In-betweenness: a geometric monotonicity property for operator means
arXiv:1011.6313 · doi:10.1016/j.laa.2011.02.051
Abstract
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.
15 pages; a preliminary version has been presented at the June 2010 ILAS Conference in Pisa, Italy