Maps stemming from the functional calculus that transform a Kubo\u2013Ando mean into another

Abstract

In this paper, we investigate maps on sets of positive operators which are induced by the continuous functional calculus and transform a Kubo–Ando mean sigma into another tau . We establish that under quite mild conditions, a mapping phi can have this property only in the trivial case, i.e. when sigma and tau are nontrivial weighted harmonic means and phi stems from a function which is a constant multiple of the generating function of such a mean. In the setting where exactly one of sigma and tau is a weighted arithmetic mean, we show that under fairly weak assumptions, the mentioned transformer property never holds. Finally, when both of sigma and tau are such a mean, it turns out that the latter property is only satisfied in the trivial case, i.e. for maps induced by affine functions.