Maps preserving the geometric mean of positive operators

Molnár, Lajos

doi:10.1090/S0002-9939-08-09749-9

Maps preserving the geometric mean of positive operators
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Proc. Amer. Math. Soc. 137 (2009), 1763-1770 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 $H$.

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