Maps preserving the geometric mean of positive operators

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 $H$.