In-betweenness: a geometric monotonicity property for operator means

Details In-betweenness: a geometric monotonicity property for operator means In-betweenness: a geometric monotonicity property for operator means

2010-11-29

arxiv.org/abs/1011.6313v1

Mathematics

Functional Analysis

15 pages; a preliminary version has been presented at the June 2010 ILAS Conference in Pisa, Italy

Scientific paper

10.1016/j.laa.2011.02.051

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.

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