Logarithmic mean of positive invertible operators

As a generalization of the logarithmic mean of positive real numbers, we establish the logarithmic mean of two positive invertible operators with two different construction schemes using Bochner integral and the convergence of skewed mean iteration. Indeed, we see that two constructions are the same for a non-weighted version with the uniform probability. Furthermore, we study several fundamental properties for the logarithmic mean of two positive invertible operators, and also investigate the logarithmic mean under tolerance relation on the open convex cone P-m of positive definite Hermitian matrices.