Submajorisation inequalities for convex and concave functions of sums of measurable operators

It is shown that non-negative, increasing, convex (respectively, concave) functions are superadditive (respectively, subadditive) with respect to submajorisation on the positive cone of the space of all τ-measurable operators affiliated with a semifinite von Neumann algebra. This extends recent results for n × n-matrices by Ando-Zhan, Kosem and Bourin-Uchiyama. This work was partially supported by the Australian Research Council.