A new weighted spectral geometric mean and properties

Abstract

In this paper, we introduce a new weighted spectral geometric mean: \begin{equation*}\label{F-mean}F_t(A,B)= (A^{-1}\sharp_t B)^{1/2} A^{2-2t} (A^{-1} \sharp_t B)^{1/2}, \quad t\in [0,1],\end{equation*} where $A$ and $B$ are positive definite matrices. We study basic properties and inequalities for $F_t(A, B)$. We also establish the Lie-Trotter formula for $F_t(A, B)$. Finally, we extend some of the results on $F_t(A, B)$ to symmetric space of noncompact types.