. Carlangelo Liverani and Maciej P. Wojtkowski, Generalization of the Hilbert metric to the space of positive definite matrices, Pacific J. Math. 166 (1994), no. 2, 339–355. MR 1313459
13. K. Löwner, ``Über monotone Matrix funktionen", Math. Z. 38 (1934), 177-216.
14. Roger D. Nussbaum, Hilbert’s projective metric and iterated nonlinear maps, Mem. Amer. Math. Soc. 75 (1988), no. 391, iv+137. MR 961211, https://doi.org/10.1090/memo/0391
15. Roger D. Nussbaum, Finsler structures for the part metric and Hilbert’s projective metric and applications to ordinary differential equations, Differential Integral Equations 7 (1994), no. 5-6, 1649–1707. MR 1269677
16. Irving Segal, Notes towards the construction of nonlinear relativistic quantum fields. III. Properties of the 𝐶*-dynamics for a certain class of interactions, Bull. Amer. Math. Soc. 75 (1969), 1390–1395. MR 251992, https://doi.org/10.1090/S0002-9904-1969-12429-8
17. A. C. Thompson, On certain contraction mappings in a partially ordered vector space., Proc. Amer. Math. soc. 14 (1963), 438–443. MR 0149237, https://doi.org/10.1090/S0002-9939-1963-0149237-7
18. Edoardo Vesentini, Invariant metrics on convex cones, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 3 (1976), no. 4, 671–696. MR 433228