Transformations on positive definite matrices preserving generalized distance measures

We substantially extend and unify former results on the structure of surjective isometries of spaces of positive definite matrices obtained in the paper [14]. The isometries there correspond to certain geodesic distances in Finsler-type structures and to a recently defined interesting metric which also follows a non-Euclidean geometry. The novelty in our present paper is that here we consider not only true metrics but also so-called generalized distance measures which are parameterized by unitarily invariant norms and continuous real functions satisfying certain conditions. Among the many possible applications, we shall see that using our new result it is easy to describe the surjective maps of the set of positive definite matrices that preserve the Stein loss or several other types of divergences. We also present results concerning similar preserver transformations defined on the subset of all complex positive definite matrices with unit determinant.