Maps Preserving Schatten p-Norms of Convex Combinations - David Li-Wei Kuo; Ming-Cheng Tsai; Ngai-Ching Wong; Jun Zhang (Abstract and Applied Analysis)
We study maps ϕ of positive operators of the Schatten p-classes (1<p<+∞), which preserve the p-norms of convex combinations, that is,   ∥tρ+(1-t)σ∥p=∥tϕ(ρ)+(1-t)ϕ(σ)∥p,  ∀ρ,σ∈𝒮p+(H)1,  t∈[0,1]. They are exactly those carrying the form ϕ(ρ)=UρU* 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.