In-betweenness, a geometrical monotonicity property for operator means

We introduce the notions of in-betweenness and monotonicity with respect to a metric for operator means. These notions can be seen as generalising their natural counterpart for scalar means, and as a relaxation of the notion of geodesity. We exhibit two classes of non-trivial means that are monotonic with respect to the Euclidean metric. We also show that all Kubo–Ando means are monotonic with respect to the trace metric, which is the natural metric for the geometric mean.