Unexpected relations which characterize operator means

Abstract:

We give some characterizations of self-adjointness and symmetricity of operator monotone functions by using the Barbour transform $f \mapsto \frac {t+f}{1+f}$ and show that there are many non-symmetric operator means between the harmonic mean $!$ and the arithmetic mean $\nabla$. Indeed, we show that there exists a non-symmetric operator mean between any two symmetric operator means.