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Thompson isometries of the space of invertible positive operators


Author: Lajos Molnár
Journal: Proc. Amer. Math. Soc. 137 (2009), 3849-3859
MSC (2000): Primary 46B28, 47B49
DOI: https://doi.org/10.1090/S0002-9939-09-09963-8
Published electronically: June 15, 2009
MathSciNet review: 2529894
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Abstract: We determine the structure of bijective isometries of the set of all invertible positive operators on a Hilbert space equipped with the Thompson metric or the Hilbert projective metric.


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  • 1. T. Ando, Topics on operator inequalities, mimeographed lecture notes, Hokkaido University, Sapporo, 1978. MR 0482378 (58:2451)
  • 2. T. Ando, On the arithmetic-geometric-harmonic mean inequality for positive definite matrices, Linear Algebra Appl. 52-53 (1983), 31-37. MR 709342 (84j:15016)
  • 3. E. Andruchow, G. Corach and D. Stojanoff, Geometrical significance of the Löwner-Heinz inequality, Proc. Amer. Math. Soc. 128 (2000), 1031-1037. MR 1636922 (2000j:46100)
  • 4. G. Birkhoff, Extensions of Jentzsch's theorem, Trans. Amer. Math. Soc. 85 (1957), 219-227. MR 0087058 (19:296a)
  • 5. G. Corach, A. Maestripieri and D. Stojanoff, Orbits of positive operators from a differentiable viewpoint, Positivity 8 (2004), 31-48. MR 2053574 (2005d:58011)
  • 6. R.J. Fleming and J.E. Jamison, Isometries on Banach Spaces: Function Spaces, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 129, CRC Press, Boca Raton, FL, 2003. MR 1957004 (2004j:46030)
  • 7. R.J. Fleming and J.E. Jamison, Isometries on Banach Spaces: Vector-valued Function Spaces and Operator Spaces, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 138, CRC Press, Boca Raton, FL, 2007. MR 2361284
  • 8. J. Fujii, Arithmetic-geometric means of operators, Math. Japon. 23 (1979), 667-669. MR 529901 (80g:47006)
  • 9. S. Gudder and G. Nagy, Sequentially independent effects, Proc. Amer. Math. Soc. 130 (2002), 1125-1130. MR 1873787 (2002i:81014)
  • 10. D. Hilbert, Neue Begaündung der Bolya-Lobatschefskyschen Geometrie, Math. Ann. 57 (1903), 137-150. MR 1511203
  • 11. F. Kubo and T. Ando, Means of positive linear operators, Math. Ann. 246 (1980), 205-224. MR 563399 (84d:47028)
  • 12. L. Molnár, Non-linear Jordan triple automorphisms of sets of self-adjoint matrices and operators, Studia Math. 173 (2006), 39-48. MR 2204461 (2006j:47059)
  • 13. L. Molnár, Maps preserving the geometric mean of positive operators, Proc. Amer. Math. Soc. 137 (2009), 1763-1770.
  • 14. L. Molnár and P. Šemrl, Non-linear commutativity preserving maps on self-adjoint operators, Quart. J. Math. 56 (2005), 589-595. MR 2182468 (2006m:47063)
  • 15. R.D. Nussbaum, Hilbert's projective metric and iterated nonlinear maps, Mem. Amer. Math. Soc. 391 (1988). MR 961211 (89m:47046)
  • 16. R.D. Nussbaum, Finsler structures for the part metric and Hilbert's projective metric and applications to ordinary differential equations, Differ. Integral Equ. 7 (1994), 1649-1707. MR 1269677 (95b:58010)
  • 17. R.D. Nussbaum and J.E. Cohen, The arithmetic-geometric mean and its generalizations for noncommuting linear operators, Ann. Sc. Norm. Super. Pisa 15 (1988), 239-308. MR 1007399 (90e:47018)
  • 18. W. Pusz and S. L. Woronowicz, Functional calculus for sesquilinear forms and the purification map, Rep. Math. Phys. 8 (1975), 159-170. MR 0420302 (54:8316)
  • 19. A.C. Thompson, On certain contraction mappings in a partially ordered vector space, Proc. Amer. Math. Soc. 14 (1963), 438-443. MR 0149237 (26:6727)
  • 20. J. Väisälä, A proof of the Mazur-Ulam theorem, Amer. Math. Monthly 110 (2003), 633-635. MR 2001155 (2004d:46021)
  • 21. A. Vogt, Maps which preserve equality of distance, Studia Math. 45 (1973), 43-48. MR 0333676 (48:12001)

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Additional Information

Lajos Molnár
Affiliation: Institute of Mathematics, University of Debrecen, P.O. Box 12, H-4010 Debrecen, Hungary
Email: molnarl@math.klte.hu

DOI: https://doi.org/10.1090/S0002-9939-09-09963-8
Received by editor(s): December 22, 2008
Received by editor(s) in revised form: March 3, 2009, and March 7, 2009
Published electronically: June 15, 2009
Additional Notes: The author was supported by the Hungarian National Foundation for Scientific Research (OTKA), Grant No. NK68040.
Communicated by: Nigel J. Kalton
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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