Inequalities and limits of weighted spectral geometric mean

A new weighted spectral geometric mean and properties

“…Generalizations to semisimple Lie groups. The readers are referred to [8,9] regarding Lie theoretic preliminaries and notations. Let G be a noncompact connected semisimple Lie group with Lie algebra g. Let Θ : G → G be a Cartan involution of G, and let K be the fixed point set of Θ, which is is an analytic subgroup of G. Denote by θ the differential map dΘ of Θ.…”

“…Denote by ≺ G the Kostant pre-order on G [14,17,8,9]. Given f, g ∈ G, by setting f ≺ G g means (see [14, p. 426]) (4.7)…”

“…They also studied several properties similar to those of the Tsallis relative operator entropy by the usual geometric mean. Recently, Gan, Liu, and Tam [8] and Gan and Tam [9] studied A t B and obtained some nice properties. A nice survey and new properties related to the weighted spectral geometric mean A t B can be found in the recent paper of Gan and Kim [7].…”

Weak log-majorization between the spectral geometric and Wasserstein means

“…We collect some useful properties of the spectral geometric mean from [9,17,18] and the Wasserstein mean [12,13,15].…”