Weighted multivariable operator means of positive definite operators

In this paper we present a new weighted, multivariable operator mean of positive definite operators over an arbitrary Hilbert space which provides us the first generally applicable extension of the classical Kubo–Ando theory of 2-variable operator means. The construction is a weighted extension of the Bini–Meini–Poloni symmetrization process originally given for the matrix geometric mean. Here to be able to consider such an iterative procedure, we need a weighted version of every Kubo–Ando mean in two variables. Therefore we also give a new construction for two arbitrary positive operators on a possibly infinite dimensional Hilbert space that provides weighted counterparts to every (not-necessarily symmetric) Kubo–Ando mean and also agrees with the most well known weighted operator means.