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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Maps preserving the geometric mean of positive operators
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by Lajos Molnár PDF
Proc. Amer. Math. Soc. 137 (2009), 1763-1770 Request permission


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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Additional Information
  • Lajos Molnár
  • Affiliation: Institute of Mathematics, University of Debrecen, P.O. Box 12, H-4010 Debrecen, Hungary
  • Email:
  • 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:
  • MathSciNet review: 2470835