Maps preserving the harmonic mean or the parallel sum of positive operators

Let H be a complex Hilbert space. The symbol A!B stands for the harmonic 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 that operation. We prove that any such transformation is implemented by an invertible bounded linear or conjugate-linear operator on H. Similar results concerning the parallel sum and the arithmetic mean in the place of the harmonic mean are also presented.