A new weighted spectral geometric mean and properties

In this paper, we introduce a new weighted spectral geometric mean: equation*F-meanFₜ (A, B) = (A^-1ₜ B) ^1/2 A^2-2t (A^-1 ₜ B) ^1/2, t 0, 1, equation* where A and B are positive definite matrices. We study basic properties and inequalities for Fₜ (A, B). We also establish the Lie-Trotter formula for Fₜ (A, B). Finally, we extend some of the results on Fₜ (A, B) to symmetric space of noncompact types.