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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 $A\# B$ 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 $H$.

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Lajos Molnár
Affiliation: Institute of Mathematics, University of Debrecen, P.O. Box 12, H-4010 Debrecen, Hungary

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.