A Matsumoto-Yor characterization for Kummer and Wishart random matrices

In the paper we resolve positively the conjecture on a characterization of matrix Kummer and Wishart laws through independence property, which was posed in [Koudou, Statist. Probab. Lett. 82 (2012), 1903–1907] . Apart from the probabilistic result, we determine the general solution of the functional equation associated to the characterization problem under weak assumptions.