Order Relations of the Wasserstein mean and the spectral geometric mean

On the space of positive definite matrices, several operator means are popular and have been studied extensively. In this paper, we investigate the near order and the Löwner order relations on the curves defined by the Wasserstein mean and the spectral geometric mean. We show that the near order $⪯$ is stronger than the eigenvalue entrywise order, and that $A{\mathrm{♮}}_{t}B⪯A{\diamond }_{t}B$ for $t\in [0,1]$. We prove the monotonicity properties of the curves originated from the Wasserstein mean and the spectral geometric mean in terms of the near order. The Löwner order properties of the Wasserstein mean and the spectral geometric mean are also explored.