Maps preserving the geometric mean of positive operators
Author:
Lajos Molnár
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
Published electronically:
December 11, 2008
MathSciNet review:
2470835
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
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 $H$.
- T. Ando, Topics on operator inequalities, Division of Applied Mathematics, Research Institute of Applied Electricity, Hokkaido University, 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
- 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 https://doi.org/10.1023/A%3A1007573216122
- 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 https://doi.org/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 Šemrl, Isomorphisms of standard operator algebras, Proc. Amer. Math. Soc. 123 (1995), no. 6, 1851–1855. MR 1242104, DOI https://doi.org/10.1090/S0002-9939-1995-1242104-8
Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 47B49, 47A64
Retrieve articles in all journals with MSC (2000): 47B49, 47A64
Additional Information
Lajos Molnár
Affiliation:
Institute of Mathematics, University of Debrecen, P.O. Box 12, H-4010 Debrecen, Hungary
Email:
molnarl@math.klte.hu
Keywords:
Geometric mean,
positive operators,
automorphism.
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
Article copyright:
© Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
