Maps preserving the geometric mean of positive operators
HTML articles powered by AMS MathViewer
- by Lajos Molnár
- Proc. Amer. Math. Soc. 137 (2009), 1763-1770
- DOI: https://doi.org/10.1090/S0002-9939-08-09749-9
- Published electronically: December 11, 2008
- PDF | Request permission
Abstract:
Let $H$ be a complex Hilbert space. The symbol $A\# B$ stands for the geometric mean of the positive bounded linear operators $A,B$ on $H$ in the sense of Ando. In this paper we describe the general form of all automorphisms of the set of positive operators with respect to the operation $\#$. We prove that if $\dim H\geq 2$, any such transformation is implemented by an invertible bounded linear or conjugate-linear operator on $H$.References
- T. Ando, Topics on operator inequalities, Hokkaido University, Research Institute of Applied Electricity, Division of Applied Mathematics, Sapporo, 1978. MR 0482378
- Ingemar Bengtsson and Karol Życzkowski, Geometry of quantum states, Cambridge University Press, Cambridge, 2006. An introduction to quantum entanglement. MR 2230995, DOI 10.1017/CBO9780511535048
- P. Busch and S. P. Gudder, Effects as functions on projective Hilbert space, Lett. Math. Phys. 47 (1999), no. 4, 329–337. MR 1693743, DOI 10.1023/A:1007573216122
- Jun-ichi Fujii, Arithmetico-geometric mean of operators, Math. Japon. 23 (1978/79), no. 6, 667–669. MR 529901
- Stan Gudder and Gabriel Nagy, Sequentially independent effects, Proc. Amer. Math. Soc. 130 (2002), no. 4, 1125–1130. MR 1873787, DOI 10.1090/S0002-9939-01-06194-9
- L. Molnár, Selected preserver problems on algebraic structures of linear operators and on function spaces, Lecture Notes in Mathematics, vol. 1895, Springer-Verlag, Berlin, 2007. MR 2267033
- Peter emrl, Isomorphisms of standard operator algebras, Proc. Amer. Math. Soc. 123 (1995), no. 6, 1851–1855. MR 1242104, DOI 10.1090/S0002-9939-1995-1242104-8
Bibliographic Information
- Lajos Molnár
- Affiliation: Institute of Mathematics, University of Debrecen, P.O. Box 12, H-4010 Debrecen, Hungary
- Email: molnarl@math.klte.hu
- Received by editor(s): November 13, 2007
- Received by editor(s) in revised form: June 27, 2008
- Published electronically: December 11, 2008
- Additional Notes: The author was supported by the Hungarian National Foundation for Scientific Research (OTKA), Grant No. T046203, NK68040 and by the Alexander von Humboldt Foundation, Germany.
- Communicated by: Marius Junge
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 137 (2009), 1763-1770
- MSC (2000): Primary 47B49, 47A64
- DOI: https://doi.org/10.1090/S0002-9939-08-09749-9
- MathSciNet review: 2470835