Operator mean and mean iterations of positive constant upper triangular matrices

Abstract

Adapting the Kubo-Ando's operator mean, we define the operator mean on the Lie group $\mathrm{CP}_{n}$ of $n \times n$ positive constant upper triangular matrices. We also study the weighted spectral geometric mean on $\mathrm{CP}_{n}$ and provide its binomial expansion. Moreover, we establish the Gauss mean and logarithmic mean on $\mathrm{CP}_n$ by proving the convergence of mean iterations. Finally, we investigate two multivariable means, the resolvent mean and the $A \# H$ mean, on $\mathrm{CP}_n$.