Paths of means for positive operators with strongly unitarily equivalent supports

Based on the Bonnabel–Sepulchre geometric mean for fixed rank matrices, we introduced paths of matrix means corresponding to the Kubo–Ando operator ones. We also observed that our mean includes the Batzies–Hüper–Machado–Silva Leite geometric means for fixed rank projection matrices. But these means are restricted to real ones and moreover it seems that it is not easy to extend them to operators on a complex (infinite dimensional) Hilbert space since these means were based on geometries for finite dimensional real spaces. In this paper, we introduce the general paths of means on a complex Hilbert space corresponding to those of the Kubo–Ando ones based on the infinite dimensional complex Grassmann geodesic in the sense of Andruchow.