Nonexpansiveness of the resolvent average

We show that the resolvent average for positive definite matrices, which interpolates between the harmonic and the arithmetic means, contracts the Thompson metric on the convex cone of positive definite matrices. Classical results depending on the nonexpansive property of the arithmetic average are considered in the context of the resolvent average. In particular, resolvent power mean approximation to the Karcher mean and Ferrante and Levy-like matrix equations are investigated.