Near-geometric mean of positive definite operators

Abstract

In this paper we study a new kind of geometric mean of positive definite operators A and B, named the near-geometric mean, of the form for t∈[0,1] (Formula presented.) arising from the determinantal inequality in diffusion tensor imaging. We show the geodesic property of near-geometric mean for the Thompson metric, the monotonicity on parameters and the boundedness of near-geometric mean. Moreover, we compare other known geometric means and Rényi mean with near-geometric mean in terms of the log-majorization.