Transformations on positive definite matrices preserving generalized distance measures

Abstract

Abstract. 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 nonEuclidean geometry. The novelty in our present paper is that here we consider not only true metrics but 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's 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.

Spectral conditions for Jordan *-isomorphisms on operator algebras

“…Related results in the context of matrix algebras appeared in our former paper [9]. In what follows we present several sorts of extensions of our previous theorem.…”

“…Since φ 0 respects the pair d , ρ of generalized distance measures, i.e., satisfies (9), it follows that…”

Continuous Jordan triple endomorphisms of P2

“…In the case where n ≥ 3, in [11, Theorem 1] we obtained a general result describing the possible structure of surjective maps on P n which preserve a generalized distance measure of a certain quite general kind. It is easy to see that, following the proof of [11, Theorem 1] and applying Theorem 2, the result in [11] remains valid also in the case where n = 2.…”

“…Our main reason for investigating those maps comes from the fact that they naturally appear in the study of surjective isometries and surjective maps preserving generalized distance measures between positive definite cones. For details see [9,10,11].In the paper [9] we have proved the following statement which appeared as Theorem 1 there. In what follows we denote by M n the algebra of all n × n complex matrices and P n stands for the cone of all positive definite matrices in M n .…”

“…Our main reason for investigating those maps comes from the fact that they naturally appear in the study of surjective isometries and surjective maps preserving generalized distance measures between positive definite cones. For details see [9,10,11].…”