Maps preserving the harmonic mean or the parallel sum of positive operators (Q1020940)

Let \(H\) be a complex Hilbert space. Denote by \(B(H)\) the algebra of all bounded linear operators on \(H\) and by \(B(H)^+\) the set of all positive operators on \(H\). For \(A, B \in B(H)^+\), the harmonic mean \(A! B\) is defined by \[ A ! B= \max \left\{X \geq 0: \left[\begin{smallmatrix} 2A & 0 \\ 0 & 2B \end{smallmatrix} \right] \geq \left[ \begin{smallmatrix} X & X\\ X & X \end{smallmatrix} \right] \right\}. \] This concept was introduced in [\textit{T.\,Ando}, ``Topics on operator inequalities'' (Sapporo:\ Hokkaido University) (1978; Zbl 0388.47024)] and developed in [\textit{F.\,Kubo} and \textit{T.\,Ando}, Math.\ Ann.\ 246, 205--224 (1980; Zbl 0412.47013)]. In the paper under review, the author obtains the following interesting result: If \(\phi: B(H)^+ \rightarrow B(H)^+\) is a bijective map satisfying \(\phi (A!B)= \phi (A)! \phi (B)\) for all \(A, B \in B(H)^+\), then \(\phi\) is of the form \(\phi (A)= SA S^*\) for all \(A \in A\), where \(S\) is an invertible bounded linear or conjugate linear operator on \(H\). The author also describes bijective maps preserving the arithmetic mean.