Maps stemming from the functional calculus that transform a Kubo-Ando mean into another (Q785653)

The author investigates maps on sets of positive operators which are induced by the continuous functional calculus and also have the property that they transform a Kubo-Ando mean \(\sigma\) into another, \(\tau\). In the presence of rather mild conditions, it is shown that a mapping \(\phi\) has such a property only in the (trivial) case, when \(\sigma\) and \(\tau\) are nontrivial weighted harmonic means and \(\phi\) arises from a function which is a constant multiple of the generating function of such a mean. If precisely one of \(\sigma\) and \(\tau\) is a weighted arithmetic mean, it is shown in the presence of fairly weak assumptions, that the mentioned transformer property does not hold. It is also shown that, if both \(\sigma\) and \(\tau\) are such means, the latter property is only satisfied for maps induced by affine functions.