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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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
Published electronically: December 11, 2008
MathSciNet review: 2470835
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Abstract: Let $ H$ be a complex Hilbert space. The symbol $ A\char93 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 $ \char93 $. We prove that if $ \dim H\geq 2$, any such transformation is implemented by an invertible bounded linear or conjugate-linear operator on $ H$.


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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: http://dx.doi.org/10.1090/S0002-9939-08-09749-9
PII: S 0002-9939(08)09749-9
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.