Maps preserving Schatten p-norms of convex combinations

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