Maps stemming from the functional calculus that transform a Kubo–Ando mean into another

Abstract

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