Operator Means of Lower Triangular Matrices

We show that every Kubo-Ando operator mean of positive definite operators exists on the solvable Lie group of lower triangular matrices with positive diagonal entries. In particular, we show that the operator geometric mean of such lower triangular matrices appears as the common limit of the iteration process of the arithmetic and harmonic means. We further show that the iteration terminates in the finite number inverted right perpendicularlog(2) minverted left perpendicular of iterations for m x m lower unitriangular matrices and present its entrywise closed form for m <= 4.