This short note is concerned with maps on the cone of positive definite matrices which preserve certain generalized distance measures. We mean sorts of distances which are connected with quantum divergences, relative entropies, or related to geometrical structures on the positive definite cone. The novelty here is that we do not assume that our maps, the so-called generalized isometries, are surjective. As we will see, in some cases, those transformations turn to be automatically surjective, while in other cases, we can determine the precise structure of all not necessarily surjective such maps. In spite of the presented results, a number of related questions are left open and proposed to be the objects of further research.
- generalized distance measures
- Positive definite matrices
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