Maps preserving Schatten p-norms of convex combinations

We study maps �� of positive operators of the Schatten p -classes (1 < p < + ��), which preserve the p -norms of convex combinations, that is, t �� + (1 - t) �� p = t �� (��) + (1 - t) �� (��) p, ��, �� p + (H) 1, t [ 0,1 ]. They are exactly those carrying the form �� (��) = U �� U for a unitary or antiunitary U. In the case p = 2, we have the same conclusion whenever it just holds �� + �� 2 = �� (��) + �� (��) 2 for all the positive Hilbert-Schmidt class operators ��, �� of norm 1. Some examples are demonstrated. ? 2014 David Li-Wei Kuo et al.